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Course 3 4-7 The Real Numbers 4-7 The Real Numbers Course 3 Warm Up Problem of the Day Lesson Presentation

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Page 1: Course 3 4-7 The Real Numbers 4-7 The Real Numbers Course 3 Warm Up Warm Up Problem of the Day Problem of the Day Lesson Presentation Lesson Presentation

Course 3

4-7 The Real Numbers

4-7 The Real Numbers

Course 3

Warm Up

Problem of the Day

Lesson Presentation

Page 2: Course 3 4-7 The Real Numbers 4-7 The Real Numbers Course 3 Warm Up Warm Up Problem of the Day Problem of the Day Lesson Presentation Lesson Presentation

Course 3

4-7 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

–4 and –3

1.4

–11.1

1. 119

2. – 15

3. 2

4. – 123

Page 3: Course 3 4-7 The Real Numbers 4-7 The Real Numbers Course 3 Warm Up Warm Up Problem of the Day Problem of the Day Lesson Presentation Lesson Presentation

Course 3

4-7 The Real Numbers

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

Page 4: Course 3 4-7 The Real Numbers 4-7 The Real Numbers Course 3 Warm Up Warm Up Problem of the Day Problem of the Day Lesson Presentation Lesson Presentation

Course 3

4-7 The Real Numbers

Learn to determine if a number is rational or irrational.

Page 5: Course 3 4-7 The Real Numbers 4-7 The Real Numbers Course 3 Warm Up Warm Up Problem of the Day Problem of the Day Lesson Presentation Lesson Presentation

Course 3

4-7 The Real Numbers

irrational numberreal numberDensity Property

Vocabulary

Page 6: Course 3 4-7 The Real Numbers 4-7 The Real Numbers Course 3 Warm Up Warm Up Problem of the Day Problem of the Day Lesson Presentation Lesson Presentation

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

Page 7: Course 3 4-7 The Real Numbers 4-7 The Real Numbers Course 3 Warm Up Warm Up Problem of the Day Problem of the Day Lesson Presentation Lesson Presentation

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

Page 8: Course 3 4-7 The Real Numbers 4-7 The Real Numbers Course 3 Warm Up Warm Up Problem of the Day Problem of the Day Lesson Presentation Lesson Presentation

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!

Page 9: Course 3 4-7 The Real Numbers 4-7 The Real Numbers Course 3 Warm Up Warm Up Problem of the Day Problem of the Day Lesson Presentation Lesson Presentation

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

Page 10: Course 3 4-7 The Real Numbers 4-7 The Real Numbers Course 3 Warm Up Warm Up Problem of the Day Problem of the Day Lesson Presentation Lesson Presentation

Course 3

4-7 The Real Numbers

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.

Page 11: Course 3 4-7 The Real Numbers 4-7 The Real Numbers Course 3 Warm Up Warm Up Problem of the Day Problem of the Day Lesson Presentation Lesson Presentation

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

Page 12: Course 3 4-7 The Real Numbers 4-7 The Real Numbers Course 3 Warm Up Warm Up Problem of the Day Problem of the Day Lesson Presentation Lesson Presentation

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

Additional Example 2: Determining the Classification of All Numbers

A.

B.

Page 13: Course 3 4-7 The Real Numbers 4-7 The Real Numbers Course 3 Warm Up Warm Up Problem of the Day Problem of the Day Lesson Presentation Lesson Presentation

Course 3

4-7 The Real Numbers

not a real number

Additional 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.

Page 14: Course 3 4-7 The Real Numbers 4-7 The Real Numbers Course 3 Warm Up Warm Up Problem of the Day Problem of the Day Lesson Presentation Lesson Presentation

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.

Page 15: Course 3 4-7 The Real Numbers 4-7 The Real Numbers Course 3 Warm Up Warm Up Problem of the Day Problem of the Day Lesson Presentation Lesson Presentation

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.

Page 16: Course 3 4-7 The Real Numbers 4-7 The Real Numbers Course 3 Warm Up Warm Up Problem of the Day Problem of the Day Lesson Presentation Lesson Presentation

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.

Page 17: Course 3 4-7 The Real Numbers 4-7 The Real Numbers Course 3 Warm Up Warm Up Problem of the Day Problem of the Day Lesson Presentation Lesson Presentation

Course 3

4-7 The Real Numbers

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

Page 18: Course 3 4-7 The Real Numbers 4-7 The Real Numbers Course 3 Warm Up Warm Up Problem of the Day Problem of the Day Lesson Presentation Lesson Presentation

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

Page 19: Course 3 4-7 The Real Numbers 4-7 The Real Numbers Course 3 Warm Up Warm Up Problem of the Day Problem of the Day Lesson Presentation Lesson Presentation

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