Course 3

4-7 The Real Numbers

Warm Up 1-28-09Each square root is between two integers. Name the two integers.

Estimate each value. Round to the nearest tenth.

10 and 11

–4 and –3

1.4

–11.1

1. 119

2. – 15

3. 2

4. – 123

Course 3

4-7 The Real Numbers

Learn to determine if a number is rational or irrational.

Course 3

4-7 The Real Numbers

irrational numberreal numberDensity Property

Vocabulary

Course 3

4-7 The Real Numbers

AnimalReptile

Biologists classify animals based on shared characteristics. The horned lizard is an animal, a reptile, a lizard, and a gecko.

You already know that some numbers can be classified as whole numbers, integers, or rational numbers. The number 2 is a whole number, an integer, and a rational number. It is also a real number.

LizardGecko

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

= 0.623

1.44 = 1.2

Course 3

4-7 The Real Numbers

Irrational numbers can only be written as decimals that do not terminate or repeat. If a whole number is not a perfect square, then its square root is an irrational number.

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

Rationals

Integers

Wholes

Naturals

Irrationals

What are the different types of numbers?Real Numbers

1

3

Course 3

4-7 The Real Numbers

9

Fill In Your Real Number Chart

Counting “Natural” Numbers: 1, 2, 3, 4, 5, 6, . . .

Whole Numbers: 0, 1, 2, 3, 4, . . .

Integers: . . . -3, -2, -1, 0, 1, 2, 3, 4. . .

Rational Numbers: 0, …1/10, …1/5, …1/4, ... 33, …1/2, …1, perfect squares

Real Numbers: all numbers

Irrationals: π, non-repeating decimal, nonperfect squares

Course 3

4-7 The Real Numbers

Classifying Real Numbers

Write all names that apply to each number (whole, integer, rational,

irrational, real)

Course 3

4-7 The Real Numbers

Example 1

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

Example 2

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

Determining the Classification of All

NumbersState if each number is rational, irrational,

or not a real number.

Course 3

4-7 The Real Numbers

21

irrational

0 3

rational

0 3

= 0

Example 3

A.

B.

Course 3

4-7 The Real Numbers

not a real number

Example 3 continued..

–4

4 9

rational

2 3

=2 3

4 9

C.

D.

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

Example 4

A.

B.

Course 3

4-7 The Real Numbers

not a real number

–7

64 81

rational

8 9

=8 9

64 81

C.

D.

Example 4 Continued…

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

Find a real number between a set of numbers

There are many solutions. Let’s try to find the solution that is halfway

between the two numbers

Course 3

4-7 The Real Numbers

Example 5

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

Example 6

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 Summary

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