Rational Numbers
Joseph Lee
Metropolitan Community College
Joseph Lee
Rational Numbers
Goals
The goals of this section are:
Joseph Lee
Rational Numbers
Goals
The goals of this section are:
1
to introduce the integers;
Joseph Lee
Rational Numbers
Goals
The goals of this section are:
1
to introduce the integers;
2
to establish a definition for a rational number;
Joseph Lee
Rational Numbers
Goals
The goals of this section are:
1
to introduce the integers;
2
to establish a definition for a rational number;
3
to be able to recognize a rational number; and
Joseph Lee
Rational Numbers
Goals
The goals of this section are:
1
to introduce the integers;
2
to establish a definition for a rational number;
3
to be able to recognize a rational number; and
4
to assert that there are numbers that are not rational.
Joseph Lee
Rational Numbers
Definition: Natural Numbers
The natural numbers, or the counting numbers, are the set of
numbers
{1, 2, 3, 4, 5, ...}.
Joseph Lee
Rational Numbers
Definition: Whole Numbers
The whole numbers consist of the natural numbers and zero, i.e.
{0, 1, 2, 3, 4, 5, ...}.
Joseph Lee
Rational Numbers
Definition: Integers
The integers consist of the whole numbers and the negative
natural numbers, i.e.
{..., −5, −4, −3, −2, −1, 0, 1, 2, 3, 4, 5, ...}.
Joseph Lee
Rational Numbers
Definition: Rational Numbers
A rational number is any number that can be written as
a
b
where a and b are integers and b 6= 0.
Joseph Lee
Rational Numbers
Example 1
Are the following numbers rational?
3
1
4
Joseph Lee
Rational Numbers
Example 1
Are the following numbers rational?
3
1
yes
4
2
−
5
6
Joseph Lee
Rational Numbers
Example 1
Are the following numbers rational?
3
1
yes
4
2
−
3
2
5
6
yes
Joseph Lee
Rational Numbers
Example 1
Are the following numbers rational?
3
1
yes
4
5
6
2
−
3
2
4
−17
yes
yes
Joseph Lee
Rational Numbers
Example 1
Are the following numbers rational?
3
1
yes
4
5
6
2
−
3
2
4
−17
5
0
yes
yes
yes
Joseph Lee
Rational Numbers
Example 1
Are the following numbers rational?
3
1
yes
4
5
6
2
−
3
2
4
−17
5
0
6
0.9
yes
yes
yes
yes
Joseph Lee
Rational Numbers
Example 1
Are the following numbers rational?
3
1
yes
4
5
6
2
−
3
2
4
−17
5
0
6
0.9
7
−3.45
yes
yes
yes
yes
yes
Joseph Lee
Rational Numbers
Example 1
Are the following numbers rational?
3
1
yes
4
5
6
2
−
3
2
4
−17
5
0
6
0.9
7
−3.45
8
2
yes
yes
yes
yes
1
3
yes
yes
yes
Joseph Lee
Rational Numbers
Is every number rational?
Are all numbers rational?
Joseph Lee
Rational Numbers
Is every number rational?
Are all numbers rational?
In other words, can you write any number as
integers (b 6= 0)?
Joseph Lee
a
b
Rational Numbers
where a and b are
Is every number rational?
Are all numbers rational?
In other words, can you write any number as
integers (b 6= 0)?
a
b
where a and b are
The answer to this question was discovered over 2000 years ago by
Greek mathematicians. The answer is no – not every number can
be written as a fraction of integers.
Joseph Lee
Rational Numbers
Definition: Irrational Numbers
An irrational number is a real number that is not rational.
Joseph Lee
Rational Numbers
Definition: Irrational Numbers
An irrational number is a real number that is not rational.
The following numbers are irrational:
√ √ √ √ √ √ √
2, 3, 5, 6, 7, 8, 10, ..., π, ...
Joseph Lee
Rational Numbers
Definition: Irrational Numbers
An irrational number is a real number that is not rational.
The following numbers are irrational:
√ √ √ √ √ √ √
2, 3, 5, 6, 7, 8, 10, ..., π, ...
We will only assert these numbers are irrational. If you are
interested in discovering why these numbers cannot be written as
fractions of integers, I would suggest taking every math class we
offer here at Metro, and then entering a 4 year university to study
mathematics.
Joseph Lee
Rational Numbers
Definition: Real Numbers
A real number is any number that can be represented on the real
number line.
Joseph Lee
Rational Numbers
Example 2.a
Determine which of the following numbers are natural numbers
(positive integers)?
√
14
−5, 17, 0, , −3.89, 11
5
Joseph Lee
Rational Numbers
Example 2.a
Determine which of the following numbers are natural numbers
(positive integers)?
√
14
−5, 17, 0, , −3.89, 11
5
Solution. 11
Joseph Lee
Rational Numbers
Example 2.b
Determine which of the following numbers are whole numbers?
√
14
−5, 17, 0, , −3.89, 11
5
Joseph Lee
Rational Numbers
Example 2.b
Determine which of the following numbers are whole numbers?
√
14
−5, 17, 0, , −3.89, 11
5
Solution. 0, 11
Joseph Lee
Rational Numbers
Example 2.c
Determine which of the following numbers are integers?
√
14
−5, 17, 0, , −3.89, 11
5
Joseph Lee
Rational Numbers
Example 2.c
Determine which of the following numbers are integers?
√
14
−5, 17, 0, , −3.89, 11
5
Solution. −5, 0, 11
Joseph Lee
Rational Numbers
Example 2.d
Determine which of the following numbers are rational?
√
14
−5, 17, 0, , −3.89, 11
5
Joseph Lee
Rational Numbers
Example 2.d
Determine which of the following numbers are rational?
√
14
−5, 17, 0, , −3.89, 11
5
Solution. −5, 0,
14
, −3.89, 11
5
Joseph Lee
Rational Numbers
Example 2.e
Determine which of the following numbers are irrational?
√
14
−5, 17, 0, , −3.89, 11
5
Joseph Lee
Rational Numbers
Example 2.e
Determine which of the following numbers are irrational?
√
14
−5, 17, 0, , −3.89, 11
5
√
Solution. 17
Joseph Lee
Rational Numbers
Example 2.f
Determine which of the following numbers are real?
√
14
−5, 17, 0, , −3.89, 11
5
Joseph Lee
Rational Numbers
Example 2.f
Determine which of the following numbers are real?
√
14
−5, 17, 0, , −3.89, 11
5
√
14
Solution. −5, 17, 0, , −3.89, 11
5
Joseph Lee
Rational Numbers
Example 3.a
Determine which of the following numbers are natural numbers
(positive integers)?
√
3
2 , −1, 0.5, 6, 2, −4.5
8
Joseph Lee
Rational Numbers
Example 3.a
Determine which of the following numbers are natural numbers
(positive integers)?
√
3
2 , −1, 0.5, 6, 2, −4.5
8
Solution. 6
Joseph Lee
Rational Numbers
Example 3.b
Determine which of the following numbers are whole numbers?
√
3
2 , −1, 0.5, 6, 2, −4.5
8
Joseph Lee
Rational Numbers
Example 3.b
Determine which of the following numbers are whole numbers?
√
3
2 , −1, 0.5, 6, 2, −4.5
8
Solution. 6
Joseph Lee
Rational Numbers
Example 3.c
Determine which of the following numbers are integers?
√
3
2 , −1, 0.5, 6, 2, −4.5
8
Joseph Lee
Rational Numbers
Example 3.c
Determine which of the following numbers are integers?
√
3
2 , −1, 0.5, 6, 2, −4.5
8
Solution. −1, 6
Joseph Lee
Rational Numbers
Example 3.d
Determine which of the following numbers are rational?
√
3
2 , −1, 0.5, 6, 2, −4.5
8
Joseph Lee
Rational Numbers
Example 3.d
Determine which of the following numbers are rational?
√
3
2 , −1, 0.5, 6, 2, −4.5
8
3
Solution. 2 , −1, 0.5, 6, −4.5
8
Joseph Lee
Rational Numbers
Example 3.e
Determine which of the following numbers are irrational?
√
3
2 , −1, 0.5, 6, 2, −4.5
8
Joseph Lee
Rational Numbers
Example 3.e
Determine which of the following numbers are irrational?
√
3
2 , −1, 0.5, 6, 2, −4.5
8
√
Solution. 2
Joseph Lee
Rational Numbers
Example 3.f
Determine which of the following numbers are real?
√
3
2 , −1, 0.5, 6, 2, −4.5
8
Joseph Lee
Rational Numbers
Example 3.f
Determine which of the following numbers are real?
√
3
2 , −1, 0.5, 6, 2, −4.5
8
√
3
Solution. 2 , −1, 0.5, 6, 2, −4.5
8
Joseph Lee
Rational Numbers