Page 152 #19-38 ANSWERS

Pre-Algebra

3-10 The Real Numbers

Student Learning Goal Chart

Lesson Reflections

3-10 LAST ONE!

Pre-Algebra Learning Goal

Students will understand rational and real numbers.

Students will understand rational and real numbers by being able to do the following:

• Learn to write rational numbers in equivalent forms (3.1)

• Learn to add and subtract decimals and rational numbers with like denominators (3.2)

• Learn to add and subtract fractions with unlike denominators (3.5)

• Learn to multiply fractions, decimals, and mixed numbers (3.3)

• Learn to divide fractions and decimals (3.4)

• Learn to solve equations with rational numbers (3.6)

• Learn to solve inequalities with rational numbers (3-7)

• Learn to find square roots (3-8)

• Learn to estimate square roots to a given number of decimal places and solve problems using square roots (3-9)

• Learn to determine if a number is rational or irrational (3-10)

Pre-Algebra

3-10 The Real Numbers

Today’s Learning Goal Assignment

Learn to determine if a number is rational or irrational.

Pre-Algebra

3-10 The Real Numbers

Pre-Algebra HW

Page 169

#1-40 all

Pre-Algebra

3-10 The Real Numbers3-10 The Real Numbers

Pre-Algebra

Warm Up

Problem of the Day

Lesson Presentation

Pre-Algebra

3-10 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.

Pre-Algebra

3-10 The Real Numbers

10 and 11

–4 and –3

1.4

–11.1

1. 119

2. – 15

3. 2

4. – 123

Pre-Algebra

3-10 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

Pre-Algebra

3-10 The Real Numbers

Today’s Learning Goal Assignment

Learn to determine if a number is rational or irrational.

Pre-Algebra

3-10 The Real Numbers

irrational numberreal numberDensity Property

Vocabulary

Pre-Algebra

3-10 The Real Numbers

AnimalsMammals

Biologists classify animals based on shared characteristics. The gray lesser mouse lemur is an animal, a mammal, a primate, and a lemur.

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.

PrimatesLemurs

Pre-Algebra

3-10 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

Pre-Algebra

3-10 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.

Helpful Hint

Pre-Algebra

3-10 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

Pre-Algebra

3-10 The Real Numbers

Additional Examples 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.

Pre-Algebra

3-10 The Real Numbers

Try This: 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

Pre-Algebra

3-10 The Real Numbers

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

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

15

irrational

0 3

rational

0 3

= 0

Additional Examples 2: Determining the Classification of All Numbers

A.

B.

Pre-Algebra

3-10 The Real Numbers

not a real number

Additional Examples 2: Determining the Classification of All Numbers

–9

4 9

rational

2 3

=2 3

4 9

C.

D.

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

Pre-Algebra

3-10 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

Try This: Examples 2

A.

B.

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

Pre-Algebra

3-10 The Real Numbers

not a real number

–7

64 81

rational

8 9

=8 9

64 81

C.

D.

Try This: Examples 2

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

Pre-Algebra

3-10 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.

Pre-Algebra

3-10 The Real Numbers

Additional Examples 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

Pre-Algebra

3-10 The Real Numbers

Try This: 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

Pre-Algebra

3-10 The Real Numbers

Lesson Quiz

Write all names that apply to each number.

1. 2. –

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

3. 4.

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

3 45.

2

4 • 9

16 2

25 0

not a real number rational

real, irrational real, integer, rational

Possible answer –2 .5 8