Lesson Real Numbers and the Number Line

Chapter 13

13-7A

BIG IDEA On a real number line, both rational and irrational

numbers can be graphed.

As we noted in Lesson 13-7, every real number is either a rational number or an irrational number. In this lesson you will explore some important properties of rational and irrational numbers.

Rational Numbers on the Number LineRecall that a rational number is a number that can be expressed as a simple fraction. When a rational number is written as a fraction, it can be rewritten as a decimal by dividing the numerator by the denominator. The result will either be terminating, such as 5 _ 8 = 0.625, or repeating, such as 10 _ 11 = 0.

__ 90 . This makes it possible to graph the number on a

number line.

Example 1Graph 0.8

− 3 on a number line.

Solution Let x = 0.8 __

3 . Change 0.8 __

3 into a fraction.

Then 10x = 8.3 _

3 x = 0.8

_ 3

? x = 7.5 Subtract

x = 7.5 _ 9 = ? _ 90 = ? _ 6 To graph ? _ 6 , divide the interval 0 to 1 into ? equal spaces. Locate the point corresponding to 0.8

__ 3 .

0 1

When two different rational numbers are graphed on a number line, there are always many rational numbers whose graphs are between them.

GUIDEDGUIDED

1 Using Algebra to Prove

Lesson 13-7A

Example 2Find a rational number between 11

_ 13 and 20 _ 23 .

Solution 1 Find a common denominator by multiplying 13 · 23, which is 299.

11 _ 13 = 253 _ 299 ; 20 _ 23 = 260 _ 299 .

So pick a fraction between 253 _ 299 and 260

_ 299 , such as 255 _ 299 .255299

260299

2023

253299

1113= =

Solution 2 Change the fractions to decimals.

11 _ 13 = 0.8461...

20 _ 23 = 0.8695...Pick a number, such as 0.85, that is between the two numbers.

0.85

2023

1113 ≈ 0.846 ≈ 0.870

In Example 2, we found a number between two rational numbers, but is it possible that there are two rational numbers so close together that no other rational number is between them? The answer is no. We prove this by showing that the mean of two numbers is always between those two numbers.

Here is the proof:

Let a and b be rational numbers with a < b. By its defi nition, the mean of a and b is a + b _ 2 . We fi rst need to show that a < a + b _ 2 . Starting with a < b we can show this using properties of rational numbers.

a < b Given

a _ 2 < b _ 2 Multiplication Property of Inequality

a _ 2 + a _ 2 < b _ 2 + a _ 2 Addition Property of Inequality

2a _ 2 < b + a _ 2 Distributive Property

a < b + a _ 2 Equal Fractions Property

a < a + b _ 2 Commutative Property

The proof that a + b _ 2 < b is similar. You are asked to complete it in Question 16. Since a < a + b _ 2 and a + b _ 2 < b, a + b _ 2 is between a and b. Therefore, it is always possible to fi nd a rational number between two distinct rational numbers.

QY1

QY1

Find a rational number between 2 _ 3 and 3 _ 4 .

QY1

Find a rational number between 2_3 and 3_4 .

QY1

Find a rational number between 2 _ 3 and 3 _ 4 .

Real Numbers and the Number Line 2

Chapter 13

Some Irrational Numbers on the Number LineEvery irrational number can be written as an infi nite, non-repeating decimal. For this reason, you cannot write their exact decimal representation. However, you can still know exactly where some irrational numbers are graphed on the number line.

MATERIALS a straightedge, a protractor, scissors

Step 1 Create a right triangle NUM with the right angle at M, MU = 1 cm, and MN = 3 cm.

Step 2 Calculate NU.

Step 3 Cut out �NUM.

Step 4 Draw a number line showing coordinates from –2 to 4. Make each unit 1 cm in length.

Step 5 Using side __

NU , graph the exact value of NU = √ ___

10 .

QY2

QuestionsCOVERING THE IDEAS

In 1–4, tell whether the statement is always true, sometimes but not always true, or never true.

1. It is possible to write an exact decimal for a rational number.

2. It is possible to write an exact decimal for an irrational number.

3. It is possible to locate a rational number exactly on a number line.

4. A real number is either rational or irrational.

In 5–7, graph each rational number on a number line.

5. 8 4 _ 7

6. -0. _

5

7. 3. __

27

In 8–10, � nd a number between the given numbers.

8. - 5 _ 7 , - 6 _ 7

9. 5 __ 12 , 7 __ 15

10. 2 3 _ 5 , 2 4 _ 5

11. Explain how you could show an exact length for √ ___

17 .

ActivityActivity

1 cm

3 cm

N

M U1 cm

3 cm

N

M U

QY2

√ ___

10 is between which two consecutive integers?

QY2

√___

√___

√10 is between which two consecutive integers?

QY2

√ ___

10 is between which two consecutive integers?

3 Using Algebra to Prove

Lesson 13-7A

APPLYING THE MATHEMATICS

12. Find a number between a. 2 _ 3 and 5 _ 6 . Call your value a.

b. 2 _ 3 and a. Call this value b.

c. 2 _ 3 and b. Call this value c.

d. 2 _ 3 and c. e. How long could you continue this pattern?

(Hint: The answer is not 26 times.)

13. a. Find AC in right �ABC at the right. b. Explain how to use the result in Part a to fi nd an exact length

for √ ___

40 . 14. a. Draw an accurate rectangle with length √

__ 8 cm and width √

__ 2 cm.

b. Use your calculator to determine the area of the rectangle.

15. H and I are graphed on the number line below.

a. Fill in the Blanks H = ? and I = ? . b. List ten rational numbers between H and I. c. List fi ve irrational numbers between H and I.

16. Complete the proof from the lesson by showing that a + b _ 2 < b. (Hint: The structure of your proof should be similar to that of the proof that a < a + b _ 2 .)

2

6

A

B C

x

2

6

A

B C

x

0 2 4

H I

0 2 4

H I

QY ANSWERS

1. Answers vary. Sample: 7 _ 10 , 8 _ 11 .

2. 3 and 4

Real Numbers and the Number Line 4