G7 Math Q1 Week 6- Rational and Irrational Numbers.ppt

Rational and Irrational
Rational and Irrational
5
7 2
1

Rational and Irrational Numbers
Rational Numbers
A rational number is any number that can be expressed as the
ratio of two integers.
Examples
All terminating and repeating decimals can be expressed in
this way so they are rational numbers.
a
b
4
5
2 2
3
=
8
3
6 =
6
1
2.7 =
27
10
0.625 =
5
8 34.56 =
3456
100
-3=
3
1
-
0.3=
1
3 0.27 = 3
11 0.142857 = 1
7
0.7=
7
10

Rational Numbers
this way so they are irrational numbers.
a
b
Show that the terminating decimals below are rational.
0.6 3.8 56.1 3.45 2.157
6
10
38
10
561
10
345
100
2157
1000
Rational

Rational Numbers
a
b
To show that a repeating decimal is rational.
Example 1
To show that 0.333… is rational.
Let x = 0.333…
10x = 3.33…
9x = 3
x = 3/9
x = 1/3
Example 2
To show that 0.4545… is rational.
Let x = 0.4545…
100x = 45.45…
99x = 45
x = 45/99
x = 5/11

Rational Numbers
a
b
Question 1
Show that 0.222… is rational.
Let x = 0.222…
10x = 2.22…
9x = 2
x = 2/9
Question 2
Show that 0.6363… is rational.
Let x = 0.6363…
100x = 63.63…
99x = 63
x = 63/99
x = 7/11

Rational Numbers
a
b
999x = 273
x = 273/999
9999x = 1234
x = 1234/9999
Question 3
Show that 0.273is rational.
Let x = 0.273
1000x = 273.273
x = 91/333
Question 4
Show that 0.1234 is rational.
Let x = 0.1234
10000x = 1234.1234

Rational Numbers
a
b
By looking at the previous examples can you spot a quick method of
determining the rational number for any given repeating decimal.
0.1234
1234
9999
0.273
273
999
0.45
45
99
0.3
3
9

Rational Numbers
a
b
0.1234
1234
9999
0.273
273
999
0.45
45
99
0.3
3
9
Write the repeating part of the decimal as the numerator and write the
denominator as a sequence of 9’s with the same number of digits as the
numerator then simplify where necessary.

Rational Numbers
a
b
1543
9999
628
999
32
99
7
9
0.1543 0.628
0.32 0.7
Write down the rational form for each of the repeating decimals below.

Squares and Roots
12 1 1 1
22
32


4
9
4
9


2
3
42
52
62
72
82
92






16
25
36
49
64
81
16
25
36
49
64
81






4
5
6
7
8
9
102
112
122



100
121
144
100
121
144



10
11
12
These are
These are
Rational Numbers.
Rational Numbers.
Why?
Why?

Irrational
a
b
Irrational Numbers
An irrational number is any number that cannot be
expressed as the ratio of two integers.
1
1
2
Pythagoras
The history of irrational numbers begins with a
discovery by the Pythagorean School in ancient
Greece. A member of the school discovered that
the diagonal of a unit square could not be
expressed as the ratio of any two whole
numbers. The motto of the school was “All is
Number” (by which they meant whole numbers).
Pythagoras believed in the absoluteness of whole
numbers and could not accept the discovery. The
member of the group that made it was Hippasus
and he was sentenced to death by drowning.
(See slide 19/20 for more history)

1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
Can you cite rational and
Can you cite rational and
irrational numbers?
irrational numbers?

a
b
Irrational Numbers
1
1
2
Pythagoras
Intuition alone may convince you that all points
on the “Real Number” line can be constructed
from just the infinite set of rational numbers,
after all between any two rational numbers we
can always find another. It took
mathematicians hundreds of years to show that
the majority of Real Numbers are in fact
irrational. The rationals and irrationals are
needed together in order to complete the
continuum that is the set of “Real Numbers”.

a
b
Irrational Numbers
1
1
2
Pythagoras
Surds are Irrational Numbers
3
1
27
1
2
1
4
1 3 
 and
We can simplify numbers such as
into rational numbers.
However, other numbers involving
roots such as those shown cannot
be reduced to a rational form.
3
12
8
2 ,
,
Any number of the form which cannot be written
as a rational number is called a surd.
n
m
All irrational numbers are non-terminating, non-repeating decimals.
Their decimal expansion form shows no pattern whatsoever.
Other irrational numbers include and e, (Euler’s number)


Combining Rationals and Irrationals
Determine whether the following are rational or irrational.
(a) 0.73 (b) (c) 0.666…. (d) 3.142 (e) .25
12
(f) (g) (h) (i) (j)
7 5
4  1
3

)
2
(3
162
1
2
3
2)
(
(j) (k) (l)
1)
3
1)(
( 

3 )
6
1)(
( 1
6 
 )
)(
( 2
1
2
1 

irrational
rational rational rational irrational
irrational irrational rational rational irrational
irrational rational rational
2

The Square Root of 2 is Irrational
1
1
This is a “reductio-ad-absurdum” proof.
To prove that is irrational
Assume the contrary: 2 is rational
That is, there exist integers p and q with no common factors such that:
2

q
p
2
2
2


q
p
even
is
p
q
p 2 2
2



(Since 2q2
is even, p2
is even so p even) So p = 2k for some k.
.
, even
is
q
p
q
q
p
as
Also
2
2
2
2
2
2




(Since p is even is even, q2
is even so q is even)
2
2
p
So q = 2m for some m.

p
q

2k
2m

p
q
have a factor of 2 in common.
This contradicts the original assumption.
is irrational. QED
(odd2
= odd)
Proof
2
2
2

√
√2, √3, √5, √6, √7,
2, √3, √5, √6, √7,
√8,√10, pi, 2√2
√8,√10, pi, 2√2

G7 Math Q1 Week 6- Rational and Irrational Numbers.ppt

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G7 Math Q1 Week 6- Rational and Irrational Numbers.ppt