![26 I The Expanding Universe of Numbers
Proposition 25 Any fundamental sequence of real numbers is convergent.
Proof If {an} is a fundamental sequence of real numbers, then {an} is bounded and,
for any ε > 0, there exists a positive integer m = m(ε) such that
−ε/2 < ap − aq < ε/2 for all p, q ≥ m.
But, by Proposition 24, the sequence {an} has a convergent subsequence {ank }. If l is
the limit of this subsequence, then there exists a positive integer N ≥ m such that
l − ε/2 < ank < l + ε/2 for nk ≥ N.
It follows that
l − ε < an < l + ε for n ≥ N.
Thus the sequence {an} converges with limit l. ✷
Proposition 25 was known to Bolzano (1817) and was clearly stated in the influ-
ential Cours d’analyse of Cauchy (1821). However, a rigorous proof was impossible
until the real numbers themselves had been precisely defined.
The Méray–Cantor method of constructing the real numbers from the
rationals is based on Proposition 25. We define two fundamental sequences {an} and
{a′
n} of rational numbers to be equivalent if an − a′
n → 0 as n → ∞. This is indeed
an equivalence relation, and we define a real number to be an equivalence class of
fundamental sequences. The set of all real numbers acquires the structure of a field if
addition and multiplication are defined by
{an} + {bn} = {an + bn}, {an} · {bn} = {anbn}.
It acquires the structure of a complete ordered field if the fundamental sequence {an} is
said to be positive when it has a positive lower bound. The field Q of rational numbers
may be regarded as a subfield of the field thus constructed by identifying the ratio-
nal number a with the equivalence class containing the constant sequence {an}, where
an = a for every n.
It is not difficult to show that an ordered field is complete if every bounded
monotonic sequence is convergent, or if every bounded sequence has a convergent
subsequence. In this sense, Propositions 22 and 24 state equivalent forms for the least
upper bound property. This is not true, however, for Proposition 25. An ordered field
need not have the least upper bound property, even though every fundamental sequence
is convergent. It is true, however, that an ordered field has the least upper bound
property if and only if it has the Archimedean property (Proposition 19) and every
fundamental sequence is convergent.
In a course of real analysis one would now define continuity and prove those
properties of continuous functions which, in the 18th century, were assumed as
‘geometrically obvious’. For example, for given a, b ∈ R with a < b, let I = [a, b] be
the interval consisting of all x ∈ R such that a ≤ x ≤ b. If f : I → R is continuous,
then it attains its supremum, i.e. there exists c ∈ I such that f (x) ≤ f (c) for every
x ∈ I. Also, if f (a) f (b) < 0, then f (d) = 0 for some d ∈ I (the intermediate-
value theorem). Real analysis is not our primary concern, however, and we do not feel
obliged to establish even those properties which we may later use.](https://image.slidesharecdn.com/11052267-250525204806-55299cc9/85/Number-Theory-An-Introduction-To-Mathematics-2nd-Edition-Coppel-42-320.jpg)
![4 Metric Spaces 29
(iii) Let E = C (I) be the set of all continuous functions f : I → R, where
I = [a, b] = {x ∈ R: a ≤ x ≤ b}
is an interval of R, and define d(g, h) = |g − h|, where
| f | = sup
a≤x≤b
| f (x)|.
(A well-known property of continuous functions ensures that f is bounded on I.)
Alternatively, one can replace the norm | f | by either
| f |1 =
$ b
a
| f (x)|dx
or
| f |2 =
"$ b
a
| f (x)|2
dx
#1/2
.
(iv) Let E = C (R) be the set of all continuous functions f : R → R and define
d(g, h) =
!
N≥1
dN (g, h)/2N
[1 + dN (g, h)],
where dN (g, h) = sup|x|≤N |g(x) − h(x)|. The triangle inequality (D3) follows from
the inequality
|α + β|/[1 + |α + β|] ≤ |α|/[1 + |α|] + |β|/[1 + |β|]
for arbitrary real numbers α, β.
The metric here has the property that d( fn, f ) → 0 if and only if fn(x) → f (x)
uniformly on every bounded subinterval of R. It may be noted that, even though E is
a vector space, the metric is not derived from a norm since, if λ ∈ R, one may have
d(λg, λh) ̸= |λ|d(g, h).
(v) Let E be the set of all measurable functions f : I → R, where I = [a, b] is an
interval of R, and define
d(g, h) =
$ b
a
|g(x) − h(x)|(1 + |g(x) − h(x)|)−1
dx.
In order to obtain (D1), we identify functions which take the same value at all points
of I, except for a set of measure zero.
Convergence with respect to this metric coincides with convergence in measure,
which plays a role in the theory of probability.
(vi) Let E = F∞
2 be the set of all infinite sequences a = (α1, α2, . . .), where αj = 0 or
1 for every j, and define d(a, a) = 0, d(a, b) = 2−k if a ̸= b, where b = (β1, β2, . . .)
and k is the least positive integer such that αk ̸= βk.](https://image.slidesharecdn.com/11052267-250525204806-55299cc9/85/Number-Theory-An-Introduction-To-Mathematics-2nd-Edition-Coppel-45-320.jpg)
![30 I The Expanding Universe of Numbers
Here the triangle inequality holds in the stronger form
d(a, b) ≤ max[d(a, c), d(c, b)].
This metric space plays a basic role in the theory of dynamical systems.
(vii) A connected graph can be given the structure of a metric space by defining the dis-
tance between two vertices to be the number of edges on the shortest path joining them.
Let E be an arbitrary metric space and {an} a sequence of elements of E. The
sequence {an} is said to converge, with limit a ∈ E, if
d(an, a) → 0 as n → ∞,
i.e. if for each real ε > 0 there is a corresponding positive integer N = N(ε) such that
d(an, a) < ε for every n ≥ N.
The limit a is uniquely determined, since if also d(an, a′) → 0, then
d(a, a′
) ≤ d(an, a) + d(an, a′
),
and the right side can be made arbitrarily small by taking n sufficiently large. We write
lim
n→∞
an = a,
or an → a as n → ∞. If the sequence {an} has limit a, then so also does any (infinite)
subsequence.
If an → a and bn → b, then d(an, bn) → d(a, b), as one sees by taking a′ = an
and b′ = bn in (∗).
The sequence {an} is said to be a fundamental sequence, or ‘Cauchy sequence’,
if for each real ε > 0 there is a corresponding positive integer N = N(ε) such that
d(am, an) < ε for all m, n ≥ N.
If {an} and {bn} are fundamental sequences then, by (∗), the sequence {d(an, bn)}
of real numbers is a fundamental sequence, and therefore convergent.
A set S ⊆ E is said to be bounded if the set of all real numbers d(a, b) with
a, b ∈ S is a bounded subset of R.
Any fundamental sequence {an} is bounded, since if
d(am, an) < 1 for all m, n ≥ N,
then
d(am, an) < 1 + δ for all m, n ∈ N,
where δ = max1≤ j<k≤N d(aj, ak).
Furthermore, any convergent sequence {an} is a fundamental sequence, as one sees
by taking a = limn→∞ an in the inequality
d(am, an) ≤ d(am, a) + d(an, a).
A metric space is said to be complete if, conversely, every fundamental sequence
is convergent.](https://image.slidesharecdn.com/11052267-250525204806-55299cc9/85/Number-Theory-An-Introduction-To-Mathematics-2nd-Edition-Coppel-46-320.jpg)
![34 I The Expanding Universe of Numbers
We will also use the notion of norm of a linear map. If A: Rn → Rm is a linear
map, its norm |A| is defined by
|A| = sup
|x|≤1
|Ax|.
Evidently
|A1 + A2| ≤ |A1| + |A2|.
Furthermore, if B : Rm → Rl is another linear map, then
|B A| ≤ |B||A|.
Hence, if m = n and |A| < 1, then the linear map I − A is invertible, its inverse being
given by the geometric series
(I − A)−1
= I + A + A2
+ · · · .
It follows that for any invertible linear map A: Rn → Rn, if B : Rn → Rn is a lin-
ear map such that |B − A| < |A−1|−1, then B is also invertible and |B−1 − A−1| → 0
as |B − A| → 0.
If ϕ : U → Rm is differentiable at x0 ∈ Rn, then it is also continuous at x0, since
|ϕ(x) − ϕ(x0)| ≤ |ϕ(x) − ϕ(x0) − ϕ′
(x0)(x − x0)| + |ϕ′
(x0)||x − x0|.
We say that ϕ is continuously differentiable in U if it is differentiable at each point of
U and if the derivative ϕ′(x) is a continuous function of x in U. The inverse function
theorem says:
Proposition 27 Let U0 be a neighbourhood of x0 ∈ Rn and let ϕ : U0 → Rn be a
continuously differentiable map for which ϕ′(x0) is invertible.
Then, for some δ > 0, the ball U = {x ∈ Rn : |x − x0| < δ} is contained in U0
and
(i) the restriction of ϕ to U is injective;
(ii) V := ϕ(U) is open, i.e. if η ∈ V , then V contains all y ∈ Rn near η;
(iii) the inverse map ψ : V → U is also continuously differentiable and, if y = ϕ(x),
then ψ′(y) is the inverse of ϕ′(x).
Proof To simplify notation, assume x0 = ϕ(x0) = 0 and write A = ϕ′(0). For any
y ∈ Rn, put
fy(x) = x + A−1
[y − ϕ(x)].
Evidently x is a fixed point of fy if and only if ϕ(x) = y. The map fy is also contin-
uously differentiable and
f ′
y(x) = I − A−1
ϕ′
(x) = A−1
[A − ϕ′
(x)].
Since ϕ′(x) is continuous, we can choose δ > 0 so that the ball U = {x ∈ Rn : |x| < δ}
is contained in U0 and
| f ′
y(x)| ≤ 1/2 for x ∈ U.](https://image.slidesharecdn.com/11052267-250525204806-55299cc9/85/Number-Theory-An-Introduction-To-Mathematics-2nd-Edition-Coppel-50-320.jpg)
![4 Metric Spaces 35
If x1, x2 ∈ U, then
| fy(x2) − fy(x1)| =
&
&
&
&
$ 1
0
f ′
((1 − t)x1 + tx2)(x2 − x1)dt
&
&
&
&
≤ |x2 − x1|/2.
It follows that fy has at most one fixed point in U. Since this holds for arbitrary y ∈ Rn,
the restriction of ϕ to U is injective.
Suppose next that η = ϕ(ξ) for some ξ ∈ U. We wish to show that, if y is near η,
the map fy has a fixed point near ξ.
Choose r = r(ξ) > 0 so that the closed ball Br = {x ∈ Rn : |x − ξ| ≤ r} is
contained in U, and fix y ∈ Rn so that |y − η| < r/2|A−1|. Then
| fy(ξ) − ξ| = |A−1
(y − η)|
≤ |A−1
||y − η| < r/2.
Hence if |x − ξ| ≤ r, then
| fy(x) − ξ| ≤ | fy(x) − fy(ξ)| + | fy(ξ) − ξ|
≤ |x − ξ|/2 + r/2 ≤ r.
Thus fy(Br) ⊆ Br. Also, if x1, x2 ∈ Br, then
| fy(x2) − fy(x1)| ≤ |x2 − x1|/2.
But Br is a complete metric space, with the same metric as Rn, since it is a closed
subset (if xn ∈ Br and xn → x in Rn, then also x ∈ Br). Consequently, by the con-
traction principle (Proposition 26), fy has a fixed point x ∈ Br. Then ϕ(x) = y, which
proves (ii).
Suppose now that y, η ∈ V . Then y = ϕ(x), η = ϕ(ξ) for unique x, ξ ∈ U. Since
| fy(x) − fy(ξ)| ≤ |x − ξ|/2
and
fy(x) − fy(ξ) = x − ξ − A−1
(y − η),
we have
|A−1
(y − η)| ≥ |x − ξ|/2.
Thus
|x − ξ| ≤ 2|A−1
||y − η|.
If F = ϕ′(ξ) and G = F−1, then
ψ(y) − ψ(η) − G(y − η) = x − ξ − G(y − η)
= −G[ϕ(x) − ϕ(ξ) − F(x − ξ)].](https://image.slidesharecdn.com/11052267-250525204806-55299cc9/85/Number-Theory-An-Introduction-To-Mathematics-2nd-Edition-Coppel-51-320.jpg)
![4 Metric Spaces 37
Proof If x(t) is a solution of the differential equation (1) which satisfies the initial
condition (2), then by integration we get
x(t0) = ξ0 +
$ t
t0
ϕ[τ, x(τ)]dτ.
Conversely, if a continuous function x(t) satisfies this relation then, since ϕ is contin-
uous, x(t) is actually differentiable and is a solution of (1) that satisfies (2). Hence we
need only show that the map F defined by
(F x)(t) = ξ0 +
$ t
t0
ϕ[τ, x(τ)]dτ
has a unique fixed point in the space of continuous functions.
There exist positive constants M, L such that
|ϕ(t, ξ)| ≤ M, |ϕ′
(t, ξ)| ≤ L
for all (t, ξ) in a neighbourhood of (t0, ξ0), which we may take to be U. If (t, ξ1) ∈ U
and (t, ξ2) ∈ U, then
|ϕ(t, ξ2) − ϕ(t, ξ1)| =
&
&
&
&
$ 1
0
ϕ′
(t, (1 − u)ξ1 + uξ2)(ξ2 − ξ1)du
&
&
&
&
≤ L|ξ2 − ξ1|.
Choose δ > 0 so that the box |t − t0| ≤ δ, |ξ − ξ0| ≤ Mδ is contained in U and
also Lδ < 1. Take I = [t0 − δ, t0 + δ] and let C (I) be the complete metric space of
all continuous functions x : I → Rn with the distance function
d(x1, x2) = sup
t∈I
|x1(t) − x2(t)|.
The constant function x0(t) = ξ0 is certainly in C (I). Let E be the subset of all
x ∈ C (I) such that x(t0) = ξ0 and d(x, x0) ≤ Mδ. Evidently if xn ∈ E and xn → x
in C (I), then x ∈ E. Hence E is also a complete metric space with the same metric.
Moreover F(E) ⊆ E, since if x ∈ E then (F x)(t0) = ξ0 and, for all t ∈ I,
|(F x)(t) − ξ0| =
&
&
&
&
$ t
t0
ϕ[τ, x(τ)]dτ
&
&
&
& ≤ Mδ.
Furthermore, if x1, x2 ∈ E, then d(F x1, F x2) ≤ Lδd(x1, x2), since for all t ∈ I,
|(F x1)(t) − (F x2)(t)| =
&
&
&
&
$ t
t0
{ϕ[τ, x1(τ)] − ϕ[τ, x2(τ)]}dτ
&
&
&
&
≤ Lδ d(x1, x2).
Since Lδ < 1, the result now follows from Proposition 26. ✷](https://image.slidesharecdn.com/11052267-250525204806-55299cc9/85/Number-Theory-An-Introduction-To-Mathematics-2nd-Edition-Coppel-53-320.jpg)
![bliss. Wide are thrown the golden gates, and as they open, ten
thousand voices are heard chaunting in chorus; "Lift up your heads,
O ye gates; and be ye lift up, ye everlasting doors; and the King of
glory shall come in. Who is this King of glory? The Lord, strong and
mighty; the Lord, mighty in battle. Lift up your heads, O ye gates;
even lift them up, ye everlasting doors; and the King of glory shall
come in. Who is this King of glory? The Lord of Hosts, he is the King
of glory." Forth from heaven's portals there issued a goodly band,
singing as they advance to meet and welcome their victorious King,
whom they convey in celestial triumph to the presence of the eternal
Father; seated on his throne of glory, he receives, with ineffable
delight and joy, this, his only-begotten, always well-beloved, but now
still more endeared Son, the Glorious Deliverer of the children of
men. Great was the joy of that illustrious day, when the eternal Son
of God, entered the city of the new Jerusalem, as the victorious
Conqueror of sin, death, and hell, whom he led as captives to adorn
his triumph, for, "having spoiled principalities and powers, he made a
show of them openly, triumphing over them, and ascended on high,
leading captivity captive." Then the eternal hills resounded to the
melodious sound of ten thousand times ten thousand voices, who
sing aloud, "Worthy is the Lamb that was slain, to receive power,
and riches, and wisdom, and strength, and honour, and glory, and
blessing." Then all in heaven said, "Blessing, and honour, and glory,
and power, be unto him that sitteth upon the throne, and unto the
Lamb, for ever, and ever." The spirits of the redeemed vie with elect
angels, in testifying their love, reverence, and gratitude to the God
of their salvation. They knew, if the eternal Son of God had not
become their surety, not one of Adam's race could ever have entered
the realms of bliss.[107]
But in the eternal council of peace, he did
covenant and promise, in the fulness of time, to become a sacrifice,
and God who knew him to be faithful, did, on the credit of that
promise, save all the Old Testament saints.[108]
Jesus had now
fulfilled that engagement; paid the full price of their redemption;
"blotted out the hand-writing of ordinances that was against them,
taking it away by nailing it to his cross." What wonder, if his return](https://image.slidesharecdn.com/11052267-250525204806-55299cc9/85/Number-Theory-An-Introduction-To-Mathematics-2nd-Edition-Coppel-60-320.jpg)
![unlearned, and many of them unlettered, Galilean fishermen. The
inhabitants of Galilee were proverbial for their dulness and stupidity;
[109]
yet these men were taught, in an instant of time, to speak, with
ease and fluency, languages whose very names, it is more than
probable, they were an hour before unable to pronounce correctly.
An opportunity was instantly offered for the apostles openly to
display their extraordinary gifts. Amidst the assembled throng were
men of sixteen different nations, to whom these poor fishermen
publicly proclaimed, in their several languages, or dialects, the
wonderful works of God. They needed no interpreter, in addressing
this motley crowd. How preposterous to accuse the apostles of
drunkenness! Truly, we should not imagine a state of inebriety the
best calculated for acquiring a knowledge of any of the learned
languages. We seldom know men, (however well their heads are
furnished,) in a state of intoxication, speak any thing except it be the
language of foolishness. Beside, it was only the third hour of the
day, (nine o'clock) the time of offering the daily morning sacrifice in
the temple, before which hour the Jews were forbidden to take any
refreshment; and, as this was a solemn festival, no doubt the
command was then more strictly observed. How mild, yet energetic,
the reply of Peter, who declared the event to be a fulfilment of the
prophecy of Joel, accomplished on the return of Jesus to glory;
"when being by the right hand of God exalted, and having received
of the Father the promise of the Holy Ghost, he had shed forth that
which they then saw and heard." The appearance of the Holy Spirit
was sufficient to prove his personality. Might not the sound from
heaven, as of a rushing mighty wind, be designed to show that the
operations of God the Holy Spirit, are like the unknown and
unexplored sources of the air. "The wind bloweth where it listeth,
and thou hearest the sound thereof; but canst not tell whence it
cometh, or whither it goeth: so is every one that is born of the
Spirit." This was a lesson taught Nicodemus by Jesus, the wisdom
and word of God.
On Shinar's plains, the Lord, to testify his divine displeasure,
confounded the language of mankind. It was a curse pronounced on](https://image.slidesharecdn.com/11052267-250525204806-55299cc9/85/Number-Theory-An-Introduction-To-Mathematics-2nd-Edition-Coppel-64-320.jpg)
![prominent feature in the latter-day glories of the Church. The Lord
has promised to gather together the dispersed in Judah, and the
outcasts of Israel. "The deliverer shall arise out of Zion, and turn
away ungodliness from Jacob." This nation, who once refused and
crucified the Messiah, shall, when partakers of this promised
blessing, "look upon him whom they have pierced, and mourn." This
promise is not confined to the Jews, but extends to the fallen race of
Adam, whom our spiritual David will make inhabitants of the new
Jerusalem, which is above, without regard to their being of Jewish or
Gentile extraction.[110]
He will not consider the trifling distinctions of
colour, language, or nation, a barrier of such importance as to
preclude their participating in his blessings.
The thing promised is an abundant outpouring of the Holy Spirit.
Adam, by his apostasy, lost the image of God stamped upon his soul
at his creation. The sentence, "in the day thou eatest thereof thou
shalt surely die," was not suffered to go unexecuted. From that
hapless hour, his soul, the most noble part, was dead to all spiritual
life, and became the abode of corroding passions and depraved
principles. He immediately shrank from holding intercourse with God,
and tried to hide himself from the presence of his benefactor. As
Adam begat a son in his own fallen likeness, all his race partake of
the same corrupt nature. We are ignorant of God and his ways. We
need divine teaching; we cannot naturally understand the things of
God, which are spiritual, the eye of our understanding being
darkened; God is not in all our thoughts; we are averse to
communion with the Father of Spirits. We despise his offers of free
grace—we prefer to be saved by our own rather than God's method
—we see no beauty in Jesus that we should desire him—we dislike
to renounce our own, and trust in his complete righteousness—we
consider his commands grievous, and the language of our soul is,
"we will not have this man to reign over us." But we are here told of
a sovereign antidote for these deep-seated moral disorders of the
soul. Here is a gracious promise of an abundant outpouring of the
Holy Spirit, whose office it is to "convince of sin, of righteousness,
and of judgment." He convinces the soul, into which he enters, of](https://image.slidesharecdn.com/11052267-250525204806-55299cc9/85/Number-Theory-An-Introduction-To-Mathematics-2nd-Edition-Coppel-67-320.jpg)
![the exceeding sinfulness of sin—that it is the evil thing which God
hates; and shows the divine law is spiritual, extending to the
thoughts and intents of the heart.[111]
He puts a cry for mercy into
the soul, destroys the natural enmity of the mind against God's plan
of salvation, and makes the object of his divine teaching willing and
anxious to partake of the Lord's bounty, and be a debtor to mercy
alone. The Holy Spirit teaches of righteousness by convincing that a
better righteousness than our own tattered rags is absolutely
necessary, ere we can see the face of God with peace. He makes the
soul willing to be clothed with the wedding garment of Jesus'
righteousness, which is the fine linen of the saints. It is
indispensable that we be clothed with this livery of the court of
Heaven, or we shall be denied admission into the mansions of the
King of Glory. Would we behold the fulfilment of this prophetic
promise, then let us direct our minds back to a survey of the glorious
scenes exhibited on the ever memorable day of Pentecost, when the
Spirit was, in so free and copious a manner, poured out from on
high. Attend to the sermon Peter preached on the day of his
ordination; mark its effects on the three thousand of the House of
David, inhabitants of Jerusalem's much-famed city. Listen to their
cry, "Men and brethren, what must we do?" Surely these were none
of the stout hearts who dared even to crucify the Lord of life and
glory? The same! yet how different their tone—how altered their
conduct! To what cause can we attribute this astonishing change in
the minds of three thousand persons in the same instant of time?
Surely it was none other than the almighty work of God the Holy
Ghost. It was his influence on the minds of these men which
produced the Spirit of grace and supplication, and taught them to
direct the anxious cry and supplicating look unto him whom they had
pierced. Was not the anguish of their souls, under a sense of their
sins, equal to the exquisite sorrow of those who bitterly bewail the
death of their first-born? However skilfully Peter might wield the
sword of the Spirit, (the word of God,) it was none other than the
God of all grace, who directed and sent it home with saving power
to the hearts and consciences of these Jerusalem sinners. Are not](https://image.slidesharecdn.com/11052267-250525204806-55299cc9/85/Number-Theory-An-Introduction-To-Mathematics-2nd-Edition-Coppel-68-320.jpg)
![would direct us to the Lord the Spirit, as the almighty guide who
pointed out the road, and taught their wandering feet to tread the
strait, the narrow way, the only path, that leads to Zion's hill. In the
Bible, that chart of life, the road is shown with clearness, and
described with accuracy. It is called faith in the finished salvation of
Christ, and obedience to his commands. The hand which drew this
path to glory, is the very same that painted the splendid canopy of
heaven. By this good old way, all the patriarchs, prophets, apostles,
martyrs, and reformers, entered the city of the Lord of Hosts. Their
guide and comforter, through this waste howling wilderness, was the
third person of the Triune-Jehovah. What countless myriads has this
almighty guide led to the mount of God, from the antediluvian
worthies, down to the happy spirit just entered into the joy of its
Lord! Like them, led by the same unerring teacher, we shall not fail
of arriving safely at the mansion of everlasting joy, for he is the only
faithful conductor[112]
to the heavenly Jerusalem; untaught by him,
none can find the path of life, but will assuredly stumble on the dark
mountains of sin and error, and run the downward road that leads to
hell.
Eternal life is the gift of God. Christ is "the way, the truth, and the
life: none can come unto God, but by him." The office of the Holy
Spirit is to instruct the ignorant, comfort the mourners in Zion, and
make us meet to be "partakers of the inheritance of the saints in
light." "If ye, being evil, know how to give good gifts unto your
children, how much more will your heavenly Father give the Holy
Spirit to them that ask him." May we be partakers of that
inestimable blessing, for without his influence on our hearts, vain will
be even the electing love of God the Father—vain the vicarious
sacrifice and imputed righteousness of Christ the Son—vain to us the
plan of salvation; and vain, all the promises of the Gospel. As well
for us, if those glad tidings of great joy, "Glory to God in the highest,
and on earth peace, good-will toward men," had not reached our
ears. Unapplied, the most sovereign remedy is useless, for then not
even Gilead's balm, can heal the dire disease.[113]
Christ will prove no
Saviour to us, unless applied to our individual case. It is the office of](https://image.slidesharecdn.com/11052267-250525204806-55299cc9/85/Number-Theory-An-Introduction-To-Mathematics-2nd-Edition-Coppel-70-320.jpg)
![CHAPTER LXII.
The Lord hath sworn and will not repent, thou art a priest for
ever, after the order of Melchizedek.—Psalm cx. 4.
In the Old Testament, we find but little recorded of Melchizedek, that
venerable priest of the most High God, who met and blessed the
patriarch Abraham as he returned victorious from the slaughter of
Chedorlaomer and the confederate kings. But from that little, we are
led to regard him as a person of distinction. To him, the great father
of the faithful and friend of God presented the tithes or tenths of the
spoil. It is from the prophetical word of the royal Psalmist, "the Lord
hath sworn and will not repent, thou art a Priest for ever, after the
order of Melchizedek," that we are taught to view this ancient priest
of God as a type: and of whom, if not of Christ? Paul, in his epistle
to the Hebrews,[114]
speaks largely on the subject; he proves the
fulfilment of the prophecy, and declares, that Christ's priestly office
was prefigured in the person of Melchizedek, to Abraham the father
of the Israelitish race. In the same epistle, we find blended the
priesthood of Aaron, in order to show the vast superiority of that of
Christ over the other two, though both instituted by God himself. But
as we find no prophecy respecting the Aaronic priesthood, we make
no further reference to that subject, in order to attend more
immediately to the words, "The Lord hath sworn, and will not
repent, thou art a priest for ever, after the order of Melchizedek."
Was this priest of the most High God honoured with the title of King
of Salem—by interpretation, King of Righteousness, and King of
Peace? Is not Jesus proclaimed King of Zion; the Lord our
Righteousness, and the Prince of Peace? Nor are these mere empty
titles, but real characters, and offices, sustained by Him, who
"abideth a priest upon his throne for ever." We have no historical
account of the parentage or descendants of Melchizedek; he is](https://image.slidesharecdn.com/11052267-250525204806-55299cc9/85/Number-Theory-An-Introduction-To-Mathematics-2nd-Edition-Coppel-72-320.jpg)
![presented to us as "without father, without mother, without descent,
having neither beginning of days, nor end of life;" but being made
like unto the Son of God, abideth a priest continually.[115]
And Christ's
priesthood was not derived by genealogy, or succession, he had
neither father or mother of the family of Aaron, from whom his
priesthood could descend. It is evident our Lord sprang "out of
Judah, of which tribe no man gave attendance at the altar;"[116]
neither did Christ die and leave it to others, by way of descent, but
was constituted a single priest, without predecessor or successor.
"He abideth a priest for ever, after the order of Melchizedek." It is
impossible for a finite mind to comprehend the eternal sonship of
the Son of God, whom the Father, before the foundation of the
world, constituted a priest for ever; and therefore, the priesthood of
Melchizedek was instituted to prefigure to us the nature of Christ's
eternal priesthood. "The Lord hath sworn and will not repent, thou
art a priest for ever, after the order of Melchizedek." These words
deserve particular attention. It is God the Father who swears to
Christ; no oath of allegiance is required from him who is constituted
our Priest. Jehovah, whose eye pierces through futurity, knew he
would be faithful in his office, and he freely and unreservedly trusted
him to maintain his divine honour and justice, and accomplish the
salvation of sinners. The high-priestly office, though honourable,
could not add to Christ's dignity; but his glorious person did confer
honour and dignity upon the sacred office, for he who is constituted
our High Priest, "is fellow to the Lord of Hosts." "Every high priest is
ordained, to offer both gifts and sacrifices," and great was the
sacrifice offered by Christ: he offered up himself; he would borrow
nothing, but was both priest, sacrifice, altar, and temple: and "by
that offering, he hath perfected for ever them that are sanctified."
"And because he continueth ever, he hath an unchangeable
priesthood;" "wherefore he is able to save them to the uttermost,
that come unto God by him, seeing he ever liveth to make
intercession for them." Blessed Jesus! thou priest of Melchizedek's
order, while we would not withhold from thee a portion of all that
thou givest us, let us not rest satisfied, till we are enabled to present](https://image.slidesharecdn.com/11052267-250525204806-55299cc9/85/Number-Theory-An-Introduction-To-Mathematics-2nd-Edition-Coppel-73-320.jpg)
![CHAPTER LXIII.
Seventy weeks are determined upon thy people and upon thy
holy city, to finish the transgression, and to make an end of
sins, and to make reconciliation for iniquity, and to bring in
everlasting righteousness, and to seal up the vision and
prophecy, and to anoint the most Holy. Know, therefore, and
understand, that from the going forth of the commandment to
restore and to build Jerusalem, unto the Messiah, the Prince,
shall be seven weeks, and three score and two weeks: the
street shall be built again, and the wall, even in troublous times.
—Daniel ix. 24, 25.
The harps of Judah were silent—the disconsolate Israelites hung
them on the willows of Babylon—no songs of Zion were heard in
that land of captivity, where, for seventy long years, they wore the
galling yoke of bondage, bereft of home and all its blessings—the
land of their forefathers in the possession of strangers—Jerusalem in
ruins—her palaces consumed—the Temple destroyed—the spot
trodden down by the Heathen—themselves exposed to the taunts of
their conquerors, and compelled to bow before the idolatrous image
of Chaldean superstition.[117]
Well might Judah's sons weep by the
waters of Babylon, whose murmurings recalled to their recollection
the stream which gushed from Horeb's mount.[118]
The remembrance
of past blessings increases the weight of present misery. How
changed their state, and changed to punish their awful rebellions
against the Lord of Sabaoth! Yet the God of Israel was not unmindful
of his promise—he cheered their drooping spirits with the assurance
of speedy deliverance from their captive state. The prayer of Daniel
entered into the ears of the Lord of Hosts—the command was given
—swiftly the angel, even Gabriel, flew to reveal his Lord's decrees
unto the mourning prophet—that "man greatly beloved" of his God.](https://image.slidesharecdn.com/11052267-250525204806-55299cc9/85/Number-Theory-An-Introduction-To-Mathematics-2nd-Edition-Coppel-75-320.jpg)
![Daniel was commissioned to foretel the deliverance of the Jews from
Babylon—the building of Jerusalem and its walls in troublous times;
and to him, Jehovah was graciously pleased to renew the promise of
the Prince, Messiah, whose appearance all the patriarchs and
prophets had foretold. The nearer that glorious epoch approached,
the more minutely was it described. The Lord gave Daniel to "know
and understand, that from the going forth of the commandment to
restore and build Jerusalem unto the Messiah, the Prince, should be
seven weeks, and three score and two weeks." The period here
styled weeks, is generally allowed to be sabbaths of years. This
appears to be the sense of the passage, for the Jews were
accustomed to reckon their time and feasts by weeks or sabbaths.
The week of days was from one seventh or sabbath day to another.
The week of years was from one seventh or sabbatical year to
another; in the seventh, or sabbatical year, they neither sowed their
fields nor pruned their vineyards; it was a sabbath of rest unto the
land.[119]
In the regulation of the year of Jubilee, they were
commanded to number "seven sabbaths of years, seven times seven
years, and the space of the seven sabbaths of years shall be to thee
forty and nine years."[120]
We therefore only follow the Mosaic rule,
(to which Moses' disciples cannot object,) if we consider these seven
weeks, and three score and two weeks, as seven times sixty-nine, or
four hundred and eighty-three years, which should be between "the
going forth of the commandment to restore and build Jerusalem
unto the Messiah, the Prince." There were four distinct decrees or
commandments granted by the kings of Persia, in favour of the
Jews, who came under the dominion of that empire by its conquest
of Babylon. This was the epoch of Daniel's vision. No sooner had
Cyrus obtained possession of Chaldea, than he issued a decree
allowing the Jews to quit the land of their captivity, and repair to
Judea to build the temple of the Lord. He also restored to them the
vessels and treasures which Nebuchadnezzar had taken from the
temple built by Solomon. On the grant of this decree,[121]
five
hundred and thirty-six years before Christ, many of the Jews
returned to their own land, and laid the foundation of the temple;](https://image.slidesharecdn.com/11052267-250525204806-55299cc9/85/Number-Theory-An-Introduction-To-Mathematics-2nd-Edition-Coppel-76-320.jpg)
![but they were hindered in the building of it by their several enemies,
who were supported in their opposition by Artaxerxes, the successor
of Cyrus. But when Darius Hystaspes ascended the throne of Persia,
he issued a decree[122]
five hundred and nineteen years before
Christ, forbidding the enemies of the Jews to interrupt the building
of the temple, and further commanded that materials requisite for
the work, and the animals, oil, and wine for the sacrifices, should be
supplied at his (the king's) cost. The third decree was granted to
Ezra, the scribe, four hundred and sixty-seven years before Christ,
by Artaxerxes Longimanus, in the seventh year of his reign, by which
he bestowed great favours upon the Jews,[123]
appointing Ezra
Governor of Judea. He permitted all the Jews to return to Jerusalem,
and commanded his treasurers beyond the river, to supply Ezra with
such things as he needed for the house of his God, even to an
hundred talents of silver, an hundred measures of wheat, an
hundred baths of wine, and an hundred baths of oil. The king and
his princes presented much silver and gold, and many vessels, and
ordered that what else might be required for the house of God,
should be supplied from the king's treasury. This is not the same
Artaxerxes who listened to the slanderous reports of the enemies of
the Jews, and stopped the building of their temple; but Artaxerxes,
surnamed Longimanus, supposed to be the person styled Ahasuerus,
in the book of Esther, whose attachment to his Israelitish consort
may account for the distinguished favours he conferred on the
people of her nation. We find the queen was present when
Nehemiah presented his petition, which was the second decree
granted by this monarch, and was the fourth and last decree, being
granted in the twentieth year of his reign, and four hundred and
fifty-four years before Christ.[124]
This was the most efficient decree,
for by it Jerusalem and its walls were built. The high resolves of the
court of Heaven were revealed; Daniel was made "to know and
understand that from the going forth of the commandment to
restore and build Jerusalem, unto the Messiah, the prince, shall be
seven weeks, and three score and two weeks, being sixty nine
weeks, or four hundred and eighty-three years. From the last, or](https://image.slidesharecdn.com/11052267-250525204806-55299cc9/85/Number-Theory-An-Introduction-To-Mathematics-2nd-Edition-Coppel-77-320.jpg)
![fourth, decree to the birth of Christ, (vide Rollin, volume 8, page
265,) is four hundred and fifty-four years, to which we add twenty-
nine years (the age at about which Christ entered on his public
ministry);[125]
these united, make the exact period of sixty-nine
weeks, or four hundred and eighty-three years. Daniel also declares
that "seventy weeks (or four hundred and ninety years) are
determined upon thy people and upon thy holy city, to finish the
transgression, and to make an end of sins, and to make
reconciliation for iniquity, and to bring in everlasting righteousness,
and to seal up the vision and prophecy, and to anoint the most
Holy." We find between the seventy weeks, or four hundred and
ninety years, and the sixty-nine weeks, or four hundred and eighty-
three years, a difference of one week, or seven years, which is the
week evidently alluded to in the twenty-seventh verse of this
chapter, in which "he shall confirm the covenant with many for one
week, &c." From the period of Christ's first entry into the ministry,
and the calling of his apostles, until his crucifixion, were three and a
half years, and, for three and a half years after that event, his
apostles continued to minister amongst the Jews. This makes a
period of seven years, (or one prophetic week,) in the midst of
which the Messiah was cut off, and "the sacrifice and oblation"
virtually ceased. The correspondence is exact: Jesus, the Messiah,
not only entered on his public ministry at the very period pointed out
ages before, but was actually cut off in the midst of the week, as
was expressly foretold. These predictions of the Prince Messiah are
peculiarly striking. The time for his appearance is marked, and the
particular objects he should effect on his coming, are described with
such minuteness, as scarcely to admit of the possibility of mistaking
his person. The grand features of his mission were so strongly
exhibited, that it was morally impossible the Messiah should appear
and not be recognised. Prejudice must have blinded the eye of that
mind which does not, on comparing the whole of the New Testament
with this prophecy, acknowledge Jesus of Nazareth to be the
Messiah. It bears the stamp of divine prescience: none but the
omniscient God could have given his features with such clearness so](https://image.slidesharecdn.com/11052267-250525204806-55299cc9/85/Number-Theory-An-Introduction-To-Mathematics-2nd-Edition-Coppel-78-320.jpg)
![over—the ransom paid—the captives of the mighty delivered—the
law was honoured—justice satisfied—God glorified—Heaven opened
—man redeemed—and hell vanquished. That was the glorious event
which types were intended to exhibit, and prophets were
commissioned to proclaim. The appointed time of the vision was
arrived—it had long tarried, but it was accomplished. The chain of
prophecy was complete—the vision was sealed[126]
—and the most
holy anointed. The God-man, Christ Jesus, anointed by his Father
king and priest of Zion, then exchanged his thorny crown for the
royal diadem—then left the sorrows of earth for the glories of his
mediatorial throne, which no enemy can touch—their opposition is
vain—he that sitteth upon the circle of the heavens, will laugh them
to scorn. Happy are they who have for their king and priest, him
whose kingdom is eternal, and priesthood unchangeable—who look
to the Redeemer of Israel as the rock of their salvation, and crown
the most holy, Lord of all. "Happy are the people that are in such a
case, yea, blessed are the people whose God is the Lord."](https://image.slidesharecdn.com/11052267-250525204806-55299cc9/85/Number-Theory-An-Introduction-To-Mathematics-2nd-Edition-Coppel-80-320.jpg)
![allude to the destruction of the city, previous to which event the
Messiah should be cut off. We hope we shall not offer any violence
to the words, if we give them this interpretation. The destruction of
Jerusalem is not the only event alluded to in this interesting
prophecy; there is one of paramount importance to the ruin of
Salem's palaces, though that involved the fate of Judah's sons. On
the other momentous fact hang the highest interests of Jew and
Gentile, bond and free, past, present, and future generations; not
only the happiness of earth, but much of the glory of heaven,
depends on its accomplishment. Without it no sweet song of
"Salvation to God and the Lamb," would have echoed amidst the
heavenly hills, none of the race of Adam would be seen worshipping
before the presence of Jehovah with the angels of light; those
melodious hymns of redemption, now chaunted by ten thousand
times ten thousand glorified Saints, had not been heard but for the
vicarious sacrifice of the Son of God,[127]
who not only covenanted,
but did actually lay down his life a ransom for sinners. When Jesus,
the Christ of God, the Prince Messiah, appeared on earth, it was not
simply to set the children of men an example of piety and virtue; we
ardently admire his glorious example, and consider his followers
bound to imitate the bright pattern he has left them; yet we dare
not believe that that was the only object he designed to accomplish
when he visited our world.[128]
No, he came as the federal Head, the
Representative and Surety of his people.[129]
He was "cut off from
the land of the living," by a violent and cruel death; yet not for
himself, not for any sin of his own,[130]
nor purposely to set us a
pattern of patience and resignation; but to discharge the debt of sin,
he had covenanted to cancel on man's account. Jehovah executed
towards him the severest justice, and permitted his crucifiers to
exercise the blackest ingratitude, and most inhuman cruelty. "O
Jerusalem, Jerusalem, thou who killest the prophets, and stonest
them that are sent unto thee, how often would the Lord have
gathered thee under his protecting care as a hen gathereth her
chickens under her wings, but ye would not." Thy awful doom was
sealed when thou didst reject the authority, and persecute unto](https://image.slidesharecdn.com/11052267-250525204806-55299cc9/85/Number-Theory-An-Introduction-To-Mathematics-2nd-Edition-Coppel-82-320.jpg)
![death Jesus the Messiah, thy prophet and benefactor, thy God and
King. The thought of thy approaching misery drew tears from the
eyes, and groans from the heart, of Incarnate Deity; yet thy children
beheld, with feelings of triumphant scorn, the sorrows and sufferings
their wanton cruelty inflicted on the Holy Jesus. But heaven marked
the impious deed.[131]
The blood of Jesus, of prophets, of apostles,
and of martyrs, called for vengeance on thy guilty land; the cry was
heard, justice remembered thy black catalogue of crimes, the King of
heaven beheld the insult offered to his beloved Son, and Jehovah
arose to punish thy rejection of Jesus the Messiah, whom "ye would
not have to reign over you." The crimes of Jerusalem were of the
blackest and most awful character, and her punishment was
tremendously dreadful.[132]
The Israelites, once the peculiar
favourites of Heaven[133]
—nursed in the lap of plenty, instructed in
the oracles of God—blessed with the temple of Jehovah—taught to
adore the God of truth whom their forefathers worshipped; this
people, who once had the Lord for their Law-giver and King,[134]
were compelled to bow beneath the oppressive power of arbitrary
despots—the law of truth was exchanged for the tyrant's mandate—
equity and justice were banished the walls of Salem, and despotism,
oppression, blasphemy, and pride, reigned within that devoted,
miserable, city. Anarchy and confusion ruled that senate and
sanctuary, once as gloriously "distinguished from the rest of the
world by the purity of its government, as by the richness and
elegance of its buildings. Jerusalem was devoted to destruction, and
she sunk beneath the accumulated horrors of war, famine, fire, and
pestilence. Internal faction and a foreign foe reduced that beauteous
city and magnificent sanctuary, to a heap of ruins. The temple fell—
not all the commands, promises, or threats of Titus, could save that
splendid edifice from destruction; the people of the prince,
regardless of their general's orders, helped to complete the work of
desolation;—but prophecy was fulfilled, Jerusalem was overwhelmed
with the flood of divine vengeance, and desolation prevailed even
unto the end of the war.](https://image.slidesharecdn.com/11052267-250525204806-55299cc9/85/Number-Theory-An-Introduction-To-Mathematics-2nd-Edition-Coppel-83-320.jpg)
![CHAPTER LXV.
And he shall confirm the covenant with many for one week; and
in the midst of the week he shall cause the sacrifice and the
oblation to cease, and for the overspreading of abominations he
shall make it desolate, even until the consummation, and that
determined shall be poured upon the desolate.—Daniel ix. 27.
Some writers consider this verse prophetical of the desolate state of
Jerusalem under Antiochus Epiphanes, that sacrilegious monarch
who impiously profaned the sanctuary of the God of Israel. By him
the temple was ransacked and despoiled of its holy vessels; its
golden ornaments pulled off; its hidden treasures seized; and an
unclean animal offered on the altar of burnt-offerings. Thus did this
impious Syrian king dare profane the altar and temple dedicated to
Jehovah. Neither was this all; Jerusalem again felt the force of his
horrid cruelty and profaneness; men, women, and children, were
either slain or taken captive; and the houses and city walls were
destroyed. The Jews were not allowed to offer burnt offerings or
sacrifices to the God of Israel—circumcision was forbidden—they
were required to profane the Sabbath, and eat the flesh of swine,
and other beasts forbidden by their law[135]
—the sanctuary dedicated
to Jehovah was called the temple of Jupiter Olympius, and his image
set up on the altar—idol temples and altars were erected throughout
all their cities—and the Holy Scriptures destroyed whenever they
were met with—and death was the fate of those who read the word
of the Lord. The most horrid and brutal cruelties were inflicted on
such as chose to obey God, rather than this Syrian monster.
Jerusalem was overspread by his abominations; desolation was
indeed poured out "upon the desolate" when Antiochus Epiphanes
held the blood stained sceptre, emblem of satanic power. Yet, closely
as these circumstances resemble the description given by the](https://image.slidesharecdn.com/11052267-250525204806-55299cc9/85/Number-Theory-An-Introduction-To-Mathematics-2nd-Edition-Coppel-85-320.jpg)
![prophet's vision, we cannot think it is the event alluded to in this
prophecy. Daniel, in the three preceding verses, speaks of the
Messiah, and the final destruction of the city and sanctuary: by
Antiochus the temple certainly was not destroyed. In the eleventh
chapter there appears a striking prophecy of the events which
happened in Jerusalem during the dominion of the Syrian tyrant, but
we cannot think he is alluded to in any part of the ninth chapter. The
first clause of this verse, "He shall confirm the covenant with many,"
cannot refer to Antiochus, but alludes to the same glorious person
mentioned in the preceding verses. The latter part of this verse may
with propriety be considered as a continuance of the prophecy of
Jerusalem's final destruction, as it occurred under Titus. To Jesus the
Messiah we direct our eyes. The one week, or the midst of the week,
(seven years half expired,) alludes to the time of his Public Ministry,
which was three years and a half; during which period he declared,
the design of his mission was to confirm the well-ordered covenant
of redemption and peace, which was drawn up in the counsels of
eternity—sealed on earth with the blood of the Incarnate God—
signed in the presence of Jehovah, angels, men, and devils—
registered in the court of Heaven—and proclaimed good and valid by
the resurrection of Jesus from the dead, and the outpouring of the
Holy Spirit.[136]
It is true, the sacrifices and oblations of the temple
service did not cease immediately on the death of Christ, they were
continued some little time after that event; but they became
unnecessary, they had lost their value, and were but idle ceremonies
and useless rights, when the thing signified was accomplished. At
best, they were only types of the Lamb of God, the blood of that one
great sacrifice, which alone "cleanseth from all sin." "It is not
possible for the blood of bulls or goats to take away sin." No, the
sacrifices and ceremonies of the Mosaic economy were only
efficacious so far as Christ, the substance, was viewed through the
shadow.[137]
In less than forty years after the death of Christ, the
sacrifices and oblations ceased, for the temple was demolished. A
spot so deeply stained with crime, needed the fire of divine
vengeance to consume it from the face of the earth: it was erected](https://image.slidesharecdn.com/11052267-250525204806-55299cc9/85/Number-Theory-An-Introduction-To-Mathematics-2nd-Edition-Coppel-86-320.jpg)
![CHAPTER LXVI.
For I will gather all nations against Jerusalem to battle; and the
city shall be taken, and the houses rifled, and the women
ravished; and half of the city shall go forth into captivity, and
the residue of the people shall not be cut off from the city.—
Zechariah xiv. 2.
Imperial Rome, to whom the world once bowed, and whose power
could command armies from "all nations," had conquered Judea, and
received from her the yearly tribute of her subjection:[138]
but,
through the oppression of the Roman governors, and the madness
of the people, the standard of revolt was planted, and the Jews
attempted to break their yoke of bondage. The Roman legions,
inured to war, and accustomed to the shout of victory, hastened to
subdue the rebellious Israelites: they passed from city to city, and
from province to province; slaughter and death marked their course;
the strife was desperate; the conflict bloody; the Jews fought like
men determined to conquer or to die: two hundred and forty-seven
thousand seven hundred were slain before their provinces were
subjugated, and an immense number made prisoners: amongst
whom was Josephus, the historian of the war, who was governor of
the two Galilees, and who defended them with skill and bravery. The
Romans, having conquered the provinces, approached to assault
Jerusalem, which was then a dreadful scene. The sound of war was
heard through all her gates; regardless of the approaching foe, the
Jews had turned their arms against each other; three several
factions were busily engaged in the work of slaughter and
destruction. Eleazar and the Zealots seized the temple; John of
Gischala and his followers occupied its out-works; and Simon, the
son of Gorias, possessed the whole of the lower, and a great part of
the upper, town. Jerusalem was built on two hills; the highest, on](https://image.slidesharecdn.com/11052267-250525204806-55299cc9/85/Number-Theory-An-Introduction-To-Mathematics-2nd-Edition-Coppel-89-320.jpg)
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- 7. W.A. Coppel Number Theory An Introduction to Mathematics Second Edition
- 8. W.A. Coppel 3 Jansz Crescent 2603 Griffith Australia ISBN 978-0-387-89485-0 e-ISBN 978-0-387-89486-7 DOI 10.1007/978-0-387-89486-7 All rights reserved. or dissimilar methodology now known or hereafter developed is forbidden. to proprietary rights. Springer Dordrecht Heidelberg London New York Springer Science+ Business Media, LLC 2009 Printed on acid-free paper This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+ Business Media, LLC, 233 Spring Street, New York, NY The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection Springer is part of Springer Science+Business Media (www.springer.com) with any form of information storage and retrieval, electronic adaptation, computer software, or by similar Library of Congress Control Number: 2009931687 Mathematics Subject Classification (2000): 11-xx, 05B20, 33E05 c ∞ Editorial board: Sheldon Axler, San Francisco State University Vincenzo Capasso, Università degli Studi di Milano Carles Casacuberta, Universitat de Barcelona Angus MacIntyre, Queen Mary, University of London Kenneth Ribet, University of California, Berkeley Claude Sabbah, CNRS, École Polytechnique Endre Süli, University of Oxford Wojbor Woyczyński, Case Western Reserve University
- 9. For Jonathan, Nicholas, Philip and Stephen
- 10. Contents Preface to the Second Edition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi Part A I The Expanding Universe of Numbers. . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 0 Sets, Relations and Mappings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1 Natural Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2 Integers and Rational Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 3 Real Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 4 Metric Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 5 Complex Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 6 Quaternions and Octonions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 7 Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 8 Rings and Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 9 Vector Spaces and Associative Algebras . . . . . . . . . . . . . . . . . . . . . . . . 64 10 Inner Product Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 11 Further Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 12 Selected References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 Additional References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 II Divisibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 1 Greatest Common Divisors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 2 The Bézout Identity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 3 Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 4 Euclidean Domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 5 Congruences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 6 Sums of Squares . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 7 Further Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 8 Selected References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 Additional References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
- 11. viii Contents III More on Divisibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 1 The Law of Quadratic Reciprocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 2 Quadratic Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 3 Multiplicative Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152 4 Linear Diophantine Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 5 Further Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174 6 Selected References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176 Additional References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178 IV Continued Fractions and Their Uses . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 1 The Continued Fraction Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 2 Diophantine Approximation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185 3 Periodic Continued Fractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191 4 Quadratic Diophantine Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195 5 The Modular Group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201 6 Non-Euclidean Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208 7 Complements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211 8 Further Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217 9 Selected References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220 Additional References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222 V Hadamard’s Determinant Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223 1 What is a Determinant? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223 2 Hadamard Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229 3 The Art of Weighing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233 4 Some Matrix Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237 5 Application to Hadamard’s Determinant Problem . . . . . . . . . . . . . . . . . 243 6 Designs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247 7 Groups and Codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251 8 Further Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256 9 Selected References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258 VI Hensel’s p-adic Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261 1 Valued Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261 2 Equivalence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265 3 Completions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268 4 Non-Archimedean Valued Fields. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273 5 Hensel’s Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277 6 Locally Compact Valued Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284 7 Further Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 290 8 Selected References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 290
- 12. Contents ix Part B VII The Arithmetic of Quadratic Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291 1 Quadratic Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291 2 The Hilbert Symbol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303 3 The Hasse–Minkowski Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312 4 Supplements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322 5 Further Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324 6 Selected References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325 VIII The Geometry of Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327 1 Minkowski’s Lattice Point Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327 2 Lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 330 3 Proof of the Lattice Point Theorem; Other Results . . . . . . . . . . . . . . . . 334 4 Voronoi Cells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342 5 Densest Packings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347 6 Mahler’s Compactness Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352 7 Further Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 357 8 Selected References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 360 Additional References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 362 IX The Number of Prime Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363 1 Finding the Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363 2 Chebyshev’s Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 367 3 Proof of the Prime Number Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 370 4 The Riemann Hypothesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377 5 Generalizations and Analogues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 384 6 Alternative Formulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 389 7 Some Further Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 392 8 Further Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 394 9 Selected References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395 Additional References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 398 X A Character Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 399 1 Primes in Arithmetic Progressions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 399 2 Characters of Finite Abelian Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . 400 3 Proof of the Prime Number Theorem for Arithmetic Progressions . . . 403 4 Representations of Arbitrary Finite Groups . . . . . . . . . . . . . . . . . . . . . . 410 5 Characters of Arbitrary Finite Groups . . . . . . . . . . . . . . . . . . . . . . . . . . 414 6 Induced Representations and Examples . . . . . . . . . . . . . . . . . . . . . . . . . 419 7 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 425 8 Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 432 9 Further Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 443 10 Selected References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 444
- 13. x Contents XI Uniform Distribution and Ergodic Theory . . . . . . . . . . . . . . . . . . . . . . . 447 1 Uniform Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 447 2 Discrepancy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 459 3 Birkhoff’s Ergodic Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 464 4 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 472 5 Recurrence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 483 6 Further Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 488 7 Selected References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 490 Additional Reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 492 XII Elliptic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 493 1 Elliptic Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 493 2 The Arithmetic-Geometric Mean . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 502 3 Elliptic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 509 4 Theta Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 517 5 Jacobian Elliptic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 525 6 The Modular Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 531 7 Further Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 536 8 Selected References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 539 XIII Connections with Number Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 541 1 Sums of Squares . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 541 2 Partitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 544 3 Cubic Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 549 4 Mordell’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 558 5 Further Results and Conjectures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 569 6 Some Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 575 7 Further Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 581 8 Selected References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 584 Additional References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 586 Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 587 Axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 591 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 592
- 14. Preface to the Second Edition Undergraduate courses in mathematics are commonly of two types. On the one hand there are courses in subjects, such as linear algebra or real analysis, with which it is considered that every student of mathematics should be acquainted. On the other hand there are courses given by lecturers in their own areas of specialization, which are intended to serve as a preparation for research. There are, I believe, several reasons why students need more than this. First, although the vast extent of mathematics today makes it impossible for any individual to have a deep knowledge of more than a small part, it is important to have some understanding and appreciation of the work of others. Indeed the sometimes surprising interrelationships and analogies between different branches of mathematics are both the basis for many of its applications and the stimulus for further develop- ment. Secondly, different branches of mathematics appeal in different ways and require different talents. It is unlikely that all students at one university will have the same interests and aptitudes as their lecturers. Rather, they will only discover what their own interests and aptitudes are by being exposed to a broader range. Thirdly, many students of mathematics will become, not professional mathematicians, but scientists, engineers or schoolteachers. It is useful for them to have a clear understanding of the nature and extent of mathematics, and it is in the interests of mathematicians that there should be a body of people in the community who have this understanding. The present book attempts to provide such an understanding of the nature and extent of mathematics. The connecting theme is the theory of numbers, at first sight one of the most abstruse and irrelevant branches of mathematics. Yet by exploring its many connections with other branches, we may obtain a broad picture. The topics chosen are not trivial and demand some effort on the part of the reader. As Euclid already said, there is no royal road. In general I have concentrated attention on those hard-won results which illuminate a wide area. If I am accused of picking the eyes out of some subjects, I have no defence except to say “But what beautiful eyes!” The book is divided into two parts. Part A, which deals with elementary number theory, should be accessible to a first-year undergraduate. To provide a foundation for subsequent work, Chapter I contains the definitions and basic properties of various mathematical structures. However, the reader may simply skim through this chapter
- 15. xii Preface and refer back to it later as required. Chapter V, on Hadamard’s determinant problem, shows that elementary number theory may have unexpected applications. Part B, which is more advanced, is intended to provide an undergraduate with some idea of the scope of mathematics today. The chapters in this part are largely indepen- dent, except that Chapter X depends on Chapter IX and Chapter XIII on Chapter XII. Although much of the content of the book is common to any introductory work on number theory, I wish to draw attention to the discussion here of quadratic fields and elliptic curves. These are quite special cases of algebraic number fields and alge- braic curves, and it may be asked why one should restrict attention to these special cases when the general cases are now well understood and may even be developed in parallel. My answers are as follows. First, to treat the general cases in full rigour requires a commitment of time which many will be unable to afford. Secondly, these special cases are those most commonly encountered and more constructive methods are available for them than for the general cases. There is yet another reason. Some- times in mathematics a generalization is so simple and far-reaching that the special case is more fully understood as an instance of the generalization. For the topics mentioned, however, the generalization is more complex and is, in my view, more fully understood as a development from the special case. At the end of each chapter of the book I have added a list of selected references, which will enable readers to travel further in their own chosen directions. Since the literature is voluminous, any such selection must be somewhat arbitrary, but I hope that mine may be found interesting and useful. The computer revolution has made possible calculations on a scale and with a speed undreamt of a century ago. One consequence has been a considerable increase in ‘experimental mathematics’—the search for patterns. This book, on the other hand, is devoted to ‘theoretical mathematics’—the explanation of patterns. I do not wish to conceal the fact that the former usually precedes the latter. Nor do I wish to conceal the fact that some of the results here have been proved by the greatest minds of the past only after years of labour, and that their proofs have later been improved and simplified by many other mathematicians. Once obtained, however, a good proof organizes and provides understanding for a mass of computational data. Often it also suggests further developments. The present book may indeed be viewed as a ‘treasury of proofs’. We concentrate attention on this aspect of mathematics, not only because it is a distinctive feature of the subject, but also because we consider its exposition is better suited to a book than to a blackboard or a computer screen. In keeping with this approach, the proofs themselves have been chosen with some care and I hope that a few may be of interest even to those who are no longer students. Proofs which depend on general principles have been given preference over proofs which offer no particular insight. Mathematics is a part of civilization and an achievement in which human beings may take some pride. It is not the possession of any one national, political or religious group and any attempt to make it so is ultimately destructive. At the present time there are strong pressures to make academic studies more ‘relevant’. At the same time, however, staff at some universities are assessed by ‘citation counts’ and people are paid for giving lectures on chaos, for example, that are demonstrably rubbish.
- 16. Preface xiii The theory of numbers provides ample evidence that topics pursued for their own intrinsic interest can later find significant applications. I do not contend that curiosity has been the only driving force. More mundane motives, such as ambition or the necessity of earning a living, have also played a role. It is also true that mathematics pursued for the sake of applications has been of benefit to subjects such as number theory; there is a two-way trade. However, it shows a dangerous ignorance of history and of human nature to promote utility at the expense of spirit. This book has its origin in a course of lectures which I gave at the Victoria University of Wellington, New Zealand, in 1975. The demands of my own research have hitherto prevented me from completing it, although I have continued to collect material. If it succeeds at all in conveying some idea of the power and beauty of math- ematics, the labour of writing it will have been well worthwhile. As with a previous book, I have to thank Helge Tverberg, who has read most of the manuscript and made many useful suggestions. The first Phalanger Press edition of this book appeared in 2002. A revised edition, which was reissued by Springer in 2006, contained a number of changes. I removed an error in the statement and proof of Proposition II.12 and filled a gap in the proof of Proposition III.12. The statements of the Weil conjectures in Chapter IX and of a result of Heath-Brown in Chapter X were modified, following comments by J.-P. Serre. I also corrected a few misprints, made many small expository changes and expanded the index. In the present edition I have made some more expository changes and have added a few references at the end of some chapters to take account of recent de- velopments. For more detailed information the Internet has the advantage over a book. The reader is referred to the American Mathematical Society’s MathSciNet (www.ams.org/mathscinet) and to The Number Theory Web maintained by Keith Matthews (www.maths.uq.edu.au/∼krm/). I am grateful to Springer for undertaking the commercial publication of my book and hope you will be also. Many of those who have contributed to the production of this new softcover edition are unknown to me, but among those who are I wish to thank especially Alicia de los Reyes and my sons Nicholas and Philip. W.A. Coppel May, 2009 Canberra, Australia
- 17. I The Expanding Universe of Numbers For many people, numbers must seem to be the essence of mathematics. Number theory, which is the subject of this book, is primarily concerned with the properties of one particular type of number, the ‘whole numbers’ or integers. However, there are many other types, such as complex numbers and p-adic numbers. Somewhat sur- prisingly, a knowledge of these other types turns out to be necessary for any deeper understanding of the integers. In this introductory chapter we describe several such types (but defer the study of p-adic numbers to Chapter VI). To embark on number theory proper the reader may proceed to Chapter II now and refer back to the present chapter, via the Index, only as occasion demands. When one studies the properties of various types of number, one becomes aware of formal similarities between different types. Instead of repeating the derivations of properties for each individual case, it is more economical – and sometimes actually clearer – to study their common algebraic structure. This algebraic structure may be shared by objects which one would not even consider as numbers. There is a pedagogic difficulty here. Usually a property is discovered in one context and only later is it realized that it has wider validity. It may be more digestible to prove a result in the context of number theory and then simply point out its wider range of validity. Since this is a book on number theory, and many properties were first discovered in this context, we feel free to adopt this approach. However, to make the statements of such generalizations intelligible, in the latter part of this chapter we describe several basic algebraic structures. We do not attempt to study these structures in depth, but restrict attention to the simplest properties which throw light on the work of later chapters. 0 Sets, Relations and Mappings The label ‘0’ given to this section may be interpreted to stand for ‘0ptional’. We collect here some definitions of a logical nature which have become part of the common lan- guage of mathematics. Those who are not already familiar with this language, and who are repelled by its abstraction, should consult this section only when the need arises. DOI: 10.1007/978-0-387-89486-7_1, © Springer Science + Business Media, LLC 2009 1 W.A. Coppel, Number Theory: An Introduction to Mathematics, Universitext,
- 18. 2 I The Expanding Universe of Numbers We will not formally define a set, but will simply say that it is a collection of objects, which are called its elements. We write a ∈ A if a is an element of the set A and a / ∈ A if it is not. A set may be specified by listing its elements. For example, A = {a, b, c} is the set whose elements are a, b, c. A set may also be specified by characterizing its elements. For example, A = {x ∈ R: x2 < 2} is the set of all real numbers x such that x2 < 2. If two sets A, B have precisely the same elements, we say that they are equal and write A = B. (If A and B are not equal, we write A ̸= B.) For example, {x ∈ R: x2 = 1} = {1, −1}. Just as it is convenient to admit 0 as a number, so it is convenient to admit the empty set ∅, which has no elements, as a set. If every element of a set A is also an element of a set B we say that A is a subset of B, or that A is included in B, or that B contains A, and we write A ⊆ B. We say that A is a proper subset of B, and write A ⊂ B, if A ⊆ B and A ̸= B. Thus ∅ ⊆ A for every set A and ∅ ⊂ A if A ̸= ∅. Set inclusion has the following obvious properties: (i) A ⊆ A; (ii) if A ⊆ B and B ⊆ A, then A = B; (iii) if A ⊆ B and B ⊆ C, then A ⊆ C. For any sets A, B, the set whose elements are the elements of A or B (or both) is called the union or ‘join’ of A and B and is denoted by A ∪ B: A ∪ B = {x : x ∈ A or x ∈ B}. The set whose elements are the common elements of A and B is called the intersection or ‘meet’ of A and B and is denoted by A ∩ B: A ∩ B = {x : x ∈ A and x ∈ B}. If A ∩ B = ∅, the sets A and B are said to be disjoint. A B A B A ∪ B A ∩ B Fig. 1. Union and Intersection.
- 19. 0 Sets, Relations and Mappings 3 It is easily seen that union and intersection have the following algebraic properties: A ∪ A = A, A ∩ A = A, A ∪ B = B ∪ A, A ∩ B = B ∩ A, (A ∪ B) ∪ C = A ∪ (B ∪ C), (A ∩ B) ∩ C = A ∩ (B ∩ C), (A ∪ B) ∩ C = (A ∩ C) ∪ (B ∩ C), (A ∩ B) ∪ C = (A ∪ C) ∩ (B ∪ C). Set inclusion could have been defined in terms of either union or intersection, since A ⊆ B is the same as A ∪ B = B and also the same as A ∩ B = A. For any sets A, B, the set of all elements of B which are not also elements of A is called the difference of B from A and is denoted by BA: BA = {x : x ∈ B and x / ∈ A}. It is easily seen that C(A ∪ B) = (CA) ∩ (CB), C(A ∩ B) = (CA) ∪ (CB). An important special case is where all sets under consideration are subsets of a given universal set X. For any A ⊆ X, we have ∅ ∪ A = A, ∅ ∩ A = ∅, X ∪ A = X, X ∩ A = A. The set XA is said to be the complement of A (in X) and may be denoted by Ac for fixed X. Evidently ∅c = X, Xc = ∅, A ∪ Ac = X, A ∩ Ac = ∅, (Ac )c = A. By taking C = X in the previous relations for differences, we obtain ‘De Morgan’s laws’: (A ∪ B)c = Ac ∩ Bc , (A ∩ B)c = Ac ∪ Bc . Since A ∩ B = (Ac ∪ Bc)c, set intersection can be defined in terms of unions and complements. Alternatively, since A ∪ B = (Ac ∩ Bc)c, set union can be defined in terms of intersections and complements. For any sets A, B, the set of all ordered pairs (a, b) with a ∈ A and b ∈ B is called the (Cartesian) product of A by B and is denoted by A × B. Similarly one can define the product of more than two sets. We mention only one special case. For any positive integer n, we write An instead of A × · · · × A for the set of all (ordered) n-tuples (a1, . . . , an) with aj ∈ A (1 ≤ j ≤ n). We call aj the j-th coordinate of the n-tuple. A binary relation on a set A is just a subset R of the product set A × A. For any a, b ∈ A, we write aRb if (a, b) ∈ R. A binary relation R on a set A is said to be
- 20. 4 I The Expanding Universe of Numbers reflexive if aRa for every a ∈ A; symmetric if bRa whenever aRb; transitive if aRc whenever aRb and bRc. It is said to be an equivalence relation if it is reflexive, symmetric and transitive. If R is an equivalence relation on a set A and a ∈ A, the equivalence class Ra of a is the set of all x ∈ A such that x Ra. Since R is reflexive, a ∈ Ra. Since R is symmetric, b ∈ Ra implies a ∈ Rb. Since R is transitive, b ∈ Ra implies Rb ⊆ Ra. It follows that, for all a, b ∈ A, either Ra = Rb or Ra ∩ Rb = ∅. A partition C of a set A is a collection of nonempty subsets of A such that each element of A is an element of exactly one of the subsets in C . Thus the distinct equivalence classes corresponding to a given equivalence relation on a set A form a partition of A. It is not difficult to see that, conversely, if C is a partition of A, then an equivalence relation R is defined on A by taking R to be the set of all (a, b) ∈ A × A for which a and b are elements of the same subset in the collection C . Let A and B be nonempty sets. A mapping f of A into B is a subset of A × B with the property that, for each a ∈ A, there is a unique b ∈ B such that (a, b) ∈ f . We write f (a) = b if (a, b) ∈ f , and say that b is the image of a under f or that b is the value of f at a. We express that f is a mapping of A into B by writing f : A → B and we put f (A) = {f (a): a ∈ A}. The term function is often used instead of ‘mapping’, especially when A and B are sets of real or complex numbers, and ‘mapping’ itself is often abbreviated to map. If f is a mapping of A into B, and if A′ is a nonempty subset of A, then the restriction of f to A′ is the set of all (a, b) ∈ f with a ∈ A′. The identity map iA of a nonempty set A into itself is the set of all ordered pairs (a, a) with a ∈ A. If f is a mapping of A into B, and g a mapping of B into C, then the composite mapping g ◦ f of A into C is the set of all ordered pairs (a, c), where c = g(b) and b = f (a). Composition of mappings is associative, i.e. if h is a mapping of C into D, then (h ◦ g) ◦ f = h ◦ (g ◦ f ). The identity map has the obvious properties f ◦ iA = f and iB ◦ f = f . Let A, B be nonempty sets and f : A → B a mapping of A into B. The mapping f is said to be ‘one-to-one’ or injective if, for each b ∈ B, there exists at most one a ∈ A such that (a, b) ∈ f . The mapping f is said to be ‘onto’ or surjective if, for each b ∈ B, there exists at least one a ∈ A such that (a, b) ∈ f . If f is both injective and surjective, then it is said to be bijective or a ‘one-to-one correspondence’. The nouns injection, surjection and bijection are also used instead of the corresponding adjectives. It is not difficult to see that f is injective if and only if there exists a mapping g : B → A such that g ◦ f = iA, and surjective if and only if there exists a mapping h : B → A such that f ◦ h = iB. Furthermore, if f is bijective, then g and h are
- 21. 1 Natural Numbers 5 unique and equal. Thus, for any bijective map f : A → B, there is a unique inverse map f −1 : B → A such that f −1 ◦ f = iA and f ◦ f −1 = iB. If f : A → B and g : B → C are both bijective maps, then g ◦ f : A → C is also bijective and (g ◦ f )−1 = f −1 ◦ g−1 . 1 Natural Numbers The natural numbers are the numbers usually denoted by 1, 2, 3, 4, 5, . . .. However, other notations are also used, e.g. for the chapters of this book. Although one notation may have considerable practical advantages over another, it is the properties of the natural numbers which are basic. The following system of axioms for the natural numbers was essentially given by Dedekind (1888), although it is usually attributed to Peano (1889): The natural numbers are the elements of a set N, with a distinguished element 1 (one) and map S : N → N, such that (N1) S is injective, i.e. if m, n ∈ N and m ̸= n, then S(m) ̸= S(n); (N2) 1 / ∈ S(N); (N3) if M ⊆ N, 1 ∈ M and S(M) ⊆ M, then M = N. The element S(n) of N is called the successor of n. The axioms are satisfied by {1, 2, 3, . . .} if we take S(n) to be the element immediately following the element n. It follows readily from the axioms that 1 is the only element of N which is not in S(N). For, if M = S(N) ∪ {1}, then M ⊆ N, 1 ∈ M and S(M) ⊆ M. Hence, by (N3), M = N. It also follows from the axioms that S(n) ̸= n for every n ∈ N. For let M be the set of all n ∈ N such that S(n) ̸= n. By (N2), 1 ∈ M. If n ∈ M and n′ = S(n) then, by (N1), S(n′) ̸= n′. Thus S(M) ⊆ M and hence, by (N3), M = N. The axioms (N1)–(N3) actually determine N up to ‘isomorphism’. We will deduce this as a corollary of the following general recursion theorem: Proposition 1 Given a set A, an element a1 of A and a map T : A → A, there exists exactly one map ϕ : N → A such that ϕ(1) = a1 and ϕ(S(n)) = T ϕ(n) for every n ∈ N. Proof We show first that there is at most one map with the required properties. Let ϕ1 and ϕ2 be two such maps, and let M be the set of all n ∈ N such that ϕ1(n) = ϕ2(n). Evidently 1 ∈ M. If n ∈ M, then also S(n) ∈ M, since ϕ1(S(n)) = Tϕ1(n) = Tϕ2(n) = ϕ2(S(n)). Hence, by (N3), M = N. That is, ϕ1 = ϕ2.
- 22. 6 I The Expanding Universe of Numbers We now show that there exists such a map ϕ. Let C be the collection of all subsets C of N × A such that (1, a1) ∈ C and such that if (n, a) ∈ C, then also (S(n), T (a)) ∈ C. The collection C is not empty, since it contains N × A. Moreover, since every set in C contains (1, a1), the intersection D of all sets C ∈ C is not empty. It is easily seen that actually D ∈ C . By its definition, however, no proper subset of D is in C . Let M be the set of all n ∈ N such that (n, a) ∈ D for exactly one a ∈ A and, for any n ∈ M, define ϕ(n) to be the unique a ∈ A such that (n, a) ∈ D. If M = N, then ϕ(1) = a1 and ϕ(S(n)) = T ϕ(n) for all n ∈ N. Thus we need only show that M = N. As usual, we do this by showing that 1 ∈ M and that n ∈ M implies S(n) ∈ M. We have (1, a1) ∈ D. Assume (1, a′) ∈ D for some a′ ̸= a1. If D′ = D{(1, a′)}, then (1, a1) ∈ D′. Moreover, if (n, a) ∈ D′ then (S(n), T (a)) ∈ D′, since (S(n), T (a)) ∈ D and (S(n), T (a)) ̸= (1, a′). Hence D′ ∈ C . But this is a contradiction, since D′ is a proper subset of D. We conclude that 1 ∈ M. Suppose now that n ∈ M and let a be the unique element of A such that (n, a) ∈ D. Then (S(n), T (a)) ∈ D, since D ∈ C . Assume that (S(n), a′′) ∈ D for some a′′ ̸= T (a) and put D′′ = D{(S(n), a′′)}. Then (S(n), T (a)) ∈ D′′ and (1, a1) ∈ D′′. For any (m, b) ∈ D′′ we have (S(m), T (b)) ∈ D. If (S(m), T (b)) = (S(n), a′′), then S(m) = S(n) and T(b) = a′′ ̸= T(a), which implies m = n and b ̸= a. Thus D contains both (n, b) and (n, a), which contradicts n ∈ M. Hence (S(m), T (b)) ̸= (S(n), a′′), and so (S(m), T (b)) ∈ D′′. But then D′′ ∈ C , which is also a contradic- tion, since D′′ is a proper subset of D. We conclude that S(n) ∈ M. ✷ Corollary 2 If the axioms (N1)–(N3) are also satisfied by a set N′ wth element 1′ and map S′ : N′ → N′, then there exists a bijective map ϕ of N onto N′ such that ϕ(1) = 1′ and ϕ(S(n)) = S′ ϕ(n) for every n ∈ N. Proof By taking A = N′, a1 = 1′ and T = S′ in Proposition 1, we see that there exists a unique map ϕ : N → N′ such that ϕ(1) = 1′ and ϕ(S(n)) = S′ ϕ(n) for every n ∈ N. By interchanging N and N′, we see also that there exists a unique map ψ : N′ → N such that ψ(1′) = 1 and ψ(S′ (n′ )) = Sψ(n′ ) for every n′ ∈ N′ . The composite map χ = ψ ◦ϕ of N into N has the properties χ(1) = 1 and χ(S(n)) = Sχ(n) for every n ∈ N. But, by Proposition 1 again, χ is uniquely determined by these properties. Hence ψ ◦ ϕ is the identity map on N, and similarly ϕ ◦ ψ is the identity map on N′. Consequently ϕ is a bijection. ✷ We can also use Proposition 1 to define addition and multiplication of natural num- bers. By Proposition 1, for each m ∈ N there exists a unique map sm : N → N such that sm(1) = S(m), sm(S(n)) = Ssm(n) for every n ∈ N.
- 23. 1 Natural Numbers 7 We define the sum of m and n to be m + n = sm(n). It is not difficult to deduce from this definition and the axioms (N1)–(N3) the usual rules for addition: for all a, b, c ∈ N, (A1) if a + c = b + c, then a = b; (cancellation law) (A2) a + b = b + a; (commutative law) (A3) (a + b) + c = a + (b + c). (associative law) By way of example, we prove the cancellation law. Let M be the set of all c ∈ N such that a + c = b + c only if a = b. Then 1 ∈ M, since sa(1) = sb(1) implies S(a) = S(b) and hence a = b. Suppose c ∈ M. If a + S(c) = b+ S(c), i.e. sa(S(c)) = sb(S(c)), then Ssa(c) = Ssb(c) and hence, by (N1), sa(c) = sb(c). Since c ∈ M, this implies a = b. Thus also S(c) ∈ M. Hence, by (N3), M = N. We now show that m + n ̸= n for all m, n ∈ N. For a given m ∈ N, let M be the set of all n ∈ N such that m + n ̸= n. Then 1 ∈ M since, by (N2), sm(1) = S(m) ̸= 1. If n ∈ M, then sm(n) ̸= n and hence, by (N1), sm(S(n)) = Ssm(n) ̸= S(n). Hence, by (N3), M = N. By Proposition 1 again, for each m ∈ N there exists a unique map pm : N → N such that pm(1) = m, pm(S(n)) = sm(pm(n)) for every n ∈ N. We define the product of m and n to be m · n = pm(n). From this definition and the axioms (N1)–(N3) we may similarly deduce the usual rules for multiplication: for all a, b, c ∈ N, (M1) if a · c = b · c, then a = b; (cancellation law) (M2) a · b = b · a; (commutative law) (M3) (a · b) · c = a · (b · c); (associative law) (M4) a · 1 = a. (identity element) Furthermore, addition and multiplication are connected by (AM1) a · (b + c) = (a · b) + (a · c). (distributive law) As customary, we will often omit the dot when writing products and we will give multiplication precedence over addition. With these conventions the distributive law becomes simply a(b + c) = ab + ac.
- 24. 8 I The Expanding Universe of Numbers We show next how a relation of order may be defined on the set N. For any m, n ∈ N, we say that m is less than n, and write m < n, if m + m′ = n for some m′ ∈ N. Evidently m < S(m) for every m ∈ N, since S(m) = m + 1. Also, if m < n, then either S(m) = n or S(m) < n. For suppose m + m′ = n. If m′ = 1, then S(m) = n. If m′ ̸= 1, then m′ = m′′ + 1 for some m′′ ∈ N and S(m) + m′′ = (m + 1) + m′′ = m + (1 + m′′ ) = m + m′ = n. Again, if n ̸= 1, then 1 < n, since the set consisting of 1 and all n ∈ N such that 1 < n contains 1 and contains S(n) if it contains n. It will now be shown that the relation ‘<’ induces a total order on N, which is compatible with both addition and multiplication: for all a, b, c ∈ N, (O1) if a < b and b < c, then a < c; (transitive law) (O2) one and only one of the following alternatives holds: a < b, a = b, b < a; (law of trichotomy) (O3) a + c < b + c if and only if a < b; (O4) ac < bc if and only if a < b. The relation (O1) follows directly from the associative law for addition. We now prove (O2). If a < b then, for some a′ ∈ N, b = a + a′ = a′ + a ̸= a. Together with (O1), this shows that at most one of the three alternatives in (O2) holds. For a given a ∈ N, let M be the set of all b ∈ N such that at least one of the three alternatives in (O2) holds. Then 1 ∈ M, since 1 < a if a ̸= 1. Suppose now that b ∈ M. If a = b, then a < S(b). If a < b, then again a < S(b), by (O1). If b < a, then either S(b) = a or S(b) < a. Hence also S(b) ∈ M. Consequently, by (N3), M = N. This completes the proof of (O2). It follows from the associative and commutative laws for addition that, if a < b, then a + c < b + c. On the other hand, by using also the cancellation law we see that if a + c < b + c, then a < b. It follows from the distributive law that, if a < b, then ac < bc. Finally, suppose ac < bc. Then a ̸= b and hence, by (O2), either a < b or b < a. Since b < a would imply bc < ac, by what we have just proved, we must actually have a < b. The law of trichotomy (O2) implies that, for given m, n ∈ N, the equation m + x = n has a solution x ∈ N only if m < n. As customary, we write a ≤ b to denote either a < b or a = b. Also, it is sometimes convenient to write b > a instead of a < b, and b ≥ a instead of a ≤ b. A subset M of N is said to have a least element m′ if m′ ∈ M and m′ ≤ m for every m ∈ M. The least element m′ is uniquely determined, if it exists, by (O2). By what we have already proved, 1 is the least element of N.
- 25. 1 Natural Numbers 9 Proposition 3 Any nonempty subset M of N has a least element. Proof Assume that some nonempty subset M of N does not have a least element. Then 1 / ∈ M, since 1 is the least element of N. Let L be the set of all l ∈ N such that l < m for every m ∈ M. Then L and M are disjoint and 1 ∈ L. If l ∈ L, then S(l) ≤ m for every m ∈ M. Since M does not have a least element, it follows that S(l) / ∈ M. Thus S(l) < m for every m ∈ M, and so S(l) ∈ L. Hence, by (N3), L = N. Since L ∩ M = ∅, this is a contradiction. ✷ The method of proof by induction is a direct consequence of the axioms defining N. Suppose that with each n ∈ N there is associated a proposition Pn. To show that Pn is true for every n ∈ N, we need only show that P1 is true and that Pn+1 is true if Pn is true. Proposition 3 provides an alternative approach. To show that Pn is true for every n ∈ N, we need only show that if Pm is false for some m, then Pl is false for some l < m. For then the set of all n ∈ N for which Pn is false has no least element and consequently is empty. For any n ∈ N, we denote by In the set of all m ∈ N such that m ≤ n. Thus I1 = {1} and S(n) / ∈ In. It is easily seen that IS(n) = In ∪ {S(n)}. Also, for any p ∈ IS(n), there exists a bijective map fp of In onto IS(n){p}. For, if p = S(n) we can take fp to be the identity map on In, and if p ∈ In we can take fp to be the map defined by fp(p) = S(n), fp(m) = m if m ∈ In{p}. Proposition 4 For any m, n ∈ N, if a map f : Im → In is injective and f (Im) ̸= In, then m < n. Proof The result certainly holds when m = 1, since I1 = {1}. Let M be the set of all m ∈ N for which the result holds. We need only show that if m ∈ M, then also S(m) ∈ M. Let f : IS(m) → In be an injective map such that f (IS(m)) ̸= In and choose p ∈ In f (IS(m)). The restriction g of f to Im is also injective and g(Im) ̸= In. Since m ∈ M, it follows that m < n. Assume S(m) = n. Then there exists a bijective map gp of IS(m){p} onto Im. The composite map h = gp ◦ f maps IS(m) into Im and is injective. Since m ∈ M, we must have h(Im) = Im. But, since h(S(m)) ∈ Im and h is injective, this is a contradiction. Hence S(m) < n and, since this holds for every f, S(m) ∈ M. ✷ Proposition 5 For any m, n ∈ N, if a map f : Im → In is not injective and f (Im) = In, then m > n. Proof The result holds vacuously when m = 1, since any map f : I1 → In is injec- tive. Let M be the set of all m ∈ N for which the result holds. We need only show that if m ∈ M, then also S(m) ∈ M.
- 26. 10 I The Expanding Universe of Numbers Let f : IS(m) → In be a map such that f (IS(m)) = In which is not injective. Then there exist p, q ∈ IS(m) with p ̸= q and f (p) = f (q). We may choose the notation so that q ∈ Im. If fp is a bijective map of Im onto IS(m){p}, then the composite map h = f ◦ fp maps Im onto In. If it is not injective then m > n, since m ∈ M, and hence also S(m) > n. If h is injective, then it is bijective and has a bijective inverse h−1 : In → Im. Since h−1(In) is a proper subset of IS(m), it follows from Proposition 4 that n < S(m). Hence S(m) ∈ M. ✷ Propositions 4 and 5 immediately imply Corollary 6 For any n ∈ N, a map f : In → In is injective if and only if it is surjec- tive. Corollary 7 If a map f : Im → In is bijective, then m = n. Proof By Proposition 4, m < S(n), i.e. m ≤ n. Replacing f by f −1, we obtain in the same way n ≤ m. Hence m = n. ✷ A set E is said to be finite if there exists a bijective map f : E → In for some n ∈ N. Then n is uniquely determined, by Corollary 7. We call it the cardinality of E and denote it by #(E). It is readily shown that if E is a finite set and F a proper subset of E, then F is also finite and #(F) < #(E). Again, if E and F are disjoint finite sets, then their union E ∪ F is also finite and #(E ∪ F) = #(E) + #(F). Furthermore, for any finite sets E and F, the product set E × F is also finite and #(E × F) = #(E) · #(F). Corollary 6 implies that, for any finite set E, a map f : E → E is injective if and only if it is surjective. This is a precise statement of the so-called pigeonhole principle. A set E is said to be countably infinite if there exists a bijective map f : E → N. Any countably infinite set may be bijectively mapped onto a proper subset F, since N is bijectively mapped onto a proper subset by the successor map S. Thus a map f : E → E of an infinite set E may be injective, but not surjective. It may also be surjective, but not injective; an example is the map f : N → N defined by f (1) = 1 and, for n ̸= 1, f (n) = m if S(m) = n. 2 Integers and Rational Numbers The concept of number will now be extended. The natural numbers 1, 2, 3, . . . suffice for counting purposes, but for bank balance purposes we require the larger set . . . , −2, −1, 0, 1, 2, . . . of integers. (From this point of view, −2 is not so ‘unnatural’.) An important reason for extending the concept of number is the greater freedom it gives us. In the realm of natural numbers the equation a + x = b has a solution if and only if b > a; in the extended realm of integers it will always have a solution. Rather than introduce a new set of axioms for the integers, we will define them in terms of natural numbers. Intuitively, an integer is the difference m − n of two natural numbers m, n, with addition and multiplication defined by (m − n) + (p − q) = (m + p) − (n + q), (m − n) · (p − q) = (mp + nq) − (mq + np).
- 27. 2 Integers and Rational Numbers 11 However, two other natural numbers m′, n′ may have the same difference as m, n, and anyway what does m − n mean if m < n? To make things precise, we proceed in the following way. Consider the set N × N of all ordered pairs of natural numbers. For any two such ordered pairs, (m, n) and (m′, n′), we write (m, n) ∼ (m′ , n′ ) if m + n′ = m′ + n. We will show that this is an equivalence relation. It follows at once from the definition that (m, n) ∼ (m, n) (reflexive law) and that (m, n) ∼ (m′, n′) implies (m′, n′) ∼ (m, n) (symmetric law). It remains to prove the transitive law: (m, n) ∼ (m′ , n′ ) and (m′ , n′ ) ∼ (m′′ , n′′ ) imply (m, n) ∼ (m′′ , n′′ ). This follows from the commutative, associative and cancellation laws for addition in N. For we have m + n′ = m′ + n, m′ + n′′ = m′′ + n′ , and hence (m + n′ ) + n′′ = (m′ + n) + n′′ = (m′ + n′′ ) + n = (m′′ + n′ ) + n. Thus (m + n′′ ) + n′ = (m′′ + n) + n′ , and so m + n′′ = m′′ + n. The equivalence class containing (1, 1) evidently consists of all pairs (m, n) with m = n. We define an integer to be an equivalence class of ordered pairs of natural numbers and, as is now customary, we denote the set of all integers by Z. Addition of integers is defined componentwise: (m, n) + (p, q) = (m + p, n + q). To justify this definition we must show that it does not depend on the choice of repre- sentatives within an equivalence class, i.e. that (m, n) ∼ (m′ , n′ ) and (p, q) ∼ (p′ , q′ ) imply (m + p, n + q) ∼ (m′ + p′ , n′ + q′ ). However, if m + n′ = m′ + n, p + q′ = p′ + q, then (m + p) + (n′ + q′ ) = (m + n′ ) + (p + q′ ) = (m′ + n) + (p′ + q) = (m′ + p′ ) + (n + q).
- 28. 12 I The Expanding Universe of Numbers It follows at once from the corresponding properties of natural numbers that, also in Z, addition satisfies the commutative law (A2) and the associative law (A3). Moreover, the equivalence class 0 (zero) containing (1,1) is an identity element for addition: (A4) a + 0 = a for every a. Furthermore, the equivalence class containing (n, m) is an additive inverse for the equivalence containing (m, n): (A5) for each a, there exists − a such that a + (−a) = 0. From these properties we can now obtain Proposition 8 For all a, b ∈ Z, the equation a + x = b has a unique solution x ∈ Z. Proof It is clear that x = (−a) + b is a solution. Moreover, this solution is unique, since if a + x = a + x′ then, by adding −a to both sides, we obtain x = x′. ✷ Proposition 8 shows that the cancellation law (A1) is a consequence of (A2)–(A5). It also immediately implies Corollary 9 For each a ∈ Z, 0 is the only element such that a+0 = a, −a is uniquely determined by a, and a = −(−a). As usual, we will henceforth write b − a instead of b + (−a). Multiplication of integers is defined by (m, n) · (p, q) = (mp + nq, mq + np). To justify this definition we must show that (m, n) ∼ (m′, n′) and (p, q) ∼ (p′, q′) imply (mp + nq, mq + np) ∼ (m′ p′ + n′ q′ , m′ q′ + n′ p′ ). From m + n′ = m′ + n, by multiplying by p and q we obtain mp + n′ p = m′ p + np, m′ q + nq = mq + n′ q, and from p + q′ = p′ + q, by multiplying by m′ and n′ we obtain m′ p + m′ q′ = m′ p′ + m′ q, n′ p′ + n′ q = n′ p + n′ q′ . Adding these four equations and cancelling the terms common to both sides, we get (mp + nq) + (m′ q′ + n′ p′ ) = (m′ p′ + n′ q′ ) + (mq + np), as required. It is easily verified that, also in Z, multiplication satisfies the commutative law (M2) and the associative law (M3). Moreover, the distributive law (AM1) holds and, if 1 is the equivalence class containing (1 + 1, 1), then (M4) also holds. (In prac- tice it does not cause confusion to denote identity elements of N and Z by the same symbol.)
- 29. 2 Integers and Rational Numbers 13 Proposition 10 For every a ∈ Z, a · 0 = 0. Proof We have a · 0 = a · (0 + 0) = a · 0 + a · 0. Adding −(a · 0) to both sides, we obtain the result. ✷ Proposition 10 could also have been derived directly from the definitions, but we prefer to view it as a consequence of the properties which have been labelled. Corollary 11 For all a, b ∈ Z, a(−b) = −(ab), (−a)(−b) = ab. Proof The first relation follows from ab + a(−b) = a · 0 = 0, and the second relation follows from the first, since c = −(−c). ✷ By the definitions of 0 and 1 we also have (AM2) 1 ̸= 0. (In fact 1 = 0 would imply a = 0 for every a, since a · 1 = a and a · 0 = 0.) We will say that an integer a is positive if it is represented by an ordered pair (m, n) with n < m. This definition does not depend on the choice of representative. For if n < m and m + n′ = m′ + n, then m + n′ < m′ + m and hence n′ < m′. We will denote by P the set of all positive integers. The law of trichotomy (O2) for natural numbers immediately implies (P1) for every a, one and only one of the following alternatives holds: a ∈ P, a = 0, −a ∈ P. We say that an integer is negative if it has the form −a, where a ∈ P, and we denote by −P the set of all negative integers. Since a = −(−a), (P1) says that Z is the disjoint union of the sets P, {0} and −P. From the property (O3) of natural numbers we immediately obtain (P2) if a ∈ P and b ∈ P, then a + b ∈ P. Furthermore, we have (P3) if a ∈ P and b ∈ P, then a · b ∈ P. To prove this we need only show that if m, n, p, q are natural numbers such that n < m and q < p, then mq + np < mp + nq. Since q < p, there exists a natural number q′ such that q+q′ = p. But then nq′ < mq′, since n < m, and hence mq + np = (m + n)q + nq′ < (m + n)q + mq′ = mp + nq.
- 30. 14 I The Expanding Universe of Numbers We may write (P2) and (P3) symbolically in the form P + P ⊆ P, P · P ⊆ P. We now show that there are no divisors of zero in Z: Proposition 12 If a ̸= 0 and b ̸= 0, then ab ̸= 0. Proof By (P1), either a or −a is positive, and either b or −b is positive. If a ∈ P and b ∈ P then ab ∈ P, by (P3), and hence ab ̸= 0, by (P1). If a ∈ P and −b ∈ P, then a(−b) ∈ P. Hence ab = −(a(−b)) ∈ −P and ab ̸= 0. Similarly if −a ∈ P and b ∈ P. Finally, if −a ∈ P and −b ∈ P, then ab = (−a)(−b) ∈ P and again ab ̸= 0. ✷ The proof of Proposition 12 also shows that any nonzero square is positive: Proposition 13 If a ̸= 0, then a2 := aa ∈ P. It follows that 1 ∈ P, since 1 ̸= 0 and 12 = 1. The set P of positive integers induces an order relation in Z. Write a < b if b − a ∈ P, so that a ∈ P if and only if 0 < a. From this definition and the properties of P it follows that the order properties (O1)–(O3) hold also in Z, and that (O4) holds in the modified form: (O4)′ if 0 < c, then ac < bc if and only if a < b. We now show that we can represent any a ∈ Z in the form a = b − c, where b, c ∈ P. In fact, if a = 0, we can take b = 1 and c = 1; if a ∈ P, we can take b = a + 1 and c = 1; and if −a ∈ P, we can take b = 1 and c = 1 − a. An element a of Z is said to be a lower bound for a subset X of Z if a ≤ x for every x ∈ X. Proposition 3 immediately implies that if a subset of Z has a lower bound, then it has a least element. For any n ∈ N, let n′ be the integer represented by (n + 1, 1). Then n′ ∈ P. We are going to study the map n → n′ of N into P. The map is injective, since n′ = m′ implies n = m. It is also surjective, since if a ∈ P is represented by (m, n), where n < m, then it is also represented by (p + 1, 1), where p ∈ N satisfies n + p = m. It is easily verified that the map preserves sums and products: (m + n)′ = m′ + n′ , (mn)′ = m′ n′ . Since 1′ = 1, it follows that S(n)′ = n′ + 1. Furthermore, we have m′ < n′ if and only if m < n. Thus the map n → n′ establishes an ‘isomorphism’ of N with P. In other words, P is a copy of N situated within Z. By identifying n with n′, we may regard N itself as a subset of Z (and stop talking about P). Then ‘natural num- ber’ is the same as ‘positive integer’ and any integer is the difference of two natural numbers. Number theory, in its most basic form, is the study of the properties of the set Z of integers. It will be considered in some detail in later chapters of this book, but to relieve the abstraction of the preceding discussion we consider here the division algorithm:
- 31. 2 Integers and Rational Numbers 15 Proposition 14 For any integers a, b with a > 0, there exist unique integers q,r such that b = qa + r, 0 ≤ r < a. Proof We consider first uniqueness. Suppose qa + r = q′ a + r′ , 0 ≤ r,r′ < a. If r < r′, then from (q − q′ )a = r′ − r, we obtain first q > q′ and then r′ − r ≥ a, which is a contradiction. If r′ < r, we obtain a contradiction similarly. Hence r = r′, which implies q = q′. We consider next existence. Let S be the set of all integers y ≥ 0 which can be represented in the form y = b − xa for some x ∈ Z. The set S is not empty, since it contains b − 0 if b ≥ 0 and b − ba if b < 0. Hence S contains a least element r. Then b = qa + r, where q,r ∈ Z and r ≥ 0. Since r − a = b − (q + 1)a and r is the least element in S, we must also have r < a. ✷ The concept of number will now be further extended to include ‘fractions’ or ‘rational numbers’. For measuring lengths the integers do not suffice, since the length of a given segment may not be an exact multiple of the chosen unit of length. Similarly for measuring weights, if we find that three identical coins balance five of the chosen unit weights, then we ascribe to each coin the weight 5/3. In the realm of integers the equation ax = b frequently has no solution; in the extended realm of rational numbers it will always have a solution if a ̸= 0. Intuitively, a rational number is the ratio or ‘quotient’ a/b of two integers a, b, where b ̸= 0, with addition and multiplication defined by a/b + c/d = (ad + cb)/bd, a/b · c/d = ac/bd. However, two other integers a′, b′ may have the same ratio as a, b, and anyway what does a/b mean? To make things precise, we proceed in much the same way as before. Put Z× = Z{0} and consider the set Z×Z× of all ordered pairs (a, b) with a ∈ Z and b ∈ Z×. For any two such ordered pairs, (a, b) and (a′, b′), we write (a, b) ∼ (a′ , b′ ) if ab′ = a′ b. To show that this is an equivalence relation it is again enough to verify that (a, b) ∼ (a′, b′) and (a′, b′) ∼ (a′′, b′′) imply (a, b) ∼ (a′′, b′′). The same calculation as before, with addition replaced by multiplication, shows that (ab′′)b′ = (a′′b)b′. Since b′ ̸= 0, it follows that ab′′ = a′′b. The equivalence class containing (0, 1) evidently consists of all pairs (0, b) with b ̸= 0, and the equivalence class containing (1, 1) consists of all pairs (b, b) with b ̸= 0. We define a rational number to be an equivalence class of elements of Z×Z× and, as is now customary, we denote the set of all rational numbers by Q.
- 32. 16 I The Expanding Universe of Numbers Addition of rational numbers is defined by (a, b) + (c, d) = (ad + cb, bd), where bd ̸= 0 since b ̸= 0 and d ̸= 0. To justify the definition we must show that (a, b) ∼ (a′ , b′ ) and (c, d) ∼ (c′ , d′ ) imply (ad + cb, bd) ∼ (a′ d′ + c′ b′ , b′ d′ ). But if ab′ = a′b and cd′ = c′d, then (ad + cb)(b′ d′ ) = (ab′ )(dd′ ) + (cd′ )(bb′ ) = (a′ b)(dd′ ) + (c′ d)(bb′ ) = (a′ d′ + c′ b′ )(bd). It is easily verified that, also in Q, addition satisfies the commutative law (A2) and the associative law (A3). Moreover (A4) and (A5) also hold, the equivalence class 0 containing (0, 1) being an identity element for addition and the equivalence class containing (−b, c) being the additive inverse of the equivalence class containing (b, c). Multiplication of rational numbers is defined componentwise: (a, b) · (c, d) = (ac, bd). To justify the definition we must show that (a, b) ∼ (a′ , b′ ) and (c, d) ∼ (c′ , d′ ) imply (ac, bd) ∼ (a′ c′ , b′ d′ ). But if ab′ = a′b and cd′ = c′d, then (ac)(b′ d′ ) = (ab′ )(cd′ ) = (a′ b)(c′ d) = (a′ c′ )(bd). It is easily verified that, also in Q, multiplication satisfies the commutative law (M2) and the associative law (M3). Moreover (M4) also holds, the equivalence class 1 containing (1, 1) being an identity element for multiplication. Furthermore, addition and multiplication are connected by the distributive law (AM1), and (AM2) also holds since (0, 1) is not equivalent to (1, 1). Unlike the situation for Z, however, every nonzero element of Q has a multiplica- tive inverse: (M5) for each a ̸= 0, there exists a−1 such that aa−1 = 1. In fact, if a is represented by (b, c), then a−1 is represented by (c, b). It follows that, for all a, b ∈ Q with a ̸= 0, the equation ax = b has a unique solution x ∈ Q, namely x = a−1b. Hence, if a ̸= 0, then 1 is the only solution of ax = a, a−1 is uniquely determined by a, and a = (a−1)−1. We will say that a rational number a is positive if it is represented by an ordered pair (b, c) of integers for which bc > 0. This definition does not depend on the choice of representative. For suppose 0 < bc and bc′ = b′c. Then bc′ ̸= 0, since b ̸= 0 and c′ ̸= 0, and hence 0 < (bc′)2. Since (bc′)2 = (bc)(b′c′) and 0 < bc, it follows that 0 < b′c′. Our previous use of P having been abandoned in favour of N, we will now denote by P the set of all positive rational numbers and by −P the set of all rational numbers
- 33. 3 Real Numbers 17 −a, where a ∈ P. From the corresponding result for Z, it follows that (P1) continues to hold in Q. We will show that (P2) and (P3) also hold. To see that the sum of two positive rational numbers is again positive, we observe that if a, b, c, d are integers such that 0 < ab and 0 < cd, then also 0 < (ab)d2 + (cd)b2 = (ad + cb)(bd). To see that the product of two positive rational numbers is again positive, we observe that if a, b, c, d are integers such that 0 < ab and 0 < cd, then also 0 < (ab)(cd) = (ac)(bd). Since (P1)–(P3) all hold, it follows as before that Propositions 12 and 13 also hold in Q. Hence 1 ∈ P and (O4)′ now implies that a−1 ∈ P if a ∈ P. If a, b ∈ P and a < b, then b−1 < a−1, since bb−1 = 1 = aa−1 < ba−1. The set P of positive elements now induces an order relation on Q. We write a < b if b − a ∈ P, so that a ∈ P if and only if 0 < a. Then the order relations (O1)–(O3) and (O4)′ continue to hold in Q. Unlike the situation for Z, however, the ordering of Q is dense, i.e. if a, b ∈ Q and a < b, then there exists c ∈ Q such that a < c < b. For example, we can take c to be the solution of (1 + 1)c = a + b. Let Z′ denote the set of all rational numbers a′ which can be represented by (a, 1) for some a ∈ Z. For every c ∈ Q, there exist a′, b′ ∈ Z′ with b′ ̸= 0 such that c = a′b′−1. In fact, if c is represented by (a, b), we can take a′ to be represented by (a, 1) and b′ by (b, 1). Instead of c = a′b′−1, we also write c = a′/b′. For any a ∈ Z, let a′ be the rational number represented by (a, 1). The map a → a′ of Z into Z′ is clearly bijective. Moreover, it preserves sums and products: (a + b)′ = a′ + b′ , (ab)′ = a′ b′ . Furthermore, a′ < b′ if and only if a < b. Thus the map a → a′ establishes an ‘isomorphism’ of Z with Z′, and Z′ is a copy of Z situated within Q. By identifying a with a′, we may regard Z itself as a subset of Q. Then any rational number is the ratio of two integers. By way of illustration, we show that if a and b are positive rational numbers, then there exists a positive integer l such that la > b. For if a = m/n and b = p/q, where m, n, p, q are positive integers, then (np + 1)a > pm ≥ p ≥ b. 3 Real Numbers It was discovered by the ancient Greeks that even rational numbers do not suffice for the measurement of lengths. If x is the length of the hypotenuse of a right-angled tri- angle whose other two sides have unit length then, by Pythagoras’ theorem, x2 = 2.
- 34. 18 I The Expanding Universe of Numbers But it was proved, probably by a disciple of Pythagoras, that there is no rational number x such that x2 = 2. (A more general result is proved in Book X, Propo- sition 9 of Euclid’s Elements.) We give here a somewhat different proof from the classical one. Assume that such a rational number x exists. Since x may be replaced by −x, we may suppose that x = m/n, where m, n ∈ N. Then m2 = 2n2. Among all pairs m, n of positive integers with this property, there exists one for which n is least. If we put p = 2n − m, q = m − n, then p and q are positive integers, since clearly n < m < 2n. But p2 = 4n2 − 4mn + m2 = 2(m2 − 2mn + n2 ) = 2q2 . Since q < n, this contradicts the minimality of n. If we think of the rational numbers as measuring distances of points on a line from a given origin O on the line (with distances on one side of O positive and distances on the other side negative), this means that, even though a dense set of points is obtained in this way, not all points of the line are accounted for. In order to fill in the gaps the concept of number will now be extended from ‘rational number’ to ‘real number’. It is possible to define real numbers as infinite decimal expansions, the rational numbers being those whose decimal expansions are eventually periodic. However, the choice of base 10 is arbitrary and carrying through this approach is awkward. There are two other commonly used approaches, one based on order and the other on distance. The first was proposed by Dedekind (1872), the second by Méray (1869) and Cantor (1872). We will follow Dedekind’s approach, since it is conceptually sim- pler. However, the second method is also important and in a sense more general. In Chapter VI we will use it to extend the rational numbers to the p-adic numbers. It is convenient to carry out Dedekind’s construction in two stages. We will first define ‘cuts’ (which are just the positive real numbers), and then pass from cuts to arbitrary real numbers in the same way that we passed from the natural numbers to the integers. Intuitively, a cut is the set of all rational numbers which represent points of the line between the origin O and some other point. More formally, we define a cut to be a nonempty proper subset A of the set P of all positive rational numbers such that (i) if a ∈ A, b ∈ P and b < a, then b ∈ A; (ii) if a ∈ A, then there exists a′ ∈ A such that a < a′. For example, the set I of all positive rational numbers a < 1 is a cut. Similarly, the set T of all positive rational numbers a such that a2 < 2 is a cut. We will denote the set of all cuts by P. For any A, B ∈ P we write A < B if A is a proper subset of B. We will show that this induces a total order on P. It is clear that if A < B and B < C, then A < C. It remains to show that, for any A, B ∈ P, one and only one of the following alternatives holds: A < B, A = B, B < A.
- 35. 3 Real Numbers 19 It is obvious from the definition by set inclusion that at most one holds. Now suppose that neither A < B nor A = B. Then there exists a ∈ AB. It follows from (i), applied to B, that every b ∈ B satisfies b < a and then from (i), applied to A, that b ∈ A. Thus B < A. Let S be any nonempty collection of cuts. A cut B is said to be an upper bound for S if A ≤ B for every A ∈ S , and a lower bound for S if B ≤ A for every A ∈ S . An upper bound for S is said to be a least upper bound or supremum for S if it is a lower bound for the collection of all upper bounds. Similarly, a lower bound for S is said to be a greatest lower bound or infimum for S if it is an upper bound for the collection of all lower bounds. Clearly, S has at most one supremum and at most one infimum. The set P has the following basic property: (P4) if a nonempty subset S has an upper bound, then it has a least upper bound. Proof Let C be the union of all sets A ∈ S . By hypothesis there exists a cut B such that A ⊆ B for every A ∈ S . Since C ⊆ B for any such B, and A ⊆ C for every A ∈ S , we need only show that C is a cut. Evidently C is a nonempty proper subset of P, since B ̸= P. Suppose c ∈ C. Then c ∈ A for some A ∈ S . If d ∈ P and d < c, then d ∈ A, since A is a cut. Furthermore c < a′ for some a′ ∈ A. Since A ⊆ C, this proves that C is a cut. ✷ In the set P of positive rational numbers, the subset T of all x ∈ P such that x2 < 2 has an upper bound, but no least upper bound. Thus (P4) shows that there is a difference between the total order on P and that on P. We now define addition of cuts. For any A, B ∈ P, let A + B denote the set of all rational numbers a + b, with a ∈ A and b ∈ B. We will show that also A + B ∈ P. Evidently A + B is a nonempty subset of P. It is also a proper subset. For choose c ∈ PA and d ∈ PB. Then, by (i), a < c for all a ∈ A and b < d for all b ∈ B. Since a + b < c + d for all a ∈ A and b ∈ B, it follows that c + d / ∈ A + B. Suppose now that a ∈ A, b ∈ B and that c ∈ P satisfies c < a + b. If c > b, then c = b + d for some d ∈ P, and d < a. Hence, by (i), d ∈ A and c = d + b ∈ A + B. Similarly, c ∈ A + B if c > a. Finally, if c ≤ a and c ≤ b, choose e ∈ P so that e < c. Then e ∈ A and c = e + f for some f ∈ P. Then f ∈ B, since f < c, and c = e + f ∈ A + B. Thus A + B has the property (i). It is trivial that A + B also has the property (ii), since if a ∈ A and b ∈ B, there exists a′ ∈ A such that a < a′ and then a +b < a′ +b. This completes the proof that A + B is a cut. It follows at once from the corresponding properties of rational numbers that addi- tion of cuts satisfies the commutative law (A2) and the associative law (A3). We consider next the connection between addition and order. Lemma 15 For any cut A and any c ∈ P, there exists a ∈ A such that a + c / ∈ A. Proof If c / ∈ A, then a + c / ∈ A for every a ∈ A, since c < a + c. Thus we may suppose c ∈ A. Choose b ∈ PA. For some positive integer n we have b < nc and hence nc / ∈ A. If n is the least positive integer such that nc / ∈ A, then n > 1 and (n − 1)c ∈ A. Consequently we can take a = (n − 1)c. ✷
- 36. 20 I The Expanding Universe of Numbers Proposition 16 For any cuts A, B, there exists a cut C such that A + C = B if and only if A < B. Proof We prove the necessity of the condition by showing that A < A + C for any cuts A, C. If a ∈ A and c ∈ C, then a < a + c. Since A + C is a cut, it follows that a ∈ A + C. Consequently A ≤ A + C, and Lemma 15 implies that A ̸= A + C. Suppose now that A and B are cuts such that A < B, and let C be the set of all c ∈ P such that c + d ∈ B for some d ∈ PA. We are going to show that C is a cut and that A + C = B. The set C is not empty. For choose b ∈ BA and then b′ ∈ B with b < b′. Then b′ = b + c′ for some c′ ∈ P, which implies c′ ∈ C. On the other hand, C ≤ B, since c + d ∈ B and d ∈ P imply c ∈ B. Thus C is a proper subset of P. Suppose c ∈ C, p ∈ P and p < c. We have c + d ∈ B for some d ∈ PA and c = p + e for some e ∈ P. Since d + e ∈ PA and p + (d + e) = c + d ∈ B, it follows that p ∈ C. Suppose now that c ∈ C, so that c + d ∈ B for some d ∈ PA. Choose b ∈ B so that c + d < b. Then b = c + d + e for some e ∈ P. If we put c′ = c + e, then c < c′. Moreover c′ ∈ C, since c′ + d = b. This completes the proof that C is a cut. Suppose a ∈ A and c ∈ C. Then c + d ∈ B for some d ∈ PA. Hence a < d. It follows that a + c < c + d, and so a + c ∈ B. Thus A + C ≤ B. It remains to show that B ≤ A+C. Pick any b ∈ B. If b ∈ A, then also b ∈ A+C, since A < A + C. Thus we now assume b / ∈ A. Choose b′ ∈ B with b < b′. Then b′ = b + d for some d ∈ P. By Lemma 15, there exists a ∈ A such that a + d / ∈ A. Moreover a < b, since b / ∈ A, and hence b = a + c for some c ∈ P. Since c + (a + d) = b + d = b′, it follows that c ∈ C. Thus b ∈ A + C and B ≤ A + C. ✷ We can now show that addition of cuts satisfies the order relation (O3). Suppose first that A < B. Then, by Proposition 16, there exists a cut D such that A + D = B. Hence, for any cut C, A + C < (A + C) + D = B + C. Suppose next that A + C < B + C. Then A ̸= B. Since B < A would imply B + C < A+C, by what we have just proved, it follows from the law of trichotomy that A < B. From (O3) and the law of trichotomy, it follows that addition of cuts satisfies the cancellation law (A1). We next define multiplication of cuts. For any A, B ∈ P, let AB denote the set of all rational numbers ab, with a ∈ A and b ∈ B. In the same way as for A + B, it may be shown that AB ∈ P. We note only that if a ∈ A, b ∈ B and c < ab, then b−1c < a. Hence b−1c ∈ A and c = (b−1c)b ∈ AB. It follows from the corresponding properties of rational numbers that multiplication of cuts satisfies the commutative law (M2) and the associative law (M3). Moreover (M4) holds, the identity element for multiplication being the cut I consisting of all positive rational numbers less than 1. We now show that the distributive law (AM1) also holds. The distributive law for rational numbers shows at once that A(B + C) ≤ AB + AC.
- 37. 3 Real Numbers 21 It remains to show that a1b + a2c ∈ A(B + C) if a1, a2 ∈ A, b ∈ B and c ∈ C. But a1b + a2c ≤ a2(b + c) if a1 ≤ a2, and a1b + a2c ≤ a1(b + c) if a2 ≤ a1. In either event it follows that a1b + a2c ∈ A(B + C). We can now show that multiplication of cuts satisfies the order relation (O4). If A < B, then there exists a cut D such that A + D = B and hence AC < AC + DC = BC. Conversely, suppose AC < BC. Then A ̸= B. Since B < A would imply BC < AC, it follows that A < B. From (O4) and the law of trichotomy (O2) it follows that multiplication of cuts satisfies the cancellation law (M1). We next prove the existence of multiplicative inverses. The proof will use the fol- lowing multiplicative analogue of Lemma 15: Lemma 17 For any cut A and any c ∈ P with c > 1, there exists a ∈ A such that ac / ∈ A. Proof Choose any b ∈ A. We may suppose bc ∈ A, since otherwise we can take a = b. Since b < bc, we have bc = b + d for some d ∈ P. By Lemma 15 we can choose a ∈ A so that a + d / ∈ A. Since b + d ∈ A, it follows that b + d < a + d, and so b < a. Hence ab−1 > 1 and a + d < a + (ab−1 )d = ab−1 (b + d) = ac. Since a + d / ∈ A, it follows that ac / ∈ A. ✷ Proposition 18 For any A ∈ P, there exists A−1 ∈ P such that AA−1 = I. Proof Let A−1 be the set of all b ∈ P such that b < c−1 for some c ∈ PA. It is easily verified that A−1 is a cut. We note only that a−1 / ∈ A−1 if a ∈ A and that, if b < c−1, then also b < d−1 for some d > c. We now show that AA−1 = I. If a ∈ A and b ∈ A−1 then ab < 1, since a ≥ b−1 would imply a > c for some c ∈ PA. Thus AA−1 ≤ I. On the other hand, if 0 < d < 1 then, by Lemma 17, there exists a ∈ A such that ad−1 / ∈ A. Choose a′ ∈ A so that a < a′, and put b = (a′)−1d. Then b < a−1d. Since a−1d = (ad−1)−1, it follows that b ∈ A−1 and consequently d = a′b ∈ AA−1. Thus I ≤ AA−1. ✷ For any positive rational number a, the set Aa consisting of all positive rational numbers c such that c < a is a cut. The map a → Aa of P into P is injective and preserves sums and products: Aa+b = Aa + Ab, Aab = Aa Ab. Moreover, Aa < Ab if and only if a < b. By identifying a with Aa we may regard P as a subset of P. It is a proper subset, since (P4) does not hold in P.
- 38. 22 I The Expanding Universe of Numbers This completes the first stage of Dedekind’s construction. In the second stage we pass from cuts to real numbers. Intuitively, a real number is the difference of two cuts. We will deal with the second stage rather briefly since, as has been said, it is completely analogous to the passage from the natural numbers to the integers. On the set P ×P of all ordered pairs of cuts an equivalence relation is defined by (A, B) ∼ (A′ , B′ ) if A + B′ = A′ + B. We define a real number to be an equivalence class of ordered pairs of cuts and, as is now customary, we denote the set of all real numbers by R. Addition and multiplication are unambiguously defined by (A, B) + (C, D) = (A + C, B + D), (A, B) · (C, D) = (AC + BD, AD + BC). They obey the laws (A2)–(A5), (M2)–(M5) and (AM1)–(AM2). A real number represented by (A, B) is said to be positive if B < A. If we denote by P′ the set of all positive real numbers, then (P1)–(P3) continue to hold with P′ in place of P. An order relation, satisfying (O1)–(O3), is induced on R by writing a < b if b − a ∈ P′. Moreover, any a ∈ R may be written in the form a = b − c, where b, c ∈ P′. It is easily seen that P is isomorphic with P′. By identifying P with P′, we may regard both P and Q as subsets of R. An element of RQ is said to be an irrational real number. Upper and lower bounds, and suprema and infima, may be defined for subsets of R in the same way as for subsets of P. Moreover, the least upper bound property (P4) continues to hold in R. By applying (P4) to the subset −S = {−a : a ∈ S } we see that if a nonempty subset S of R has a lower bound, then it has a greatest lower bound. The least upper bound property implies the so-called Archimedean property: Proposition 19 For any positive real numbers a, b, there exists a positive integer n such that na > b. Proof Assume, on the contrary, that na ≤ b for every n ∈ N. Then b is an upper bound for the set {na : n ∈ N}. Let c be a least upper bound for this set. From na ≤ c for every n ∈ N we obtain (n + 1)a ≤ c for every n ∈ N. But this implies na ≤ c − a for every n ∈ N. Since c − a < c and c is a least upper bound, we have a contradiction. ✷ Proposition 20 For any real numbers a, b with a < b, there exists a rational number c such that a < c < b. Proof Suppose first that a ≥ 0. By Proposition 19 there exists a positive integer n such that n(b − a) > 1. Then b > a + n−1. There exists also a positive integer m such that mn−1 > a. If m is the least such positive integer, then (m − 1)n−1 ≤ a and hence mn−1 ≤ a + n−1 < b. Thus we can take c = mn−1. If a < 0 and b > 0 we can take c = 0. If a < 0 and b ≤ 0, then −b < d < −a for some rational d and we can take c = −d. ✷ Proposition 21 For any positive real number a, there exists a unique positive real number b such that b2 = a.
- 39. 3 Real Numbers 23 Proof Let S be the set of all positive real numbers x such that x2 ≤ a. The set S is not empty, since it contains a if a ≤ 1 and 1 if a > 1. If y > 0 and y2 > a, then y is an upper bound for S. In particular, 1 + a is an upper bound for S. Let b be the least upper bound for S. Then b2 = a, since b2 < a would imply (b + 1/n)2 < a for sufficiently large n > 0 and b2 > a would imply (b − 1/n)2 > a for sufficiently large n > 0. Finally, if c2 = a and c > 0, then c = b, since (c − b)(c + b) = c2 − b2 = 0. ✷ The unique positive real number b in the statement of Proposition 21 is said to be a square root of a and is denoted by √ a or a1/2. In the same way it may be shown that, for any positive real number a and any positive integer n, there exists a unique positive real number b such that bn = a, where bn = b · · · b (n times). We say that b is an n-th root of a and write b = n √ a or a1/n. A set is said to be a field if two binary operations, addition and multiplication, are defined on it with the properties (A2)–(A5), (M2)–(M5) and (AM1)–(AM2). A field is said to be ordered if it contains a subset P of ‘positive’ elements with the properties (P1)–(P3). An ordered field is said to be complete if, with the order induced by P, it has the property (P4). Propositions 19–21 hold in any complete ordered field, since only the above prop- erties were used in their proofs. By construction, the set R of all real numbers is a complete ordered field. In fact, any complete ordered field F is isomorphic to R, i.e. there exists a bijective map ϕ : F → R such that, for all a, b ∈ F, ϕ(a + b) = ϕ(a) + ϕ(b), ϕ(ab) = ϕ(a)ϕ(b), and ϕ(a) > 0 if and only if a ∈ P. We sketch the proof. Let e be the identity element for multiplication in F and, for any positive integer n, let ne = e + · · · + e (n summands). Since F is ordered, ne is positive and so has a multiplicative inverse. For any rational number m/n, where m, n ∈ Z and n > 0, write (m/n)e = m(ne)−1 if m > 0, = −(−m)(ne)−1 if m < 0, and = 0 if m = 0. The elements (m/n)e form a subfield of F isomorphic to Q and we define ϕ((m/n)e) = m/n. For any a ∈ F, we define ϕ(a) to be the least upper bound of all rational numbers m/n such that (m/n)e ≤ a. One verifies first that the map ϕ : F → R is bijective and that ϕ(a) < ϕ(b) if and only if a < b. One then deduces that ϕ preserves sums and products. Actually, any bijective map ϕ : F → R which preserves sums and products is also order-preserving. For, by Proposition 21, b > a if and only if b − a = c2 for some c ̸= 0, and then ϕ(b) − ϕ(a) = ϕ(b − a) = ϕ(c2 ) = ϕ(c)2 > 0. Those whose primary interest lies in real analysis may define R to be a complete ordered field and omit the tour through N, Z, Q and P. That is, one takes as axioms the 14 properties above which define a complete ordered field and simply assumes that they are consistent.
- 40. 24 I The Expanding Universe of Numbers The notion of convergence can be defined in any totally ordered set. A sequence {an} is said to converge, with limit l, if for any l′,l′′ such that l′ < l < l′′, there exists a positive integer N = N(l′,l′′) such that l′ < an < l′′ for every n ≥ N. The limit l of the convergent sequence {an} is clearly uniquely determined; we write lim n→∞ an = l, or an → l as n → ∞. It is easily seen that any convergent sequence is bounded, i.e. it has an upper bound and a lower bound. A trivial example of a convergentsequence is the constant sequence {an}, where an = a for every n; its limit is again a. In the set R of real numbers, or in any totally ordered set in which each bounded sequence has a least upper bound and a greatest lower bound, the definition of conver- gence can be reformulated. For, let {an} be a bounded sequence. Then, for any positive integer m, the subsequence {an}n≥m has a greatest lower bound bm and a least upper bound cm: bm = inf n≥m an, cm = sup n≥m an. The sequences {bm}m≥1 and {cm}m≥1 are also bounded and, for any positive integer m, bm ≤ bm+1 ≤ cm+1 ≤ cm. If we define the lower limit and upper limit of the sequence {an} by lim n→∞ an := sup m≥1 bm, lim n→∞ an := inf m≥1 cm, then limn→∞an ≤ limn→∞an, and it is readily shown that limn→∞ an = l if and only if lim n→∞ an = l = lim n→∞ an. A sequence {an} is said to be nondecreasing if an ≤ an+1 for every n and nonin- creasing if an+1 ≤ an for every n. It is said to be monotonic if it is either nondecreasing or nonincreasing. Proposition 22 Any bounded monotonic sequence of real numbers is convergent. Proof Let {an} be a bounded monotonic sequence and suppose, for definiteness, that it is nondecreasing: a1 ≤ a2 ≤ a3 ≤ · · · . In this case, in the notation used above we have bm = am and cm = c1 for every m. Hence lim n→∞ an = sup m≥1 am = c1 = lim n→∞ an. ✷ Proposition 22 may be applied to the centuries-old algorithm for calculating square roots, which is commonly used today in pocket calculators. Take any real number a > 1 and put x1 = (1 + a)/2.
- 41. 3 Real Numbers 25 Then x1 > 1 and x2 1 > a, since (a − 1)2 > 0. Define the sequence {xn} recursively by xn+1 = (xn + a/xn)/2 (n ≥ 1). It is easily verified that if xn > 1 and x2 n > a, then xn+1 > 1, x2 n+1 > a and xn+1 < xn. Since the inequalities hold for n = 1, it follows that they hold for all n. Thus the sequence {xn} is nonincreasing and bounded, and therefore convergent. If xn → b, then a/xn → a/b and xn+1 → b. Hence b = (b+a/b)/2, which simplifies to b2 = a. We consider now sequences of real numbers which are not necessarily monotonic. Lemma 23 Any sequence {an} of real numbers has a monotonic subsequence. Proof Let M be the set of all positive integers m such that am ≥ an for every n > m. If M contains infinitely many positive integers m1 < m2 < · · · , then {amk } is a nonincreasing subsequence of {an}. If M is empty or finite, there is a positive integer n1 such that no positive integer n ≥ n1 is in M. Then an2 > an1 for some n2 > n1, an3 > an2 for some n3 > n2, and so on. Thus {ank } is a nondecreasing subsequence of {an}. ✷ It is clear from the proof that Lemma 23 also holds for sequences of elements of any totally ordered set. In the case of R, however, it follows at once from Lemma 23 and Proposition 22 that Proposition 24 Any bounded sequence of real numbers has a convergent subse- quence. Proposition 24 is often called the Bolzano–Weierstrass theorem. It was stated by Bolzano (c. 1830) in work which remained unpublished until a century later. It became generally known through the lectures of Weierstrass (c. 1874). A sequence {an} of real numbers is said to be a fundamental sequence, or ‘Cauchy sequence’, if for each ε > 0 there exists a positive integer N = N(ε) such that −ε < ap − aq < ε for all p, q ≥ N. Any fundamental sequence {an} is bounded, since any finite set is bounded and aN − ε < ap < aN + ε for p ≥ N. Also, any convergent sequence is a fundamental sequence. For suppose an → l as n → ∞. Then, for any ε > 0, there exists a positive integer N such that l − ε/2 < an < l + ε/2 for every n ≥ N. It follows that −ε < ap − aq < ε for p ≥ q ≥ N. The definitions of convergent sequence and fundamental sequence, and the preced- ing result that ‘convergent’ implies ‘fundamental’, hold also for sequences of rational numbers, and even for sequences with elements from any ordered field. However, for sequences of real numbers there is a converse result:
- 42. 26 I The Expanding Universe of Numbers Proposition 25 Any fundamental sequence of real numbers is convergent. Proof If {an} is a fundamental sequence of real numbers, then {an} is bounded and, for any ε > 0, there exists a positive integer m = m(ε) such that −ε/2 < ap − aq < ε/2 for all p, q ≥ m. But, by Proposition 24, the sequence {an} has a convergent subsequence {ank }. If l is the limit of this subsequence, then there exists a positive integer N ≥ m such that l − ε/2 < ank < l + ε/2 for nk ≥ N. It follows that l − ε < an < l + ε for n ≥ N. Thus the sequence {an} converges with limit l. ✷ Proposition 25 was known to Bolzano (1817) and was clearly stated in the influ- ential Cours d’analyse of Cauchy (1821). However, a rigorous proof was impossible until the real numbers themselves had been precisely defined. The Méray–Cantor method of constructing the real numbers from the rationals is based on Proposition 25. We define two fundamental sequences {an} and {a′ n} of rational numbers to be equivalent if an − a′ n → 0 as n → ∞. This is indeed an equivalence relation, and we define a real number to be an equivalence class of fundamental sequences. The set of all real numbers acquires the structure of a field if addition and multiplication are defined by {an} + {bn} = {an + bn}, {an} · {bn} = {anbn}. It acquires the structure of a complete ordered field if the fundamental sequence {an} is said to be positive when it has a positive lower bound. The field Q of rational numbers may be regarded as a subfield of the field thus constructed by identifying the ratio- nal number a with the equivalence class containing the constant sequence {an}, where an = a for every n. It is not difficult to show that an ordered field is complete if every bounded monotonic sequence is convergent, or if every bounded sequence has a convergent subsequence. In this sense, Propositions 22 and 24 state equivalent forms for the least upper bound property. This is not true, however, for Proposition 25. An ordered field need not have the least upper bound property, even though every fundamental sequence is convergent. It is true, however, that an ordered field has the least upper bound property if and only if it has the Archimedean property (Proposition 19) and every fundamental sequence is convergent. In a course of real analysis one would now define continuity and prove those properties of continuous functions which, in the 18th century, were assumed as ‘geometrically obvious’. For example, for given a, b ∈ R with a < b, let I = [a, b] be the interval consisting of all x ∈ R such that a ≤ x ≤ b. If f : I → R is continuous, then it attains its supremum, i.e. there exists c ∈ I such that f (x) ≤ f (c) for every x ∈ I. Also, if f (a) f (b) < 0, then f (d) = 0 for some d ∈ I (the intermediate- value theorem). Real analysis is not our primary concern, however, and we do not feel obliged to establish even those properties which we may later use.
- 43. 4 Metric Spaces 27 4 Metric Spaces The notion of convergence is meaningful not only for points on a line, but also for points in space, where there is no natural relation of order. We now reformulate our previous definition, so as to make it more generally applicable. The absolute value |a| of a real number a is defined by |a| = a if a ≥ 0, |a| = −a if a < 0. It is easily seen that absolute values have the following properties: |0| = 0, |a| > 0 if a ̸= 0; |a| = | − a|; |a + b| ≤ |a| + |b|. The first two properties follow at once from the definition. To prove the third, we ob- serve first that a + b ≤ |a| + |b|, since a ≤ |a| and b ≤ |b|. Replacing a by −a and b by −b, we obtain also −(a + b) ≤ |a| + |b|. But |a + b| is either a + b or −(a + b). The distance between two real numbers a and b is defined to be the real number d(a, b) = |a − b|. From the preceding properties of absolute values we obtain their counterparts for dis- tances: (D1) d(a, a) = 0, d(a, b) > 0 if a ̸= b; (D2) d(a, b) = d(b, a); (D3) d(a, b) ≤ d(a, c) + d(c, b). The third property is known as the triangle inequality, since it may be interpreted as saying that, in any triangle, the length of one side does not exceed the sum of the lengths of the other two. Fréchet (1906) recognized these three properties as the essential characteristics of any measure of distance and introduced the following general concept. A set E is a metric space if with each ordered pair (a, b) of elements of E there is associated a real number d(a, b), so that the properties (D1)–(D3) hold for all a, b, c ∈ E. We note first some simple consequences of these properties. For all a, b, a′, b′ ∈ E we have |d(a, b) − d(a′ , b′ )| ≤ d(a, a′ ) + d(b, b′ ) (∗) since, by (D2) and (D3), d(a, b) ≤ d(a, a′ ) + d(a′ , b′ ) + d(b, b′ ), d(a′ , b′ ) ≤ d(a, a′ ) + d(a, b) + d(b, b′ ). Taking b = b′ in (∗), we obtain from (D1), |d(a, b) − d(a′ , b)| ≤ d(a, a′ ). (∗∗)
- 44. 28 I The Expanding Universe of Numbers In any metric space there is a natural topology. A subset G of a metric space E is open if for each x ∈ G there is a positive real number δ = δ(x) such that G also contains the whole open ball βδ(x) = {y ∈ E : d(x, y) < δ}. A set F ⊆ E is closed if its complement EF is open. For any set A ⊆ E, its closure Ā is the intersection of all closed sets containing it, and its interior int A is the union of all open sets contained in it. A subset F of E is connected if it is not contained in the union of two open subsets of E whose intersections with F are disjoint and nonempty. A subset F of E is (se- quentially) compact if every sequence of elements of F has a subsequence converging to an element of F (and locally compact if this holds for every bounded sequence of elements of F). A map f : X → Y from one metric space X to another metric space Y is contin- uous if, for each open subset G of Y, the set of all x ∈ X such that f (x) ∈ G is an open subset of X. The two properties stated at the end of §3 admit far-reaching gen- eralizations for continuous maps between subsets of metric spaces, namely that under a continuous map the image of a compact set is again compact, and the image of a connected set is again connected. There are many examples of metric spaces: (i) Let E = Rn be the set of all n-tuples a = (α1, . . . , αn) of real numbers and define d(b, c) = |b − c|, where b − c = (β1 − γ1, . . . , βn − γn) if b = (β1, . . . , βn) and c = (γ1, . . . , γn), and |a| = max 1≤ j≤n |αj |. Alternatively, one can replace the norm |a| by either |a|1 = n ! j=1 |αj | or |a|2 = " n ! j=1 |αj |2 #1/2 . In the latter case, d(b, c) is the Euclidean distance between b and c. The triangle in- equality in this case follows from the Cauchy–Schwarz inequality: for any real numbers βj , γj ( j = 1, . . . , n) " n ! j=1 βj γj #2 ≤ " n ! j=1 β2 j #" n ! j=1 γ 2 j # . (ii) Let E = Fn 2 be the set of all n-tuples a = (α1, . . . , αn), where αj = 0 or 1 for each j, and define the Hamming distance d(b, c) between b = (β1, . . . , βn) and c = (γ1, . . . , γn) to be the number of j such that βj ̸= γj . This metric space plays a basic role in the theory of error-correcting codes.
- 45. 4 Metric Spaces 29 (iii) Let E = C (I) be the set of all continuous functions f : I → R, where I = [a, b] = {x ∈ R: a ≤ x ≤ b} is an interval of R, and define d(g, h) = |g − h|, where | f | = sup a≤x≤b | f (x)|. (A well-known property of continuous functions ensures that f is bounded on I.) Alternatively, one can replace the norm | f | by either | f |1 = $ b a | f (x)|dx or | f |2 = "$ b a | f (x)|2 dx #1/2 . (iv) Let E = C (R) be the set of all continuous functions f : R → R and define d(g, h) = ! N≥1 dN (g, h)/2N [1 + dN (g, h)], where dN (g, h) = sup|x|≤N |g(x) − h(x)|. The triangle inequality (D3) follows from the inequality |α + β|/[1 + |α + β|] ≤ |α|/[1 + |α|] + |β|/[1 + |β|] for arbitrary real numbers α, β. The metric here has the property that d( fn, f ) → 0 if and only if fn(x) → f (x) uniformly on every bounded subinterval of R. It may be noted that, even though E is a vector space, the metric is not derived from a norm since, if λ ∈ R, one may have d(λg, λh) ̸= |λ|d(g, h). (v) Let E be the set of all measurable functions f : I → R, where I = [a, b] is an interval of R, and define d(g, h) = $ b a |g(x) − h(x)|(1 + |g(x) − h(x)|)−1 dx. In order to obtain (D1), we identify functions which take the same value at all points of I, except for a set of measure zero. Convergence with respect to this metric coincides with convergence in measure, which plays a role in the theory of probability. (vi) Let E = F∞ 2 be the set of all infinite sequences a = (α1, α2, . . .), where αj = 0 or 1 for every j, and define d(a, a) = 0, d(a, b) = 2−k if a ̸= b, where b = (β1, β2, . . .) and k is the least positive integer such that αk ̸= βk.
- 46. 30 I The Expanding Universe of Numbers Here the triangle inequality holds in the stronger form d(a, b) ≤ max[d(a, c), d(c, b)]. This metric space plays a basic role in the theory of dynamical systems. (vii) A connected graph can be given the structure of a metric space by defining the dis- tance between two vertices to be the number of edges on the shortest path joining them. Let E be an arbitrary metric space and {an} a sequence of elements of E. The sequence {an} is said to converge, with limit a ∈ E, if d(an, a) → 0 as n → ∞, i.e. if for each real ε > 0 there is a corresponding positive integer N = N(ε) such that d(an, a) < ε for every n ≥ N. The limit a is uniquely determined, since if also d(an, a′) → 0, then d(a, a′ ) ≤ d(an, a) + d(an, a′ ), and the right side can be made arbitrarily small by taking n sufficiently large. We write lim n→∞ an = a, or an → a as n → ∞. If the sequence {an} has limit a, then so also does any (infinite) subsequence. If an → a and bn → b, then d(an, bn) → d(a, b), as one sees by taking a′ = an and b′ = bn in (∗). The sequence {an} is said to be a fundamental sequence, or ‘Cauchy sequence’, if for each real ε > 0 there is a corresponding positive integer N = N(ε) such that d(am, an) < ε for all m, n ≥ N. If {an} and {bn} are fundamental sequences then, by (∗), the sequence {d(an, bn)} of real numbers is a fundamental sequence, and therefore convergent. A set S ⊆ E is said to be bounded if the set of all real numbers d(a, b) with a, b ∈ S is a bounded subset of R. Any fundamental sequence {an} is bounded, since if d(am, an) < 1 for all m, n ≥ N, then d(am, an) < 1 + δ for all m, n ∈ N, where δ = max1≤ j<k≤N d(aj, ak). Furthermore, any convergent sequence {an} is a fundamental sequence, as one sees by taking a = limn→∞ an in the inequality d(am, an) ≤ d(am, a) + d(an, a). A metric space is said to be complete if, conversely, every fundamental sequence is convergent.
- 47. 4 Metric Spaces 31 By generalizing the Méray–Cantor method of extending the rational numbers to the real numbers, Hausdorff (1913) showed that any metric space can be embedded in a complete metric space. To state his result precisely, we introduce some definitions. A subset F of a metric space E is said to be dense in E if, for each a ∈ E and each real ε > 0, there exists some b ∈ F such that d(a, b) < ε. A map σ from one metric space E to another metric space E′ is necessarily injec- tive if it is distance-preserving, i.e. if d′ (σ(a), σ(b)) = d(a, b) for all a, b ∈ E. If the map σ is also surjective, then it is said to be an isometry and the metric spaces E and E′ are said to be isometric. A metric space Ē is said to be a completion of a metric space E if Ē is complete and E is isometric to a dense subset of Ē. It is easily seen that any two completions of a given metric space are isometric. Hausdorff’s result says that any metric space E has a completion Ē. We sketch the proof. Define two fundamental sequences {an} and {a′ n} in E to be equivalent if lim n→∞ d(an, a′ n) = 0. It is easily shown that this is indeed an equivalence relation. Moreover, if the funda- mental sequences {an}, {bn} are equivalent to the fundamental sequences {a′ n}, {b′ n} respectively, then lim n→∞ d(an, bn) = lim n→∞ d(a′ n, b′ n). We can give the set Ē of all equivalence classes of fundamental sequences the structure of a metric space by defining d̄({an}, {bn}) = lim n→∞ d(an, bn). For each a ∈ E, let ā be the equivalence class in Ē which contains the fundamental sequence {an} such that an = a for every n. Since d̄(ā, b̄) = d(a, b) for all a, b ∈ E, E is isometric to the set E′ = {ā : a ∈ E}. It is not difficult to show that E′ is dense in Ē and that Ē is complete. Which of the previous examples of metric spaces are complete? In example (i), the completeness of Rn with respect to the first definition of distance follows directly from the completeness of R. It is also complete with respect to the two alternative definitions of distance, since a sequence which converges with respect to one of the three metrics also converges with respect to the other two. Indeed it is easily shown that, for every a ∈ Rn, |a| ≤ |a|2 ≤ |a|1 and |a|1 ≤ n1/2 |a|2, |a|2 ≤ n1/2 |a|.
- 48. 32 I The Expanding Universe of Numbers In example (ii), the completeness of Fn 2 is trivial, since any fundamental sequence is ultimately constant. In example (iii), the completeness of C (I) with respect to the first definition of distance follows from the completeness of R and the fact that the limit of a uniformly convergent sequence of continuous functions is again a continuous function. However, C (I) is not complete with respect to either of the two alternative defini- tions of distance. It is possible also for a sequence to converge with respect to the two alternative definitions of distance, but not with respect to the first definition. Similarly, a sequence may converge in the first alternative metric, but not even be a fundamental sequence in the second. The completions of the metric space C (I) with respect to the two alternative met- rics may actually be identified with spaces of functions. The completion for the first alternative metric is the set L(I) of all Lebesgue measurable functions f : I → R such that $ b a | f (x)|dx < ∞, functions which take the same value at all points of I, except for a set of measure zero, being identified. The completion L2(I) for the second alternative metric is obtained by replacing % b a | f (x)|dx by % b a | f (x)|2dx in this statement. It may be shown that the metric spaces of examples (iv)–(vi) are all complete. In example (vi), the strong triangle inequality implies that {an} is a fundamental sequence if (and only if) d(an+1, an) → 0 as n → ∞. Let E be an arbitrary metric space and f : E → E a map of E into itself. A point x̄ ∈ E is said to be a fixed point of f if f (x̄) = x̄. A useful property of complete metric spaces is the following contraction principle, which was first established in the present generality by Banach (1922), but was previously known in more concrete situations. Proposition 26 Let E be a complete metric space and let f : E → E be a map of E into itself. If there exists a real number θ, with 0 < θ < 1, such that d( f (x′ ), f (x′′ )) ≤ θd(x′ , x′′ ) for all x′ , x′′ ∈ E, then the map f has a unique fixed point x̄ ∈ E. Proof It is clear that there is at most one fixed point, since 0 ≤ d(x′, x′′) ≤ θd(x′, x′′) implies x′ = x′′. To prove that a fixed point exists we use the method of successive approximations. Choose any x0 ∈ E and define the sequence {xn} recursively by xn = f (xn−1) (n ≥ 1). For any k ≥ 1 we have d(xk+1, xk) = d( f (xk), f (xk−1)) ≤ θd(xk, xk−1). Applying this k times, we obtain d(xk+1, xk) ≤ θk d(x1, x0).
- 49. 4 Metric Spaces 33 Consequently, if n > m ≥ 0, d(xn, xm) ≤ d(xn, xn−1) + d(xn−1, xn−2) + · · · + d(xm+1, xm) ≤ (θn−1 + θn−2 + · · · + θm )d(x1, x0) ≤ θm (1 − θ)−1 d(x1, x0), since 0 < θ < 1. It follows that {xn} is a fundamental sequence and so a convergent sequence, since E is complete. If x̄ = limn→∞ xn, then d( f (x̄), x̄) ≤ d( f (x̄), xn+1) + d(xn+1, x̄) ≤ θd(x̄, xn) + d(x̄, xn+1). Since the right side can be made less than any given positive real number by taking n large enough, we must have f (x̄) = x̄. The proof shows also that, for any m ≥ 0, d(x̄, xm) ≤ θm (1 − θ)−1 d(x1, x0). ✷ The contraction principle is surprisingly powerful, considering the simplicity of its proof. We give two significant applications: an inverse function theorem and an exis- tence theorem for ordinary differential equations. In both cases we will use the notion of differentiability for functions of several real variables. The unambitious reader may simply take n = 1 in the following discussion (so that ‘invertible’ means ‘nonzero’). Functions of several variables are important, however, and it is remarkable that the proper definition of differentiability in this case was first given by Stolz (1887). A map ϕ : U → Rm, where U ⊆ Rn is a neighbourhood of x0 ∈ Rn (i.e., U contains some open ball {x ∈ Rn : |x − x0| < ρ}), is said to be differentiable at x0 if there exists a linear map A: Rn → Rm such that |ϕ(x) − ϕ(x0) − A(x − x0)|/|x − x0| → 0 as |x − x0| → 0. (The inequalities between the various norms show that it is immaterial which norm is used.) The linear map A, which is then uniquely determined, is called the derivative of ϕ at x0 and will be denoted by ϕ′(x0). This definition is a natural generalization of the usual definition when m = n = 1, since it says that the difference ϕ(x0 + h) − ϕ(x0) admits the linear approximation Ah for |h| → 0. Evidently, if ϕ1 and ϕ2 are differentiable at x0, then so also is ϕ = ϕ1 + ϕ2 and ϕ′ (x0) = ϕ′ 1(x0) + ϕ′ 2(x0). It also follows directly from the definition that derivatives satisfy the chain rule: If ϕ : U → Rm, where U is a neighbourhood of x0 ∈ Rn, is differentiable at x0, and if ψ : V → Rl, where V is a neighbourhood of y0 = ϕ(x0) ∈ Rm, is differentiable at y0, then the composite map χ = ψ ◦ ϕ : U → Rl is differentiable at x0 and χ′ (x0) = ψ′ (y0)ϕ′ (x0), the right side being the composite linear map.
- 50. 34 I The Expanding Universe of Numbers We will also use the notion of norm of a linear map. If A: Rn → Rm is a linear map, its norm |A| is defined by |A| = sup |x|≤1 |Ax|. Evidently |A1 + A2| ≤ |A1| + |A2|. Furthermore, if B : Rm → Rl is another linear map, then |B A| ≤ |B||A|. Hence, if m = n and |A| < 1, then the linear map I − A is invertible, its inverse being given by the geometric series (I − A)−1 = I + A + A2 + · · · . It follows that for any invertible linear map A: Rn → Rn, if B : Rn → Rn is a lin- ear map such that |B − A| < |A−1|−1, then B is also invertible and |B−1 − A−1| → 0 as |B − A| → 0. If ϕ : U → Rm is differentiable at x0 ∈ Rn, then it is also continuous at x0, since |ϕ(x) − ϕ(x0)| ≤ |ϕ(x) − ϕ(x0) − ϕ′ (x0)(x − x0)| + |ϕ′ (x0)||x − x0|. We say that ϕ is continuously differentiable in U if it is differentiable at each point of U and if the derivative ϕ′(x) is a continuous function of x in U. The inverse function theorem says: Proposition 27 Let U0 be a neighbourhood of x0 ∈ Rn and let ϕ : U0 → Rn be a continuously differentiable map for which ϕ′(x0) is invertible. Then, for some δ > 0, the ball U = {x ∈ Rn : |x − x0| < δ} is contained in U0 and (i) the restriction of ϕ to U is injective; (ii) V := ϕ(U) is open, i.e. if η ∈ V , then V contains all y ∈ Rn near η; (iii) the inverse map ψ : V → U is also continuously differentiable and, if y = ϕ(x), then ψ′(y) is the inverse of ϕ′(x). Proof To simplify notation, assume x0 = ϕ(x0) = 0 and write A = ϕ′(0). For any y ∈ Rn, put fy(x) = x + A−1 [y − ϕ(x)]. Evidently x is a fixed point of fy if and only if ϕ(x) = y. The map fy is also contin- uously differentiable and f ′ y(x) = I − A−1 ϕ′ (x) = A−1 [A − ϕ′ (x)]. Since ϕ′(x) is continuous, we can choose δ > 0 so that the ball U = {x ∈ Rn : |x| < δ} is contained in U0 and | f ′ y(x)| ≤ 1/2 for x ∈ U.
- 51. 4 Metric Spaces 35 If x1, x2 ∈ U, then | fy(x2) − fy(x1)| = & & & & $ 1 0 f ′ ((1 − t)x1 + tx2)(x2 − x1)dt & & & & ≤ |x2 − x1|/2. It follows that fy has at most one fixed point in U. Since this holds for arbitrary y ∈ Rn, the restriction of ϕ to U is injective. Suppose next that η = ϕ(ξ) for some ξ ∈ U. We wish to show that, if y is near η, the map fy has a fixed point near ξ. Choose r = r(ξ) > 0 so that the closed ball Br = {x ∈ Rn : |x − ξ| ≤ r} is contained in U, and fix y ∈ Rn so that |y − η| < r/2|A−1|. Then | fy(ξ) − ξ| = |A−1 (y − η)| ≤ |A−1 ||y − η| < r/2. Hence if |x − ξ| ≤ r, then | fy(x) − ξ| ≤ | fy(x) − fy(ξ)| + | fy(ξ) − ξ| ≤ |x − ξ|/2 + r/2 ≤ r. Thus fy(Br) ⊆ Br. Also, if x1, x2 ∈ Br, then | fy(x2) − fy(x1)| ≤ |x2 − x1|/2. But Br is a complete metric space, with the same metric as Rn, since it is a closed subset (if xn ∈ Br and xn → x in Rn, then also x ∈ Br). Consequently, by the con- traction principle (Proposition 26), fy has a fixed point x ∈ Br. Then ϕ(x) = y, which proves (ii). Suppose now that y, η ∈ V . Then y = ϕ(x), η = ϕ(ξ) for unique x, ξ ∈ U. Since | fy(x) − fy(ξ)| ≤ |x − ξ|/2 and fy(x) − fy(ξ) = x − ξ − A−1 (y − η), we have |A−1 (y − η)| ≥ |x − ξ|/2. Thus |x − ξ| ≤ 2|A−1 ||y − η|. If F = ϕ′(ξ) and G = F−1, then ψ(y) − ψ(η) − G(y − η) = x − ξ − G(y − η) = −G[ϕ(x) − ϕ(ξ) − F(x − ξ)].
- 52. 36 I The Expanding Universe of Numbers Hence |ψ(y) − ψ(η) − G(y − η)|/|y − η| ≤ 2|A−1 ||G||ϕ(x) − ϕ(ξ) − F(x − ξ)|/|x − ξ|. If |y − η| → 0, then |x − ξ| → 0 and the right side tends to 0. Consequently ψ is differentiable at η and ψ′(η) = G = F−1. Thus ψ is differentiable in U and, a fortiori, continuous. In fact ψ is continuously differentiable, since F is a continuous function of ξ (by hypothesis), since ξ = ψ(η) is a continuous function of η, and since F−1 is a continuous function of F. ✷ To bring out the meaning of Proposition 27 we add some remarks: (i) The invertibility of ϕ′(x0) is necessary for the existence of a differentiable inverse map, but not for the existence of a continuous inverse map. For example, the contin- uously differentiable map ϕ : R → R defined by ϕ(x) = x3 is bijective and has the continuous inverse ψ(y) = y1/3, although ϕ′(0) = 0. (ii) The hypothesis that ϕ is continuously differentiable cannot be totally dispensed with. For example, the map ϕ : R → R defined by ϕ(x) = x + x2 sin(1/x) if x ̸= 0, ϕ(0) = 0, is everywhere differentiable and ϕ′(0) ̸= 0, but ϕ is not injective in any neighbourhood of 0. (iii) The inverse map may not be defined throughout U0. For example, the map ϕ : R2 → R2 defined by ϕ1(x1, x2) = x2 1 − x2 2, ϕ2(x1, x2) = 2x1x2, is everywhere continuously differentiable and has an invertible derivative at every point except the origin. Thus the hypotheses of Proposition 27 are satisfied in any connected open set U0 ⊆ R2 which does not contain the origin, and yet ϕ(1, 1) = ϕ(−1, −1). It was first shown by Cauchy (c. 1844) that, under quite general conditions, an ordinary differential equation has local solutions. The method of successive approxi- mations (i.e., the contraction principle) was used for this purpose by Picard (1890): Proposition 28 Let t0 ∈ R, ξ0 ∈ Rn and let U be a neighbourhood of (t0, ξ0) in R × Rn. If ϕ : U → Rn is a continuous map with a derivative ϕ′ with respect to x that is continuous in U, then the differential equation dx/dt = ϕ(t, x) (1) has a unique solution x(t) which satisfies the initial condition x(t0) = ξ0 (2) and is defined in some interval |t − t0| ≤ δ, where δ > 0.
- 53. 4 Metric Spaces 37 Proof If x(t) is a solution of the differential equation (1) which satisfies the initial condition (2), then by integration we get x(t0) = ξ0 + $ t t0 ϕ[τ, x(τ)]dτ. Conversely, if a continuous function x(t) satisfies this relation then, since ϕ is contin- uous, x(t) is actually differentiable and is a solution of (1) that satisfies (2). Hence we need only show that the map F defined by (F x)(t) = ξ0 + $ t t0 ϕ[τ, x(τ)]dτ has a unique fixed point in the space of continuous functions. There exist positive constants M, L such that |ϕ(t, ξ)| ≤ M, |ϕ′ (t, ξ)| ≤ L for all (t, ξ) in a neighbourhood of (t0, ξ0), which we may take to be U. If (t, ξ1) ∈ U and (t, ξ2) ∈ U, then |ϕ(t, ξ2) − ϕ(t, ξ1)| = & & & & $ 1 0 ϕ′ (t, (1 − u)ξ1 + uξ2)(ξ2 − ξ1)du & & & & ≤ L|ξ2 − ξ1|. Choose δ > 0 so that the box |t − t0| ≤ δ, |ξ − ξ0| ≤ Mδ is contained in U and also Lδ < 1. Take I = [t0 − δ, t0 + δ] and let C (I) be the complete metric space of all continuous functions x : I → Rn with the distance function d(x1, x2) = sup t∈I |x1(t) − x2(t)|. The constant function x0(t) = ξ0 is certainly in C (I). Let E be the subset of all x ∈ C (I) such that x(t0) = ξ0 and d(x, x0) ≤ Mδ. Evidently if xn ∈ E and xn → x in C (I), then x ∈ E. Hence E is also a complete metric space with the same metric. Moreover F(E) ⊆ E, since if x ∈ E then (F x)(t0) = ξ0 and, for all t ∈ I, |(F x)(t) − ξ0| = & & & & $ t t0 ϕ[τ, x(τ)]dτ & & & & ≤ Mδ. Furthermore, if x1, x2 ∈ E, then d(F x1, F x2) ≤ Lδd(x1, x2), since for all t ∈ I, |(F x1)(t) − (F x2)(t)| = & & & & $ t t0 {ϕ[τ, x1(τ)] − ϕ[τ, x2(τ)]}dτ & & & & ≤ Lδ d(x1, x2). Since Lδ < 1, the result now follows from Proposition 26. ✷
- 54. 38 I The Expanding Universe of Numbers Proposition 28 only guarantees the local existence of solutions, but this is in the nature of things. For example, if n = 1, the unique solution of the differential equation dx/dt = x2 such that x(t0) = ξ0 > 0 is given by x(t) = {1 − (t − t0)ξ0}−1 ξ0. Thus the solution is defined only for t < t0+ξ−1 0 , even though the differential equation itself has exemplary behaviour everywhere. To illustrate Proposition 28, take n = 1 and let E(t) be the solution of the (linear) differential equation dx/dt = x (3) which satisfies the initial condition E(0) = 1. Then E(t) is defined for |t| < R, for some R > 0. If |τ| < R/2 and x1(t) = E(t + τ), then x1(t) is the solution of the differential equation (3) which satisfies the initial condition x1(0) = E(τ). But x2(t) = E(τ)E(t) satisfies the same differential equation and the same initial condi- tion. Hence we must have x1(t) = x2(t) for |t| < R/2, i.e. E(t + τ) = E(t)E(τ). (4) In particular, E(t)E(−t) = 1, E(2t) = E(t)2 . The last relation may be used to extend the definition of E(t), so that it is continuously differentiable and a solution of (3) also for |t| < 2R. It follows that the solution E(t) is defined for all t ∈ R and satisfies the addition theorem (4) for all t, τ ∈ R. It is instructive to carry through the method of successive approximations explicitly in this case. If we take x0(t) to be the constant 1, then x1(t) = 1 + $ t 0 x0(τ)dτ = 1 + t, x2(t) = 1 + $ t 0 x1(τ)dτ = 1 + t + t2 /2, · · · . By induction we obtain, for every n ≥ 1, xn(t) = 1 + t + t2 /2! + · · · + tn /n!. Since xn(t) → E(t) as n → ∞, we obtain for the solution E(t) the infinite series representation E(t) = 1 + t + t2 /2! + t3 /3! + · · · , valid actually for every t ∈ R. In particular, e := E(1) = 1 + 1 + 1/2! + 1/3! + · · · = 2.7182818 . . ..
- 55. Exploring the Variety of Random Documents with Different Content
- 56. discipline was the most severe in the world. Even if the soldiers had fallen asleep whilst watching the entrance of the sepulchre, it appears impossible for a number of persons to remove so ponderous a stone without considerable noise and bustle, or to pass among the guards without awaking some of them. But even allowing the body to have been gone whilst they slept, how could they possibly know, that it was the disciples who had taken it? But is it at all probable, that a few timid disciples, who had fled from their Master on his first apprehension, should now dare to go, in the face of a guard of Roman soldiers, justly famed for their courage, and attempt to steal, and much more to carry off, the body! Let it be observed, that though the disciples had hoped Jesus "had been he who would have redeemed Israel;" yet, when they saw him laid in the grave, all their hopes that he was the Messiah fled, for the minds of the disciples were strongly tainted by the Jewish prejudice, that the Messiah's would be a temporal kingdom. Their dreams of earthly splendour now vanished, and they were about to return to their occupations in common life; in fact, some had done so. Is it reasonable to imagine that the others would engage in a plan fraught with danger, for the sake of obtaining the body of one, in whom they began to imagine themselves deceived? Besides, what advantage could they hope to gain by such a scheme? What end was it designed to answer? They could not expect to keep the act concealed; and if discovered, they were fully convinced it would bring upon them the severest punishment. But if, as the soldiers proclaimed, the disciples did steal him away, why are these handful of fishermen allowed to retain possession? Why did not the Chief Priest, at the head of the Jewish Sanhedrim, supported by the Roman authority, instantly compel them to surrender the body? Why are not these men of Galilee brought to a judicial tribunal, examined, and openly punished, that the truth of the soldiers' tale may bear even the appearance of fact? Surely this neglect is most extraordinary in men who had shown such vigilant care over the body when in the tomb. The more we examine the conduct of the parties, the more inconsistent does the Jewish tale appear. It is evident, the disciples were as ignorant as the rest of the nation, as to what the resurrection from the dead
- 57. should mean. Jesus had again and again preached the doctrine, yet they were at the first as backward as his enemies to believe the fact, and discovered much unbelief on the first tidings of the great event. The incredulity of all of them is a strong presumption, that as they did not expect Jesus to rise from the grave, so neither did they steal the body, and falsely proclaim their Master risen. We have a still further confirmation of the fact from the events that followed. In the interval of forty days, between his resurrection and ascension, Jesus appeared to many of his disciples, and showed himself alive by many infallible proofs; the women who went early to their Lord's sepulchre, were first honoured with the sight of the risen Redeemer. He afterwards appeared to the two sorrowing disciples as they walked to Emmaus, then to the eleven as they sat at meat with the doors closed, and, eight days after, he again appeared to them, when the incredulous Thomas exclaimed, "My Lord and my God!" He also showed himself to the seven disciples who were fishing at the sea of Tiberius; after that, he was seen of above five hundred brethren at once; and, though some had fallen asleep, yet, when the Apostle wrote, the greater part were then alive, and could testify to the truth of these things. How "vain the watch, the stone, the seal!" the grave could not contain the prisoner. Jesus burst the bands of death, and arose the triumphant victor. It was necessary that he, as the Head and Representative of his church, should conquer death and the grave for them. He died "that through death he might destroy him that had the power of death, that is, the devil." He laid in the grave that he might subdue the power of the grave. He, as a surety, became subject unto death as a part of the curse; but, having paid the full ransom, justice demanded his release. Having satisfied the demands of the law, it was right that he should be honourably acquitted. Though "delivered for our offences, he must be raised again for our justification." The resurrection proves his atonement was accepted by God as fully adequate to all the requirements of justice, and declares him to be the Son of God with power. It is by reason of the incapacity of the damned in hell, to take in the full measure of God's wrath due to them for their sins, that their punishment, though it be eternal, yet never satisfies; because
- 58. they can never endure all as Christ could, and did; theirs is truly less than what Christ underwent; and, therefore, his punishment ought not in justice to be eternal, as theirs, because he could more fully satisfy God's wrath in a few hours than they could to all eternity. By his complete satisfaction, the costly, inestimable price of redemption is paid, and the sinner's surety released from all the claims of the Law and justice. "Christ is risen from the dead, and become the first fruits of them that slept." Do we not hear him exclaim, "Thy dead men shall live together; with my dead body shall they arise. Awake and sing, ye that dwell in the dust." "I will ransom them from the power of the grave; I will redeem them from death. O death, I will be thy plagues; O grave, I will be thy destruction." May we not join in happy chorus, "O death, where is thy sting? O grave, where is thy victory? The sting of death is sin; and the strength of sin is the law. But, thanks be to God, which giveth us the victory through our Lord Jesus Christ."
- 59. CHAPTER LIX. Thou hast ascended on high, thou hast led captivity captive; thou hast received gifts for men; yea, for the rebellious also, that the Lord God might dwell amongst them.—Psalm lxviii. 18. We find amid the records of the Old Testament, very distinguished honour was conferred by God on two illustrious personages, whom he was pleased to exempt from the common lot of humanity, and admit into the Celestial City, by a new, and, till then, untrodden path. Their way led not across the dark valley of the shadow of death; they entered Canaan without passing the banks of Jordan's stormy waters. God was pleased to translate the bodies of Enoch and Elijah to heaven, without an execution of the sentence "dust thou art, and unto dust shalt thou return." This was assuredly a high mark of favour; but we are in this verse presented with an event, in comparison with which, the cases of Enoch and Elijah sink into insignificance. It is a description of the return of a great and mighty conqueror, who, surrounded by the trophies of his victories, appears at court to receive the thanks and rewards his services so well deserve. And who is this mighty conqueror? It is Jesus! See him surrounded by the little band of faithful followers, on whom he bestows his parting blessing; having bidden them an affectionate farewell, he, with conscious majesty, mounts the air, and soars beyond the eagle's path, through the vast extent of space. Though he goes forth unattended, it is not long a secret that the victorious Saviour is on his way to the heavenly kingdom; for the myriads of spirits, who are anxiously watching his motions, no sooner observe that he bends his course toward the Celestial City, but they instantly proclaim the joyful news to its inhabitants; who, with holy impatience, are all anxious to fly on the wings of love and adoration to meet and welcome this illustrious Conqueror back to the realms of
- 60. bliss. Wide are thrown the golden gates, and as they open, ten thousand voices are heard chaunting in chorus; "Lift up your heads, O ye gates; and be ye lift up, ye everlasting doors; and the King of glory shall come in. Who is this King of glory? The Lord, strong and mighty; the Lord, mighty in battle. Lift up your heads, O ye gates; even lift them up, ye everlasting doors; and the King of glory shall come in. Who is this King of glory? The Lord of Hosts, he is the King of glory." Forth from heaven's portals there issued a goodly band, singing as they advance to meet and welcome their victorious King, whom they convey in celestial triumph to the presence of the eternal Father; seated on his throne of glory, he receives, with ineffable delight and joy, this, his only-begotten, always well-beloved, but now still more endeared Son, the Glorious Deliverer of the children of men. Great was the joy of that illustrious day, when the eternal Son of God, entered the city of the new Jerusalem, as the victorious Conqueror of sin, death, and hell, whom he led as captives to adorn his triumph, for, "having spoiled principalities and powers, he made a show of them openly, triumphing over them, and ascended on high, leading captivity captive." Then the eternal hills resounded to the melodious sound of ten thousand times ten thousand voices, who sing aloud, "Worthy is the Lamb that was slain, to receive power, and riches, and wisdom, and strength, and honour, and glory, and blessing." Then all in heaven said, "Blessing, and honour, and glory, and power, be unto him that sitteth upon the throne, and unto the Lamb, for ever, and ever." The spirits of the redeemed vie with elect angels, in testifying their love, reverence, and gratitude to the God of their salvation. They knew, if the eternal Son of God had not become their surety, not one of Adam's race could ever have entered the realms of bliss.[107] But in the eternal council of peace, he did covenant and promise, in the fulness of time, to become a sacrifice, and God who knew him to be faithful, did, on the credit of that promise, save all the Old Testament saints.[108] Jesus had now fulfilled that engagement; paid the full price of their redemption; "blotted out the hand-writing of ordinances that was against them, taking it away by nailing it to his cross." What wonder, if his return
- 61. was hailed with rapturous delight; his presence could not fail of adding fresh joy to the happy spirits of the redeemed in glory. Yes! Jesus has "ascended on high, he has led captivity captive, and received gifts for men." It is as the God-Man, it is in his human nature, that he is said to receive gifts; for, as God, all is his in common with the Father. It is in the office of Mediator, that he has "all power given him in heaven and on earth." It is as God-Man, that the Father set him "at his right hand, in the heavenly places; far above all principality, and power, and might, and dominion, and every name that is named, not only in this world, but also in that which is to come; and hath put all things under his feet, and gave him to be the head over all things to the church." He is made the great Almoner of heaven, and he disposes of his gifts to the children of earth. He has received freely, and he gives freely,—witness the showers of ascension gifts, on the day of Pentecost. He then, as the apostle quotes the words, "gave gifts to men, yea, to the rebellious also, that the Lord God might dwell among them." But while we view Christ as glorified, let us not fail to connect the scenes of Gethsemane and Calvary. The new song in heaven, to which their golden harps are ever tuned, is to the praise of him "who was slain, and has redeemed us to God by his blood, out of every kindred, and tongue, and people, and nation; and has made us unto our God kings and priests for ever."
- 62. CHAPTER LX. And it shall come to pass afterward, that I will pour out my spirit upon all flesh; and your sons and your daughters shall prophesy, your old men shall dream dreams, and your young men shall see visions: and also upon the servants and upon the handmaids in those days will I pour out my spirit.—Joel ii. 28, 29. That part of the prophet Joel from which this verse is selected, is highly interesting; and although not strictly prophetical of the person of the Messiah, yet it is so closely connected that it cannot be severed without injury to the whole. In fact, it serves as a test, whereby we may prove if Jesus be in truth that Messiah, of whom "Moses and the prophets did write." The "afterward" here noticed, alludes to the coming of the Messiah, after which great day of the Lord, the promise here made, of a glorious outpouring of the spirit, was to be fulfilled. It will be alike easy and delightful, to trace its accomplishment. The Holy Spirit, from the earliest ages of the world, has shed his sacred influences over the church; but no visible or open display of that divine person, God the Holy Ghost, had ever been made. That great event was reserved until after the Messiah's appearance; and, when that illustrious person had publicly manifested himself to the world, then was this promise to be fulfilled. Jesus declared himself to be the second person, in the revealed order of the Holy Trinity—the eternal Son of God—Christ the Messiah; and in such character he promised, when returned to glory, to send down the Holy Spirit. Again and again did Jesus direct his disciples to expect that event. On the last great day of the feast, he publicly proclaimed in the temple its near approach, and promised its fulfilment; "for the Holy Ghost was not yet given, because that Jesus was not yet glorified." When the faithful disciples
- 63. were overwhelmed with grief, on learning from their beloved Master that he was shortly to leave them, Jesus cheered their drooping spirits with the promise of another Comforter, even the Spirit of truth; whom he would send from the Father. To reconcile them still more to his departure, he told them "it was expedient for them that he should go away," for, "if he went not away the Comforter would not come; but if he departed, he would send him unto them." After his resurrection, Jesus again taught the disciples to expect this great event, and on the morning of his ascension he repeated his promise, adding, as it would not be many days hence, they should tarry at Jerusalem until its accomplishment. After the ascension of Jesus, the disciples were so fully persuaded that he was the Christ of God, that they continued daily assembled together, waiting for the fulfilment of the great promise made to them by their risen Lord. It will be remembered, that all the Israelitish males were commanded to appear, three times in the year, before the Lord at Jerusalem, at the feasts of Passover, Pentecost, and Tabernacles. The feast of Pentecost or weeks, was celebrated fifty days after the Passover. It was at the first great Jewish festival, the Passover, that Jesus was crucified. He arose from the dead on the third day, and as forty days intervened between his resurrection and return to glory, there could be only seven days from his ascension until the feast of Pentecost. It was on the morning of the ever-memorable day of Pentecost, the disciples being all of one accord, in one place; that "suddenly there came a sound from heaven, as of a rushing mighty wind, and filled all the house, where they were assembled; and there appeared cloven tongues, like as of fire, and sat upon each of them, and they were all filled with the Holy Ghost, and began to speak with other tongues, as the Spirit gave them utterance." Such a miraculous event was soon noised abroad, and multitudes crowd to learn the fact. As the Holy Spirit was graciously pleased to make this open display of his person and godhead, at one of the great Jewish festivals, the number of strangers who usually resorted to Jerusalem at that season, either for the purposes of worship or trade, became witnesses of the miraculous gifts bestowed on those hitherto
- 64. unlearned, and many of them unlettered, Galilean fishermen. The inhabitants of Galilee were proverbial for their dulness and stupidity; [109] yet these men were taught, in an instant of time, to speak, with ease and fluency, languages whose very names, it is more than probable, they were an hour before unable to pronounce correctly. An opportunity was instantly offered for the apostles openly to display their extraordinary gifts. Amidst the assembled throng were men of sixteen different nations, to whom these poor fishermen publicly proclaimed, in their several languages, or dialects, the wonderful works of God. They needed no interpreter, in addressing this motley crowd. How preposterous to accuse the apostles of drunkenness! Truly, we should not imagine a state of inebriety the best calculated for acquiring a knowledge of any of the learned languages. We seldom know men, (however well their heads are furnished,) in a state of intoxication, speak any thing except it be the language of foolishness. Beside, it was only the third hour of the day, (nine o'clock) the time of offering the daily morning sacrifice in the temple, before which hour the Jews were forbidden to take any refreshment; and, as this was a solemn festival, no doubt the command was then more strictly observed. How mild, yet energetic, the reply of Peter, who declared the event to be a fulfilment of the prophecy of Joel, accomplished on the return of Jesus to glory; "when being by the right hand of God exalted, and having received of the Father the promise of the Holy Ghost, he had shed forth that which they then saw and heard." The appearance of the Holy Spirit was sufficient to prove his personality. Might not the sound from heaven, as of a rushing mighty wind, be designed to show that the operations of God the Holy Spirit, are like the unknown and unexplored sources of the air. "The wind bloweth where it listeth, and thou hearest the sound thereof; but canst not tell whence it cometh, or whither it goeth: so is every one that is born of the Spirit." This was a lesson taught Nicodemus by Jesus, the wisdom and word of God. On Shinar's plains, the Lord, to testify his divine displeasure, confounded the language of mankind. It was a curse pronounced on
- 65. Babel's tower; but at Pentecost, the Holy Spirit was pleased to use the diversity of language as a witness of his almighty power and Godhead; when he publicly and solemnly ordained the apostles ministers of the everlasting Gospel, and endowed them with extraordinary gifts, as the first ambassadors of Christ, sent forth to publish unto all nations the glad tidings of great joy. Might we not be tempted, when viewing the immoral and profane amusements of Whitsuntide, to imagine it an annual feast holden to Venus or Bacchus; instead of (as at first designed) a solemn festival, intended to commemorate the visible descent of the Spirit of Purity? Certainly the general character of the public assemblies, at that season, bears a much nearer resemblance to the sports holden in honour of the deified heroes in heathen mythology, than to the pure and spiritual nature of the Divine Person, whose first public appearance in our world it was wished annually to celebrate. What would the early disciples of Christ feel, could they behold the sad perversion of this sacred festival!
- 66. CHAPTER LXI. And I will pour upon the House of David, and upon the inhabitants of Jerusalem, the Spirit of grace and of supplications; and they shall look upon me whom they have pierced, and they shall mourn for him as one mourneth for his only son, and shall be in bitterness for him as one that is in bitterness for his first born.—Zech. xii. 10. The Prophet Zechariah here presents to our view one of the richest jewels in the treasury of God's promises. It sparkles clear and bright amid the records of divine truth. All earth's richest treasures cannot offer an adequate remuneration for the withdrawment of this precious promise. The words deserve our most careful examination. We will therefore consider the person here promising; the persons to whom the promise is made; the thing promised; and search for proofs of its fulfilment. The person here promising is the God-Man, Christ Jesus, for the words are, "I will pour, &c. &c., and they shall look upon me, whom they have pierced, and mourn." We never find God the Father using such language as this when speaking of his disobedient creatures. God is justly displeased at man's apostasy. His law is dishonoured, his works defaced and injured by sin. Yet God, as God, cannot be the subject of pain and sorrow, he is beyond their reach. But if we look at the God-Man, Christ Jesus, we behold his sacred head pierced with a thorny crown, his hands and feet with nails of iron, his side with the soldier's spear, and his soul with the wrath of God. He who suffered thus on earth, did, as God, make this gracious promise. The persons to whom this promise literally applies, are the Jews, whose restoration as a nation to the divine favour, will form a
- 67. prominent feature in the latter-day glories of the Church. The Lord has promised to gather together the dispersed in Judah, and the outcasts of Israel. "The deliverer shall arise out of Zion, and turn away ungodliness from Jacob." This nation, who once refused and crucified the Messiah, shall, when partakers of this promised blessing, "look upon him whom they have pierced, and mourn." This promise is not confined to the Jews, but extends to the fallen race of Adam, whom our spiritual David will make inhabitants of the new Jerusalem, which is above, without regard to their being of Jewish or Gentile extraction.[110] He will not consider the trifling distinctions of colour, language, or nation, a barrier of such importance as to preclude their participating in his blessings. The thing promised is an abundant outpouring of the Holy Spirit. Adam, by his apostasy, lost the image of God stamped upon his soul at his creation. The sentence, "in the day thou eatest thereof thou shalt surely die," was not suffered to go unexecuted. From that hapless hour, his soul, the most noble part, was dead to all spiritual life, and became the abode of corroding passions and depraved principles. He immediately shrank from holding intercourse with God, and tried to hide himself from the presence of his benefactor. As Adam begat a son in his own fallen likeness, all his race partake of the same corrupt nature. We are ignorant of God and his ways. We need divine teaching; we cannot naturally understand the things of God, which are spiritual, the eye of our understanding being darkened; God is not in all our thoughts; we are averse to communion with the Father of Spirits. We despise his offers of free grace—we prefer to be saved by our own rather than God's method —we see no beauty in Jesus that we should desire him—we dislike to renounce our own, and trust in his complete righteousness—we consider his commands grievous, and the language of our soul is, "we will not have this man to reign over us." But we are here told of a sovereign antidote for these deep-seated moral disorders of the soul. Here is a gracious promise of an abundant outpouring of the Holy Spirit, whose office it is to "convince of sin, of righteousness, and of judgment." He convinces the soul, into which he enters, of
- 68. the exceeding sinfulness of sin—that it is the evil thing which God hates; and shows the divine law is spiritual, extending to the thoughts and intents of the heart.[111] He puts a cry for mercy into the soul, destroys the natural enmity of the mind against God's plan of salvation, and makes the object of his divine teaching willing and anxious to partake of the Lord's bounty, and be a debtor to mercy alone. The Holy Spirit teaches of righteousness by convincing that a better righteousness than our own tattered rags is absolutely necessary, ere we can see the face of God with peace. He makes the soul willing to be clothed with the wedding garment of Jesus' righteousness, which is the fine linen of the saints. It is indispensable that we be clothed with this livery of the court of Heaven, or we shall be denied admission into the mansions of the King of Glory. Would we behold the fulfilment of this prophetic promise, then let us direct our minds back to a survey of the glorious scenes exhibited on the ever memorable day of Pentecost, when the Spirit was, in so free and copious a manner, poured out from on high. Attend to the sermon Peter preached on the day of his ordination; mark its effects on the three thousand of the House of David, inhabitants of Jerusalem's much-famed city. Listen to their cry, "Men and brethren, what must we do?" Surely these were none of the stout hearts who dared even to crucify the Lord of life and glory? The same! yet how different their tone—how altered their conduct! To what cause can we attribute this astonishing change in the minds of three thousand persons in the same instant of time? Surely it was none other than the almighty work of God the Holy Ghost. It was his influence on the minds of these men which produced the Spirit of grace and supplication, and taught them to direct the anxious cry and supplicating look unto him whom they had pierced. Was not the anguish of their souls, under a sense of their sins, equal to the exquisite sorrow of those who bitterly bewail the death of their first-born? However skilfully Peter might wield the sword of the Spirit, (the word of God,) it was none other than the God of all grace, who directed and sent it home with saving power to the hearts and consciences of these Jerusalem sinners. Are not
- 69. the other triumphs of the Spirit worthy of regard, when five thousand are made willing cordially to embrace Christ crucified? May we not, by the way, observe, that the reception of the Gospel by such numbers so immediately after the ascension of Jesus, proved the truth of the facts recorded by the apostles, of the life, death, resurrection, and ascension of Christ? Many, no doubt, of these early converts of Christianity, had been eye-witnesses of several of the events, and all had an opportunity of discovering the deception, if there had existed any, in the apostles' narrative. But no sooner are they persuaded to compare the Old Testament prophecies concerning the Messiah, with all the circumstances in the history of Jesus of Nazareth, than they anxiously desire to be enlisted under the banners of the cross. Unable to resist the force of truth, they join the persecuted adherents of the crucified Jesus, and cast in their lot with his despised followers, although "a sect every where spoken against." When were converts to Christianity most numerous? Was it not when there existed the best possible opportunity of detecting the least imposition or falsehood, on the part of the writers of the New Testament? Let it not be forgotten that those early converts were neither won by the arm of worldly power, nor bribed by proffered gold. On the contrary, no sooner did they embrace the Gospel, but they were met at the very threshold by ignominy and persecution in every varied and frightful form, sufficiently terrific to deter all but men really convinced of the truth, and swayed by its sacred influence. But we must not confine the accomplishment of this promise entirely to the days of Pentecost, although it then assumed a more splendid and attractive appearance, than it has done in these latter times. Yet through each succeeding age, the Lord the Spirit has not been unmindful of his covenant engagements. Could we draw aside the veil that separates between us and the holy of holies—could we obtain a glimpse of the inhabitants of the New Jerusalem which is above, and inquire of the goodly number that surround the throne of God and the Lamb, Who was the faithful instructor and guide, that taught them to walk in the way that led to everlasting life? they
- 70. would direct us to the Lord the Spirit, as the almighty guide who pointed out the road, and taught their wandering feet to tread the strait, the narrow way, the only path, that leads to Zion's hill. In the Bible, that chart of life, the road is shown with clearness, and described with accuracy. It is called faith in the finished salvation of Christ, and obedience to his commands. The hand which drew this path to glory, is the very same that painted the splendid canopy of heaven. By this good old way, all the patriarchs, prophets, apostles, martyrs, and reformers, entered the city of the Lord of Hosts. Their guide and comforter, through this waste howling wilderness, was the third person of the Triune-Jehovah. What countless myriads has this almighty guide led to the mount of God, from the antediluvian worthies, down to the happy spirit just entered into the joy of its Lord! Like them, led by the same unerring teacher, we shall not fail of arriving safely at the mansion of everlasting joy, for he is the only faithful conductor[112] to the heavenly Jerusalem; untaught by him, none can find the path of life, but will assuredly stumble on the dark mountains of sin and error, and run the downward road that leads to hell. Eternal life is the gift of God. Christ is "the way, the truth, and the life: none can come unto God, but by him." The office of the Holy Spirit is to instruct the ignorant, comfort the mourners in Zion, and make us meet to be "partakers of the inheritance of the saints in light." "If ye, being evil, know how to give good gifts unto your children, how much more will your heavenly Father give the Holy Spirit to them that ask him." May we be partakers of that inestimable blessing, for without his influence on our hearts, vain will be even the electing love of God the Father—vain the vicarious sacrifice and imputed righteousness of Christ the Son—vain to us the plan of salvation; and vain, all the promises of the Gospel. As well for us, if those glad tidings of great joy, "Glory to God in the highest, and on earth peace, good-will toward men," had not reached our ears. Unapplied, the most sovereign remedy is useless, for then not even Gilead's balm, can heal the dire disease.[113] Christ will prove no Saviour to us, unless applied to our individual case. It is the office of
- 71. the Holy Spirit, to take of the things of Christ and show them unto us. Faith is the hand by which we grasp Christ crucified. That saving faith, by which we apprehend the finished salvation of Jesus, and make it our own, is a grace wrought in the heart by the operation of the Spirit of God. Far better would it be for the children of men, if the sun were turned into darkness, the moon into blood, and all the stars of heaven withdraw their shining; than that this glorious promise of the outpouring of the Spirit, should be blotted from the book of God's remembrance! May that blessed morning shortly dawn, "when all shall know the Lord!" Hasten, glorious Immanuel, that bright day, when "the whole earth shall be full of the knowledge of the Lord, as the waters cover the sea."
- 72. CHAPTER LXII. The Lord hath sworn and will not repent, thou art a priest for ever, after the order of Melchizedek.—Psalm cx. 4. In the Old Testament, we find but little recorded of Melchizedek, that venerable priest of the most High God, who met and blessed the patriarch Abraham as he returned victorious from the slaughter of Chedorlaomer and the confederate kings. But from that little, we are led to regard him as a person of distinction. To him, the great father of the faithful and friend of God presented the tithes or tenths of the spoil. It is from the prophetical word of the royal Psalmist, "the Lord hath sworn and will not repent, thou art a Priest for ever, after the order of Melchizedek," that we are taught to view this ancient priest of God as a type: and of whom, if not of Christ? Paul, in his epistle to the Hebrews,[114] speaks largely on the subject; he proves the fulfilment of the prophecy, and declares, that Christ's priestly office was prefigured in the person of Melchizedek, to Abraham the father of the Israelitish race. In the same epistle, we find blended the priesthood of Aaron, in order to show the vast superiority of that of Christ over the other two, though both instituted by God himself. But as we find no prophecy respecting the Aaronic priesthood, we make no further reference to that subject, in order to attend more immediately to the words, "The Lord hath sworn, and will not repent, thou art a priest for ever, after the order of Melchizedek." Was this priest of the most High God honoured with the title of King of Salem—by interpretation, King of Righteousness, and King of Peace? Is not Jesus proclaimed King of Zion; the Lord our Righteousness, and the Prince of Peace? Nor are these mere empty titles, but real characters, and offices, sustained by Him, who "abideth a priest upon his throne for ever." We have no historical account of the parentage or descendants of Melchizedek; he is
- 73. presented to us as "without father, without mother, without descent, having neither beginning of days, nor end of life;" but being made like unto the Son of God, abideth a priest continually.[115] And Christ's priesthood was not derived by genealogy, or succession, he had neither father or mother of the family of Aaron, from whom his priesthood could descend. It is evident our Lord sprang "out of Judah, of which tribe no man gave attendance at the altar;"[116] neither did Christ die and leave it to others, by way of descent, but was constituted a single priest, without predecessor or successor. "He abideth a priest for ever, after the order of Melchizedek." It is impossible for a finite mind to comprehend the eternal sonship of the Son of God, whom the Father, before the foundation of the world, constituted a priest for ever; and therefore, the priesthood of Melchizedek was instituted to prefigure to us the nature of Christ's eternal priesthood. "The Lord hath sworn and will not repent, thou art a priest for ever, after the order of Melchizedek." These words deserve particular attention. It is God the Father who swears to Christ; no oath of allegiance is required from him who is constituted our Priest. Jehovah, whose eye pierces through futurity, knew he would be faithful in his office, and he freely and unreservedly trusted him to maintain his divine honour and justice, and accomplish the salvation of sinners. The high-priestly office, though honourable, could not add to Christ's dignity; but his glorious person did confer honour and dignity upon the sacred office, for he who is constituted our High Priest, "is fellow to the Lord of Hosts." "Every high priest is ordained, to offer both gifts and sacrifices," and great was the sacrifice offered by Christ: he offered up himself; he would borrow nothing, but was both priest, sacrifice, altar, and temple: and "by that offering, he hath perfected for ever them that are sanctified." "And because he continueth ever, he hath an unchangeable priesthood;" "wherefore he is able to save them to the uttermost, that come unto God by him, seeing he ever liveth to make intercession for them." Blessed Jesus! thou priest of Melchizedek's order, while we would not withhold from thee a portion of all that thou givest us, let us not rest satisfied, till we are enabled to present
- 74. "our bodies and souls a reasonable sacrifice, holy and acceptable unto God."
- 75. CHAPTER LXIII. Seventy weeks are determined upon thy people and upon thy holy city, to finish the transgression, and to make an end of sins, and to make reconciliation for iniquity, and to bring in everlasting righteousness, and to seal up the vision and prophecy, and to anoint the most Holy. Know, therefore, and understand, that from the going forth of the commandment to restore and to build Jerusalem, unto the Messiah, the Prince, shall be seven weeks, and three score and two weeks: the street shall be built again, and the wall, even in troublous times. —Daniel ix. 24, 25. The harps of Judah were silent—the disconsolate Israelites hung them on the willows of Babylon—no songs of Zion were heard in that land of captivity, where, for seventy long years, they wore the galling yoke of bondage, bereft of home and all its blessings—the land of their forefathers in the possession of strangers—Jerusalem in ruins—her palaces consumed—the Temple destroyed—the spot trodden down by the Heathen—themselves exposed to the taunts of their conquerors, and compelled to bow before the idolatrous image of Chaldean superstition.[117] Well might Judah's sons weep by the waters of Babylon, whose murmurings recalled to their recollection the stream which gushed from Horeb's mount.[118] The remembrance of past blessings increases the weight of present misery. How changed their state, and changed to punish their awful rebellions against the Lord of Sabaoth! Yet the God of Israel was not unmindful of his promise—he cheered their drooping spirits with the assurance of speedy deliverance from their captive state. The prayer of Daniel entered into the ears of the Lord of Hosts—the command was given —swiftly the angel, even Gabriel, flew to reveal his Lord's decrees unto the mourning prophet—that "man greatly beloved" of his God.
- 76. Daniel was commissioned to foretel the deliverance of the Jews from Babylon—the building of Jerusalem and its walls in troublous times; and to him, Jehovah was graciously pleased to renew the promise of the Prince, Messiah, whose appearance all the patriarchs and prophets had foretold. The nearer that glorious epoch approached, the more minutely was it described. The Lord gave Daniel to "know and understand, that from the going forth of the commandment to restore and build Jerusalem unto the Messiah, the Prince, should be seven weeks, and three score and two weeks." The period here styled weeks, is generally allowed to be sabbaths of years. This appears to be the sense of the passage, for the Jews were accustomed to reckon their time and feasts by weeks or sabbaths. The week of days was from one seventh or sabbath day to another. The week of years was from one seventh or sabbatical year to another; in the seventh, or sabbatical year, they neither sowed their fields nor pruned their vineyards; it was a sabbath of rest unto the land.[119] In the regulation of the year of Jubilee, they were commanded to number "seven sabbaths of years, seven times seven years, and the space of the seven sabbaths of years shall be to thee forty and nine years."[120] We therefore only follow the Mosaic rule, (to which Moses' disciples cannot object,) if we consider these seven weeks, and three score and two weeks, as seven times sixty-nine, or four hundred and eighty-three years, which should be between "the going forth of the commandment to restore and build Jerusalem unto the Messiah, the Prince." There were four distinct decrees or commandments granted by the kings of Persia, in favour of the Jews, who came under the dominion of that empire by its conquest of Babylon. This was the epoch of Daniel's vision. No sooner had Cyrus obtained possession of Chaldea, than he issued a decree allowing the Jews to quit the land of their captivity, and repair to Judea to build the temple of the Lord. He also restored to them the vessels and treasures which Nebuchadnezzar had taken from the temple built by Solomon. On the grant of this decree,[121] five hundred and thirty-six years before Christ, many of the Jews returned to their own land, and laid the foundation of the temple;
- 77. but they were hindered in the building of it by their several enemies, who were supported in their opposition by Artaxerxes, the successor of Cyrus. But when Darius Hystaspes ascended the throne of Persia, he issued a decree[122] five hundred and nineteen years before Christ, forbidding the enemies of the Jews to interrupt the building of the temple, and further commanded that materials requisite for the work, and the animals, oil, and wine for the sacrifices, should be supplied at his (the king's) cost. The third decree was granted to Ezra, the scribe, four hundred and sixty-seven years before Christ, by Artaxerxes Longimanus, in the seventh year of his reign, by which he bestowed great favours upon the Jews,[123] appointing Ezra Governor of Judea. He permitted all the Jews to return to Jerusalem, and commanded his treasurers beyond the river, to supply Ezra with such things as he needed for the house of his God, even to an hundred talents of silver, an hundred measures of wheat, an hundred baths of wine, and an hundred baths of oil. The king and his princes presented much silver and gold, and many vessels, and ordered that what else might be required for the house of God, should be supplied from the king's treasury. This is not the same Artaxerxes who listened to the slanderous reports of the enemies of the Jews, and stopped the building of their temple; but Artaxerxes, surnamed Longimanus, supposed to be the person styled Ahasuerus, in the book of Esther, whose attachment to his Israelitish consort may account for the distinguished favours he conferred on the people of her nation. We find the queen was present when Nehemiah presented his petition, which was the second decree granted by this monarch, and was the fourth and last decree, being granted in the twentieth year of his reign, and four hundred and fifty-four years before Christ.[124] This was the most efficient decree, for by it Jerusalem and its walls were built. The high resolves of the court of Heaven were revealed; Daniel was made "to know and understand that from the going forth of the commandment to restore and build Jerusalem, unto the Messiah, the prince, shall be seven weeks, and three score and two weeks, being sixty nine weeks, or four hundred and eighty-three years. From the last, or
- 78. fourth, decree to the birth of Christ, (vide Rollin, volume 8, page 265,) is four hundred and fifty-four years, to which we add twenty- nine years (the age at about which Christ entered on his public ministry);[125] these united, make the exact period of sixty-nine weeks, or four hundred and eighty-three years. Daniel also declares that "seventy weeks (or four hundred and ninety years) are determined upon thy people and upon thy holy city, to finish the transgression, and to make an end of sins, and to make reconciliation for iniquity, and to bring in everlasting righteousness, and to seal up the vision and prophecy, and to anoint the most Holy." We find between the seventy weeks, or four hundred and ninety years, and the sixty-nine weeks, or four hundred and eighty- three years, a difference of one week, or seven years, which is the week evidently alluded to in the twenty-seventh verse of this chapter, in which "he shall confirm the covenant with many for one week, &c." From the period of Christ's first entry into the ministry, and the calling of his apostles, until his crucifixion, were three and a half years, and, for three and a half years after that event, his apostles continued to minister amongst the Jews. This makes a period of seven years, (or one prophetic week,) in the midst of which the Messiah was cut off, and "the sacrifice and oblation" virtually ceased. The correspondence is exact: Jesus, the Messiah, not only entered on his public ministry at the very period pointed out ages before, but was actually cut off in the midst of the week, as was expressly foretold. These predictions of the Prince Messiah are peculiarly striking. The time for his appearance is marked, and the particular objects he should effect on his coming, are described with such minuteness, as scarcely to admit of the possibility of mistaking his person. The grand features of his mission were so strongly exhibited, that it was morally impossible the Messiah should appear and not be recognised. Prejudice must have blinded the eye of that mind which does not, on comparing the whole of the New Testament with this prophecy, acknowledge Jesus of Nazareth to be the Messiah. It bears the stamp of divine prescience: none but the omniscient God could have given his features with such clearness so
- 79. many ages before. This portrait of the Messiah, which bears so exact a resemblance to Jesus, was in the possession of the Jews, at least five hundred years before that glorious person was exhibited to the world, a God incarnate. Jesus declares himself to be the long promised Messiah—his claim rests on no slight or doubtful evidence—he came at the very precise time it was foretold the Messiah should appear to the people and the holy city. Christ's ministry was among the people of the Jews—Judea was the land of his nativity—the scene of his labours—the witness of his miracles—he was born at Bethlehem, near Jerusalem, and crucified just "without the gate" of the holy city. On Calvary "he finished the transgressions, and made an end of sin, and make reconciliation for iniquity." There the God-man, Christ Jesus, offered up his life a ransom for the guilty—there the surety of the Church paid the full price for her redemption, and made peace by the blood of his cross—there "he suffered the just for the unjust to bring sinners unto God." He took away "the hand-writing of ordinances that was against us, taking them out of the way by nailing them to the cross"—there he removed the iniquity of the land in one day, and so completely "finished the transgression," by suffering the punishment due for his people's sins, that when they are "sought for they shall not be found"—there he paid the full price of their redemption, he cancelled the bond, and made peace and reconciliation with offended justice. He "brought in an everlasting righteousness, and not only suffered the penalty due for their transgressions of God's law, 'which is holy, just, and good,' but, as the head of the Church, he obeyed all the precepts of the moral law; which he exalted and made honourable. Perfect was the obedience wrought out—complete was the righteousness brought in by the incarnate Deity, the Lord our righteousness, which is from everlasting to everlasting "unto all and upon all that believe, for there is no difference." Amidst the awful gloom on Calvary's mount, was heard the cry "it is finished!" It was the conqueror's shout— victory was achieved—Satan was vanquished—the sting of death was taken away—the power of the grave destroyed—the conflict was
- 80. over—the ransom paid—the captives of the mighty delivered—the law was honoured—justice satisfied—God glorified—Heaven opened —man redeemed—and hell vanquished. That was the glorious event which types were intended to exhibit, and prophets were commissioned to proclaim. The appointed time of the vision was arrived—it had long tarried, but it was accomplished. The chain of prophecy was complete—the vision was sealed[126] —and the most holy anointed. The God-man, Christ Jesus, anointed by his Father king and priest of Zion, then exchanged his thorny crown for the royal diadem—then left the sorrows of earth for the glories of his mediatorial throne, which no enemy can touch—their opposition is vain—he that sitteth upon the circle of the heavens, will laugh them to scorn. Happy are they who have for their king and priest, him whose kingdom is eternal, and priesthood unchangeable—who look to the Redeemer of Israel as the rock of their salvation, and crown the most holy, Lord of all. "Happy are the people that are in such a case, yea, blessed are the people whose God is the Lord."
- 81. CHAPTER LXIV. And after three score and two weeks shall Messiah be cut off, but not for himself: and the people of the prince that shall come shall destroy the city and the sanctuary; and the end thereof shall be with a flood, and unto the end of the war desolations are determined.—Daniel ix. 26. This vision of Daniel appears involved in considerable obscurity, by the diversity of time alluded to in the several parts of the prophecy, and renders it difficult to prove its exact accomplishment. But we hope we have shown in the preceding part, that it does not militate against "the truth as it is in Jesus," it rather tends to strengthen the testimony, by affording an additional opportunity of proving, from sacred and profane history, the fulfilment of the great event. The proof of its accomplishment does not rest on the insulated fact, but is established by a chain of evidence, derived from the annals of nations. For, whichever of the decrees we take, it is clear from ancient chronology, that the period alluded to is passed, and the Messiah did appear not far from the time named by any decree. As we have attempted to prove the fulfilment of the first part of the prophetic vision, it may not be improper if we now endeavour to show that the remaining part of this interesting prophecy has also been accomplished. "After threescore and two weeks shall Messiah be cut off, but not for himself: and the people of the prince that shall come shall destroy the city and the sanctuary." "Secret things belong unto God; but things that are revealed, to you and your children." We cannot ascertain to a certainty when the seventy-two weeks commence, but it is evident they terminate at the cutting off of the Messiah. From the words "And the end thereof shall be with a flood, and unto the end of the war desolations are determined," it appears, also, to
- 82. allude to the destruction of the city, previous to which event the Messiah should be cut off. We hope we shall not offer any violence to the words, if we give them this interpretation. The destruction of Jerusalem is not the only event alluded to in this interesting prophecy; there is one of paramount importance to the ruin of Salem's palaces, though that involved the fate of Judah's sons. On the other momentous fact hang the highest interests of Jew and Gentile, bond and free, past, present, and future generations; not only the happiness of earth, but much of the glory of heaven, depends on its accomplishment. Without it no sweet song of "Salvation to God and the Lamb," would have echoed amidst the heavenly hills, none of the race of Adam would be seen worshipping before the presence of Jehovah with the angels of light; those melodious hymns of redemption, now chaunted by ten thousand times ten thousand glorified Saints, had not been heard but for the vicarious sacrifice of the Son of God,[127] who not only covenanted, but did actually lay down his life a ransom for sinners. When Jesus, the Christ of God, the Prince Messiah, appeared on earth, it was not simply to set the children of men an example of piety and virtue; we ardently admire his glorious example, and consider his followers bound to imitate the bright pattern he has left them; yet we dare not believe that that was the only object he designed to accomplish when he visited our world.[128] No, he came as the federal Head, the Representative and Surety of his people.[129] He was "cut off from the land of the living," by a violent and cruel death; yet not for himself, not for any sin of his own,[130] nor purposely to set us a pattern of patience and resignation; but to discharge the debt of sin, he had covenanted to cancel on man's account. Jehovah executed towards him the severest justice, and permitted his crucifiers to exercise the blackest ingratitude, and most inhuman cruelty. "O Jerusalem, Jerusalem, thou who killest the prophets, and stonest them that are sent unto thee, how often would the Lord have gathered thee under his protecting care as a hen gathereth her chickens under her wings, but ye would not." Thy awful doom was sealed when thou didst reject the authority, and persecute unto
- 83. death Jesus the Messiah, thy prophet and benefactor, thy God and King. The thought of thy approaching misery drew tears from the eyes, and groans from the heart, of Incarnate Deity; yet thy children beheld, with feelings of triumphant scorn, the sorrows and sufferings their wanton cruelty inflicted on the Holy Jesus. But heaven marked the impious deed.[131] The blood of Jesus, of prophets, of apostles, and of martyrs, called for vengeance on thy guilty land; the cry was heard, justice remembered thy black catalogue of crimes, the King of heaven beheld the insult offered to his beloved Son, and Jehovah arose to punish thy rejection of Jesus the Messiah, whom "ye would not have to reign over you." The crimes of Jerusalem were of the blackest and most awful character, and her punishment was tremendously dreadful.[132] The Israelites, once the peculiar favourites of Heaven[133] —nursed in the lap of plenty, instructed in the oracles of God—blessed with the temple of Jehovah—taught to adore the God of truth whom their forefathers worshipped; this people, who once had the Lord for their Law-giver and King,[134] were compelled to bow beneath the oppressive power of arbitrary despots—the law of truth was exchanged for the tyrant's mandate— equity and justice were banished the walls of Salem, and despotism, oppression, blasphemy, and pride, reigned within that devoted, miserable, city. Anarchy and confusion ruled that senate and sanctuary, once as gloriously "distinguished from the rest of the world by the purity of its government, as by the richness and elegance of its buildings. Jerusalem was devoted to destruction, and she sunk beneath the accumulated horrors of war, famine, fire, and pestilence. Internal faction and a foreign foe reduced that beauteous city and magnificent sanctuary, to a heap of ruins. The temple fell— not all the commands, promises, or threats of Titus, could save that splendid edifice from destruction; the people of the prince, regardless of their general's orders, helped to complete the work of desolation;—but prophecy was fulfilled, Jerusalem was overwhelmed with the flood of divine vengeance, and desolation prevailed even unto the end of the war.
- 85. CHAPTER LXV. And he shall confirm the covenant with many for one week; and in the midst of the week he shall cause the sacrifice and the oblation to cease, and for the overspreading of abominations he shall make it desolate, even until the consummation, and that determined shall be poured upon the desolate.—Daniel ix. 27. Some writers consider this verse prophetical of the desolate state of Jerusalem under Antiochus Epiphanes, that sacrilegious monarch who impiously profaned the sanctuary of the God of Israel. By him the temple was ransacked and despoiled of its holy vessels; its golden ornaments pulled off; its hidden treasures seized; and an unclean animal offered on the altar of burnt-offerings. Thus did this impious Syrian king dare profane the altar and temple dedicated to Jehovah. Neither was this all; Jerusalem again felt the force of his horrid cruelty and profaneness; men, women, and children, were either slain or taken captive; and the houses and city walls were destroyed. The Jews were not allowed to offer burnt offerings or sacrifices to the God of Israel—circumcision was forbidden—they were required to profane the Sabbath, and eat the flesh of swine, and other beasts forbidden by their law[135] —the sanctuary dedicated to Jehovah was called the temple of Jupiter Olympius, and his image set up on the altar—idol temples and altars were erected throughout all their cities—and the Holy Scriptures destroyed whenever they were met with—and death was the fate of those who read the word of the Lord. The most horrid and brutal cruelties were inflicted on such as chose to obey God, rather than this Syrian monster. Jerusalem was overspread by his abominations; desolation was indeed poured out "upon the desolate" when Antiochus Epiphanes held the blood stained sceptre, emblem of satanic power. Yet, closely as these circumstances resemble the description given by the
- 86. prophet's vision, we cannot think it is the event alluded to in this prophecy. Daniel, in the three preceding verses, speaks of the Messiah, and the final destruction of the city and sanctuary: by Antiochus the temple certainly was not destroyed. In the eleventh chapter there appears a striking prophecy of the events which happened in Jerusalem during the dominion of the Syrian tyrant, but we cannot think he is alluded to in any part of the ninth chapter. The first clause of this verse, "He shall confirm the covenant with many," cannot refer to Antiochus, but alludes to the same glorious person mentioned in the preceding verses. The latter part of this verse may with propriety be considered as a continuance of the prophecy of Jerusalem's final destruction, as it occurred under Titus. To Jesus the Messiah we direct our eyes. The one week, or the midst of the week, (seven years half expired,) alludes to the time of his Public Ministry, which was three years and a half; during which period he declared, the design of his mission was to confirm the well-ordered covenant of redemption and peace, which was drawn up in the counsels of eternity—sealed on earth with the blood of the Incarnate God— signed in the presence of Jehovah, angels, men, and devils— registered in the court of Heaven—and proclaimed good and valid by the resurrection of Jesus from the dead, and the outpouring of the Holy Spirit.[136] It is true, the sacrifices and oblations of the temple service did not cease immediately on the death of Christ, they were continued some little time after that event; but they became unnecessary, they had lost their value, and were but idle ceremonies and useless rights, when the thing signified was accomplished. At best, they were only types of the Lamb of God, the blood of that one great sacrifice, which alone "cleanseth from all sin." "It is not possible for the blood of bulls or goats to take away sin." No, the sacrifices and ceremonies of the Mosaic economy were only efficacious so far as Christ, the substance, was viewed through the shadow.[137] In less than forty years after the death of Christ, the sacrifices and oblations ceased, for the temple was demolished. A spot so deeply stained with crime, needed the fire of divine vengeance to consume it from the face of the earth: it was erected
- 87. for the worship of the God of Israel, but was turned into the seat of iniquity and profaneness. The horrid enormities observed in the temple of Juggernaut scarcely surpassed the impious practices exercised within the Jewish sanctuary. When Titus, the Roman general, approached the walls of the city, it more resembled the court of Mars and Bacchus, than the temple of Jehovah; the drunkard's voice—the clash of arms—the shouts of the victor—the cries of the vanquished—and the groans of the dying, echoed through that magnificent pile; human blood flowed in its courts, and sprinkled its altars and its walls. Jerusalem was a scene of slaughter; but it was not a war to support the glorious cause of freedom; nor were they fighting to repel the foreign foe, or shedding their blood to defend their beloved homes, and the still dearer objects of affection, around which the warm heart clings with fondest thought amidst the scene of danger and of death, and for whose preservation the weakest arm grows desperate, and the feeblest mind resolves to conquer or to die. But theirs was no such glorious contest; no—civil war had reared her hydra head; the horrid yell of intestine discord rang through Salem's courts, and echoed round her walls; that infernal power bursts the bands of brotherhood, severs the closest ties, dissolves the strongest link of union, and makes the man a monster. The sword of her own sons deluged Jerusalem with Jewish blood; the fire which destroyed her houses was kindled by her own children; death and destruction reigned through all her palaces; the city groaned beneath a three-fold faction, when the Roman legions approached her walls to complete the horrid scene of slaughter. The temple was the head-quarters of Eleazar and the Zealots; they had in their possession the stores of first fruits and offerings, and were frequently in a state of intoxication; but when not drunken with wine, they thirsted for the blood of their countrymen, and issued from their strong hold, to assault John and his party, who lay intrenched in the out-works of the temple. The ruin of Jerusalem is attributed to the horrid enormities of the Zealot faction: surely that was the summit of wickedness, when the priests sold themselves to work iniquity, and the temple of the Lord was the seat of their crimes. That was "the overspreading of abomination,"
- 88. and it continued until the sanctuary was consumed, and "ruin was poured upon the desolators." It was the iniquitous practices of the Jews, rather than the Roman eagle, which profaned the courts of the Lord's House: the conquerors did not plant their standard to insult, but with a wish to preserve, the temple from total ruin and destruction.
- 89. CHAPTER LXVI. For I will gather all nations against Jerusalem to battle; and the city shall be taken, and the houses rifled, and the women ravished; and half of the city shall go forth into captivity, and the residue of the people shall not be cut off from the city.— Zechariah xiv. 2. Imperial Rome, to whom the world once bowed, and whose power could command armies from "all nations," had conquered Judea, and received from her the yearly tribute of her subjection:[138] but, through the oppression of the Roman governors, and the madness of the people, the standard of revolt was planted, and the Jews attempted to break their yoke of bondage. The Roman legions, inured to war, and accustomed to the shout of victory, hastened to subdue the rebellious Israelites: they passed from city to city, and from province to province; slaughter and death marked their course; the strife was desperate; the conflict bloody; the Jews fought like men determined to conquer or to die: two hundred and forty-seven thousand seven hundred were slain before their provinces were subjugated, and an immense number made prisoners: amongst whom was Josephus, the historian of the war, who was governor of the two Galilees, and who defended them with skill and bravery. The Romans, having conquered the provinces, approached to assault Jerusalem, which was then a dreadful scene. The sound of war was heard through all her gates; regardless of the approaching foe, the Jews had turned their arms against each other; three several factions were busily engaged in the work of slaughter and destruction. Eleazar and the Zealots seized the temple; John of Gischala and his followers occupied its out-works; and Simon, the son of Gorias, possessed the whole of the lower, and a great part of the upper, town. Jerusalem was built on two hills; the highest, on
- 90. which stood the temple, was called the upper town, and the other the lower: between these lay a valley covered with houses; the suburbs of the city were extensive, and encircled by a wall; two other walls also surrounded Jerusalem, the interior one of remarkable strength. Neither of the three factious parties had any just claim to supremacy or power, though all contended for dominion, and fought for plunder. The Zealots were the smallest party, but, from their situation, possessed the advantage: they sallied from their strong holds to attack John, who seized every opportunity of assaulting Simon; thus John maintained a double war, and was often obliged to divide his forces, being attacked by Eleazar and Simon at the same time. In these furious contests, no age or sex was spared; the slaughter was dreadful. When either party was repelled, the other set fire to the building, without any distinction. Regardless of their contents, they consumed granaries and store- houses, which contained a stock of corn and other necessaries of life, sufficient to maintain the inhabitants during a siege of many years; but nearly the whole was burnt, and this circumstance made way for a calamity more horrid than even war itself. Famine soon showed her meagre form, and all classes felt the dreadful effects of a scarcity of food. Such was the miserable state of Jerusalem when the Roman general Titus (son of the reigning emperor, Vespasian,) prepared to attack the city. The sight of a powerful foreign foe at their gates, with all the artillery of war, could not quell the factions within; it is true, when closely pressed by the Romans, the three parties joined to repel the common enemy, but no sooner had they breathing time, than the spirit of contention arose, and they resumed the slaughter of each other: thus they maintained a fierce contest with the besiegers, and, at the same time, seized every opportunity of destroying each other. The misery of the city was soon beyond precedent, from the dreadful effects of famine, the price of provisions became exorbitant, and, when no longer offered for sale, the houses were entered and searched, and the wretched owners tortured till they confessed where the slender pittance was concealed; at length the distress became so great, that persons parted with the whole of their property to obtain a bushel of wheat,
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