The Real Numbers4-8

Warm UpWarm Up

Lesson PresentationLesson Presentation

Problem of the DayProblem of the Day

Lesson QuizzesLesson Quizzes

The Real Numbers4-8

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

–4 and –3

1.4

–11.1

1. 119

2. – 15

3. 2

4. – 123

The Real Numbers4-8

Problem of the Day

The circumference of a circle is approximately 3.14 times its diameter. A circular path 1 meter wide has an inner diameter of 100 meters. How much farther is it around the outer edge of the path than the inner edge?

6.28 m

The Real Numbers4-8

MA.8.A.6.2 …Compare mathematical expressions involving real numbers and radical expressions.

Sunshine State Standards

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irrational numberreal numberDensity Property

Vocabulary

The Real Numbers4-8

AnimalsVertebrates

Biologists classify animals based on shared characteristics. The cardinal is an animal, a vertebrate, a bird, and a passerine.

You already know that some numbers can be classified as whole numbers, integers, or rational numbers.

Birds

Passerines

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Recall that rational numbers can be written as fractions and as decimals that either terminate or repeat.

3 = 3.84 5

= 0.623

1.44 = 1.2

The Real Numbers4-8

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 ≈1.41421356237…

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

Caution!

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

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Additional 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.

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Check It Out: Example 1

Write all names that apply to each number.

9

natural, whole, integer, rational, real

35

rational, real

rational, real

A.

B.

C.

9 = 3;

–

3.724

The Real Numbers4-8

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

0 3

rational

0 3

= 0

Additional Example 2: Determining the Classification of All Numbers

A.

not a real number

–4B.

The Real Numbers4-8

Additional Example 2: Determining the Classification of All Numbers

4 9

rational

2 3

=2 3

4 9C.

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

The Real Numbers4-8

–7

√–7 is not a real number because it is the square root of a negative number.

6 0

is not a real number because division by 0 is not

defined.

Check It Out: Example 2

A.

B.

State if the number is rational, irrational, or not a real number. Justify your answer.

–

–6 0

The Real Numbers4-8

= √64 = 8 is a rational number.

√59 + 5C.

Check It Out: Example 2

State if the number is rational, irrational, or not a real number. Justify your answer.

√59 + 5

The Real Numbers4-8

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.

The Real Numbers4-8

Additional 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

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Check It Out: Example 3

1 8

5 + 52 8

83 8=

Find a real number between 5 and 5 .

2 8

1 8

1 2

1 2

83 8

=

3 16

= 5

Possible answer: 3 16

5

The Real Numbers4-8

Standard Lesson Quiz

Lesson Quizzes

Lesson Quiz for Student Response Systems

The Real Numbers4-8Lesson Quiz

Write all names that apply to each number.

1. 2. –

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

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; by 0

rational; 2 3 is 6

real, irrational real, integer, rational

Possible answer –2 .5 8

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1. Identify all names that apply to .

A. real, irrational

B. real, rational

C. not real, irrational

D. not real, rational

Lesson Quiz for Student Response Systems

The Real Numbers4-8

2. Identify the name that applies to .

A. irrational

B. rational

C. not a real number

D. none

Lesson Quiz for Student Response Systems

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3. Identify a real number between .

A. –4

B.

C.

D.

Lesson Quiz for Student Response Systems