Evaluating Algebraic Expressions

4-8 The Real Numbers

Warm Up

California StandardsCalifornia Standards

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Evaluating Algebraic Expressions

4-8 The Real Numbers

Warm UpEach square root is between two integers. Name the two integers.

Use a calculator to find each value. Round to the nearest tenth.

10 and 11

3 and 4

1.4

11.1

1. 119

2. 15

3. 2

4. 123

Evaluating Algebraic Expressions

4-8 The Real Numbers

NS1.4 Differentiate between rational and irrational numbers.

California Standards

Evaluating Algebraic Expressions

4-8 The Real Numbers

real numberirrational numberDensity Property

Vocabulary

Evaluating Algebraic Expressions

4-8 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

Evaluating Algebraic Expressions

4-8 The Real Numbers

The set of real numbers is all numbers that can be written on a number line. It consists of the set of rational numbers and the set of irrational numbers.

Irrational numbersRational numbers

Real Numbers

Integers

Wholenumbers

Evaluating Algebraic Expressions

4-8 The Real Numbers

Recall that rational numbers can be written as the quotient of two integers (a fraction) or as either terminating or repeating decimals.

3 = 3.84 5

= 0.623

1.44 = 1.2

Evaluating Algebraic Expressions

4-8 The Real Numbers

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

Caution!

Irrational numbers can be written only as decimals that do not terminate or repeat. They cannot be written as the quotient of two integers. If a whole number is not a perfect square, then its square root is an irrational number. For example, 2 is not a perfect square, so 2 is irrational.

Evaluating Algebraic Expressions

4-8 The Real NumbersAdditional Example 1: Classifying Real Numbers

Write all classifications 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.

Evaluating Algebraic Expressions

4-8 The Real NumbersCheck It Out! Example 1

Write all classifications 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

Evaluating Algebraic Expressions

4-8 The Real Numbers

A fraction with a denominator of 0 is undefined because you cannot divide by zero. So it is not a number at all.

Evaluating Algebraic Expressions

4-8 The Real Numbers

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

21

irrational

0 3

rational

0 3

= 0

Additional Example 2: Determining the Classification of All Numbers

A.

B.

Evaluating Algebraic Expressions

4-8 The Real Numbers

not a real number

Additional Example 2: Determining the Classification of All Numbers

4 0

C.

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

Evaluating Algebraic Expressions

4-8 The Real Numbers

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

23

irrational

9 0

undefined, so not a real number

Check It Out! Example 2

A.

B.

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

Evaluating Algebraic Expressions

4-8 The Real Numbers

64 81

rational

8 9

=8 9

64 81

C.

Check It Out! Example 2

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

Evaluating Algebraic Expressions

4-8 The Real Numbers

The Density Property of real numbers states that between any two real numbers is another real number. This property is not true when you limit yourself to whole numbers or integers. For instance, there is no integer between –2 and –3.

Evaluating Algebraic Expressions

4-8 The Real NumbersAdditional 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

Check: Use a graph.

Evaluating Algebraic Expressions

4-8 The Real NumbersCheck 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

Check: Use a graph.

Evaluating Algebraic Expressions

4-8 The Real NumbersLesson Quiz

Write all classifications 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