Lesson 1.2 the set of real numbers

Real Numbers
In mathematics, a is a
value that represents a quantity along a
continuous line which what you call a
.
-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6

Natural
N ⊂ U∴
Let be the
set of
.
Let be the
set of
.

Natural Numbers
These are numbers that are used for
counting. The set of number starts with 1.
Ex.
N = {1, 2, 3, 4, 5, 6, ...}
-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6
Number Line

Natural Numbers
These are numbers that are used for
counting. The set of number starts with 1.
Ex.
N = {1, 2, 3, 4, 5, 6, ...}
-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6
Number Line
1 2 3 4 5 6

Subsets of Real Numbers
Name Description Examples
Natural
Numbers
N = { 1, 2, 3, 4, 5, 6, ... }
These numbers are used for counting
2, 3, 4, 17, 31, 127

1. The set of natural numbers.
2. A = { x | x < 16, x is a natural number }
3. B = { x | 13 < x < 17, x is a natural number }
List the elements of the
following sets.
N = { 1, 2, 3, 4, ... }
A = { 1, 2, 3, 4, ... 15 }
B = { 14, 15, 16 }

Whole
Natural
Let be set
of
.
N ⊂ W
N ⊂ U
W ⊂ U
∴

Whole Numbers
are also natural
numbers that are formed by adding zero(0)
to the set.
Ex.
W = { 0, 1, 2, 3, 4, 5, 6, ...}
-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6
Number Line

41 2 3
Whole Numbers
are also natural
numbers that are formed by adding zero(0)
to the set.
Ex.
W = { 0, 1, 2, 3, 4, 5, 6, ...}
-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6
Number Line
5 6

Whole
Numbers
W = { 0, 1, 2, 3, 4, 5, ... }
These numbers are formed by adding 0 to
the set of natural numbers.
0, 2, 3, 4, 17, 31,
127
Natural
Numbers
N = { 1, 2, 3, 4, 5, 6, ... }
2, 3, 4, 17, 31, 127

1. The set of whole numbers.
2. A = { x | x < 99, x is a whole number}
3. B = { x | x ≤ 5, x is a whole number}
following sets.
W = { 0, 1, 2, 3, ... }
A = { 0, 1, 2, 3, ... 98 }
B = { 0, 1, 2, 3, 4, 5 }

Integers
Whole
Natural
Let be the
set of
.
N ⊂ W
N ⊂ I
N ⊂ U
W ⊂ I
W ⊂ U
I ⊂ U
∴

41 2 3-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6
Number Line
5 6
Integers
They are formed by adding the of
the natural numbers to the set of whole
numbers.
Ex.
I = {…, -4, -3, -2, -1, 0, 1, 2, 3, 4, …}

Whole
Numbers
W = { 0, 1, 2, 3, 4, 5, ... }
These numbers are formed by adding 0 to
the set of natural numbers.
0, 2, 3, 5, 8, 17
Natural
Numbers
N = { 1, 2, 3, 4, 5, 6, ... }
2, 3, 5, 8, 17
Integers
I = { ... , -4, -3, -2, -1, 0, 1, 2, 3, 4, ... }
They are formed by adding the negatives of
the natural number to the set of whole
numbers.
-19, -8, -5, -3, -2, 0,
2, 3, 5, 8, 17

1. The set of integers.
2. A = { x | 2 < x < -3, x is an integer }
3. B = { x | x ≥ -3, x is an integer }
following sets.
I = { ... , -3, -2, -1, 0, 1, 2, 3, ... }
A = { }
B = { -3, -2, -1, 0, 1 ... }

Rational Numbers
Integers
Whole
Natural
Let be the
set of
.
N ⊂ W
N ⊂ I
N ⊂ Q
N ⊂ U
W ⊂ I
W ⊂ Q
W ⊂ U
I ⊂ Q
I ⊂ U
∴ Q ⊂ U

Rational Number
The set of
is the set of all
numbers which can be
expressed in the formed:
The decimal representation of a rational
number or .
𝑎
𝑏Where a and b are
, b ≠ 0

Rational Number
The set of
is the set of all
number or .
𝑎
, b ≠ 0
𝑎
𝑏
1
8
Example 1:

Rational Number
The set of
is the set of all
Where a and b are
, b ≠ 0
−19
−5
𝑎
𝑏
Example 2:

Rational Number
The set of
is the set of all
Where a and b are
, b ≠ 0
40
−8
𝑎
𝑏
Example 3:

Rational Number
The set of
is the set of all
Where a and b are
, b ≠ 0
15
1
15=𝑎
𝑏
Example 4:

Rational Number
The set of
is the set of all
Where a and b are
, b ≠ 0
−3
1
-3=𝑎
𝑏
Example 5:

Rational Number
The set of
is the set of all
Where a and b are
, b ≠ 0
𝑎
𝑏
9 = 3
Example 6:

Rational Number
The set of
is the set of all
Where a and b are
, b ≠ 0
𝑎
𝑏
9 = 3=
3
1
Example 6:

Rational Number
The set of
is the set of all
Where a and b are
, b ≠ 0
𝑎
𝑏
25 = 5
Example 7:

5
1
Rational Number
The set of
is the set of all
Where a and b are
, b ≠ 0
𝑎
𝑏
525 ==
Therefore, we can
say that all
can be
expressed as rational
numbers.
Example 7:

0
5
Rational Number
The set of
is the set of all
Where a and b are
, b ≠ 0
𝑎
𝑏
5
0
Example 8:

Terminating and
Repeating Decimals
a decimal number that ends with a
remainder of zero. Ex. 1.25, 0.75, 1.5
a decimal number whose answer will have
one or more digits in a pattern that repeats
indefinitely. Ex. 0.33, 0.166, 0.55

Rational Number
The set of
is the set of all
number or .
𝑎
, b ≠ 0
The set of
is the set of all numbers
which can be expressed in
the formed
𝑎
𝑏
Where a and b are
, b ≠ 0
.75 =
3
4
Example 9:

Rational Number
number or .
0.25 =
1
4
The set of
the formed
𝑎
𝑏
Where a and b are
, b ≠ 0
Example 10:

Rational Number
number or .
0.5 =
1
2
The set of
the formed
𝑎
𝑏
Where a and b are
, b ≠ 0
Example 11:

Rational Number
number or .
1.5 =
3
2
The set of
the formed
𝑎
𝑏
Where a and b are
, b ≠ 0
Example 12:

Rational Number
number or .
0.33 =
1
3
The set of
the formed
𝑎
𝑏
Where a and b are
, b ≠ 0
Example 13:

Rational Number
number or .
0.66 =
2
3
The set of
the formed
𝑎
𝑏
Where a and b are
, b ≠ 0
Example 14:

Rational Number
number or .
1.66 =
1
6
The set of
the formed
𝑎
𝑏
Where a and b are
, b ≠ 0
Example 15:

Rational Numbers
More examples of rational numbers
1. 4
2. -27
3. -9
4. -32
5. 0.25
6. -0.5
7. 100
8. 169
9. 345
10. 0
=
4
1
=
−27
1
=
−9
1
=
−32
1
=
1
4
= −
1
2
=
10
1
=
13
1
=
345
1
=
0
1

-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6
Number Line
Plot the approximate location on the number
line of the following rational numbers.
1.
4
1 = 4

-29 -28 -27 -26 -25 -24 -23 -22 -21 -20 -19 -18 -17
Number Line
2.
−27
1 = -27

3.
0
1 = 0
-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6
Number Line

4. 36 = 6
-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6
Number Line

5. 4 = 2
-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6
Number Line

-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6
Number Line
6.
−3
4 = -0.75

-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6
Number Line
7.
7
4 = 1.75

-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6
Number Line
8.
−5
3
= -1.66

-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6
Number Line
9.
−5
−5 = 1

-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6
Number Line
10.
9
0 = Undefined and it is not
a rational number.

Whole Numbers
W = { 0, 1, 2, 3, 4, 5, ... }
These numbers are formed by adding 0 to the set
of natural numbers.
0, 2, 3, 5, 8, 17
Natural Numbers
N = { 1, 2, 3, 4, 5, 6, ... }
2, 3, 5, 8, 17
Integers
I = { ... , -4, -3, -2, -1, 0, 1, 2, 3, 4, ... }
They are formed by adding the negatives of the
natural number to the set of whole numbers.
-19, -8, -5, -3, -2, 0, 2,
3, 5, 8, 17
Rational
Numbers
The set of rational numbers is the set of all
numbers which can be expressed in the formed
𝑎
𝑏
, where a and b are integers, b ≠ 0. The
decimal representation of a rational number
either terminates or repeats
-19, -8, -5, -3, -2, 0, 2,
3, 5, 8, 17

Rational Numbers
Integers
Whole
Natural
Let be the
set of
N ⊂ W
N ⊂ I
N ⊂ Q
N ⊂ U
W ⊂ I
W ⊂ Q
W ⊂ U
I ⊂ Q
I ⊂ U
∴ Q ⊂ U
P ⊂ U
Irrational Numbers

Irrational Number
The set of is the set of
numbers whose decimal representations are
nor .
These numbers be
expressed as a
.
𝑎
𝑏

Irrational Number
The set of is the set of
nor .
𝑎
𝑏
The set of is the set of numbers whose
decimal representations are nor
.
𝑎
𝑏
These numbers be expressed as a
.
1.414213562373095048801688724
2097
Example 1:
These numbers be
expressed as a
.

Irrational Number
.
𝑎
𝑏
.
1.414213562373095048801688724
2097
Example 1:
-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6
Number Line

Irrational Number
.
.
Example 2:
6.782329983125268139064556326
626
Number Line
𝑎
𝑏
-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7

Irrational Number
1.73205080756887729352
744634150593 =
Example 3:
.
𝑎
𝑏
.
-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6
Number Line

Irrational Number
3.60551275463989293119
221267470513=
Example 4:
.
𝑎
𝑏
.
-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6
Number Line

Irrational Number
6.08276253029821968899
9684245202137=
Example 5:
.
𝑎
𝑏
.
-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6
Number Line

Irrational Number
1.41421356237309504880
168872420972 =
Example 6:
.
𝑎
𝑏
.
-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6
Number Line

Irrational Number
2
Notice that:
3, 13,37,
These numbers are not perfect squares. Their
equivalent decimals are nor
and cannot be expressed quotient of
integers.
Therefore, we can say that numbers that are
can be considered as
.

Irrational Number
3.14159265358979323846
26433832795
is also considered as an
for its decimal representation are
or and it cannot be
expressed as .
𝜋 =
And for the last example:

Whole Numbers
W = { 0, 1, 2, 3, 4, 5, ... }
These numbers are formed by adding 0 to the set
of natural numbers.
0, 2, 3, 5, 8, 17
Natural Numbers
N = { 1, 2, 3, 4, 5, 6, ... }
2, 3, 5, 8, 17
Integers
I = { ... , -4, -3, -2, -1, 0, 1, 2, 3, 4, ... }
They are formed by adding the negatives of the
natural number to the set of whole numbers.
-19, -8, -5, -3, -2, 0, 2,
3, 5, 8, 17
Rational
Numbers
The set of rational numbers is the set of all
numbers which can be expressed in the formed
𝑎
𝑏
, where a and b are integers, b ≠ 0. The
decimal representation of a rational number
either terminates or repeats
-19=
−19
1
, -8=
−8
1
, -5=
−8
1
,
-3, -2, 0, 2, 3, 5, 8, 17,
3
5
= 0.6,
−2
3
=-0.666
Irrational
Numbers
The set of irrational numbers is the set of
neither terminating nor repeating. These
numbers cannot be expressed as a quotient of
integers.
- 2 = 1.414214
3 = 1.73205
Π = 3.1416

Rational Numbers
Integers
Whole
Natural
Irrational Numbers

Check the set(s) to which each
number belongs.
Set -12 21
−3
7
0 5 3.45
Natural Numbers
Whole Numbers
Integers
Rational Numbers
Irrational Numbers
Real Numbers

E-Math 7
Vocabulary and Concepts
Practice and Application
Test I, II, III, IV
Page 15 -16

Lesson 1.2 the set of real numbers

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Lesson 1.2 the set of real numbers