Course 3

4-7 The Real NumbersN1.h How Do I Distinguish Between Rational And Irrational Numbers?

Course 3

Lesson Presentation

Course 3

4-7 The Real Numbers

Learn to determine if a number is rational or irrational.

Course 3

4-7 The Real Numbers

real numberrational numberirrational number

Vocabulary

Course 3

4-7 The Real Numbers

Recall that rational numbers can be written as fractions. Rational numbers can also be written as decimals that either terminate or repeat.3 = 3.84

5= 0.62

31.44 = 1.2

Course 3

4-7 The Real Numbers

Irrational numbers cannot be written as a fraction & can only be written as decimals that do not terminate or repeat (NON-TERMINATING/NON-REPEATING DECIMALS. The square root of a non-perfect square is an irrational number.

For example,

2 ≈1.4142135623730950488016…

A repeating decimal may not appear to repeat on a calculator, because calculators show a finite number of digits.

Caution!

Course 3

4-7 The Real Numbers

The set of real numbers consists of the set of rational numbers and the set of irrational numbers.

Irrational numbersRational numbers

Real Numbers

Integers

Wholenumbers

Course 3

4-7 The Real Numbers

Example 1: Classifying Real Numbers

Write all names that apply to each number.

5 is a whole number that is not a perfect square.

5

irrational, real

–12.75 is a terminating decimal.–12.75rational, real

16 2

whole, integer, rational, real

= = 24 2

16 2

A.

B.

C.

Course 3

4-7 The Real Numbers

Check It Out: Example 1

Write all names that apply to each number.

9

whole, integer, rational, real

–35.9 is a terminating decimal.–35.9rational, real

81 3

whole, integer, rational, real

= = 39 3

81 3

A.

B.

C.

9 = 3

Course 3

4-7 The Real Numbers

State if each number is rational, irrational, or not a real number.

21

irrational

0 3

rational

0 3

= 0

Example 2: Determining the Classification of All Numbers

A.

B.

Course 3

4-7 The Real Numbers

not a real number

Example 2: Determining the Classification of All Numbers

–4

4 9

rational

2 3

=2 3

4 9

C.

D.

State if each number is rational, irrational, or not a real number.

Course 3

4-7 The Real Numbers

23 is a whole number that is not a perfect square.

23

irrational

9 0

not a number, so not a real number

Check It Out: Example 2

A.

B.

State if each number is rational, irrational, or not a real number.

Course 3

4-7 The Real Numbers

not a real number

–7

64 81

rational

8 9

=8 9

64 81

C.

D.

Check It Out: Example 2

State if each number is rational, irrational, or not a real number.

Course 3

4-7 The Real Numbers

The Density Property of real numbers states that between any two real numbers is another real number. This property is also true for rational numbers, but not for whole numbers or integers. For instance, there is no integer between –2 and –3.

Course 3

4-7 The Real Numbers

Example 3: Applying the Density Property of Real Numbers

2 5

3 + 3 ÷ 23 5

There are many solutions. One solution is halfway between the two numbers. To find it, add the numbers and divide by 2.

5 5

= 6 ÷ 21 2

= 7 ÷ 2 = 3

31 2

3 3 31 5

2 5 43 33

54 5

Find a real number between 3 and 3 .

3 5

2 5

A real number between 3 and 3 is 3 .3 5

2 5

1 2

Course 3

4-7 The Real Numbers

Check It Out: Example 3

3 7

4 + 4 ÷ 24 7

There are many solutions. One solution is halfway between the two numbers. To find it, add the numbers and divide by 2.

7 7= 8 ÷ 2

1 2= 9 ÷ 2 = 4

41 2

4 44 4 4 42 7

3 7

4 7

5 7

1 7

6 7

Find a real number between 4 and 4 .

4 7

3 7

A real number between 4 and 4 is 4 .4 7

3 7

1 2

Course 3

4-7 The Real NumbersLesson Quiz

Write all names that apply to each number.

1. 2. –

State if each number is rational, irrational, or not a real number.

3. 4.

Find a real number between –2 and –2 .3 8

3 4

5.

2

4 • 9

16 2

25 0

not a real number rational

real, irrational real, integer, rational

Possible answer –2 .5 8