Chapter 13 Decimal 13-7B Irrational Numbersd75gtjwn62jkj.cloudfront.net/lessons/algebra-grade-8/SMP...Irrational Numbers Chapter 13 13-7B BIG IDEA Decimal approximations of irrational

Lesson Decimal Approximations of Irrational Numbers

Chapter 13

13-7B

BIG IDEA Decimal approximations of irrational numbers help

to place them in order and graph them on a number line.

The number π = 3.141592653589793238…, is an irrational number, so its decimal is infi nite and does not repeat. Using its decimal, it is possible to fi nd an interval containing π of any positive width. By using a small enough width you can approximate the value of π to any degree of accuracy.

Example 1Find an interval of the given width containing π.

a. One

b. 0.1

c. One ten-thousandth

d. One millionth

Solution

a. Write the decimal approximation of π to one more decimal place than the width. The decimal begins 3.1…. Because 3 < 3.1 < 4, 3 < π < 4 is an interval of width 1 containing π.

b. The decimal approximation of π begins 3.14…. Because ? < 3.14 < 3.2, ? < π < 3.2 is an interval of width 0.1 containing π.

c. The decimal approximation of π begins 3.14159…. Because ? < 3.14159 < ? , ? < π < ? is an interval of width ? containing π.

d. The decimal approximation of π begins ? …. ? < π < ? is an interval of width ? containing π.

GUIDEDGUIDED

1 Using Algebra to Prove

Lesson 13-7B

You can picture smaller and smaller intervals on the number line.

π

π

π

0 1 2 3 4 5 6 7 8 9 10

3 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 4

3.1 3.11 3.12 3.13 3.14 3.15 3.16 3.17 3.18 3.19 3.2

whole number interval

tenth

hundredth

While it is impossible to write all the digits of π, you can place it in an interval to make the estimate as close as you would like to its actual value.

Calculating with Irrational NumbersYou can estimate the answer to a calculation with irrational numbers by fi nding an interval of the desired accuracy containing each irrational number and using those approximations to complete the calculation.

Example 2Estimate √

__ 5 + √

__ 2 + 2 to the nearest hundredth.

Solution A calculator gives: √ __

5 + √ __

2 + 2 = 5.6502815... so,

√ � 5 + √ � 2 + 2 ≈ 5.65 .

Check √ __

5 is between 2 and 3, or 2 < √ __

5 < 3. √ __

2 is between 1 and 2, or 1 < √

__ 2 < 2. Add the sides of the inequalities:

2 < √ � 5 < 3 + 1 < √ � 2 < 2 + 2 = 2 = 2

5 < √ � 5 + √ � 2 + 2 < 7This checks, since 5.65 is between 5 and 7.

Adding, subtracting, multiplying, and dividing rational numbers always results in another rational number (remember that dividing by 0 is forbidden). What happens when you calculate with irrational numbers?

Decimal Approximations of Irrational Numbers 2

Chapter 13

Example 3Which is larger,

a. √ —

9 - √ —

4 or √ —

9 - 4 ?

b. √ —

30 - √ —

14 or √ —

30 - 14 ?

Solutions

a. √ —

9 = 3 and √ —

4 = 2, so √ —

9 - √ —

4 = 1. √ —

9 - 4 = √ —

5 , and √ —

5 > √ —

4 = 2, so √

— 9 - √

— 4 < √

— 9 - 4 .

b. √ —

30 ≈ 5.4772 and √ —

14 ≈ 3.7417, so √ —

30 - √ —

14 ≈ 1.7355. √ —

30 - 14 = √ —

16 = 4, so √

— 30 - √

— 14 < √

— 30 - 14 .

QuestionsCOVERING THE IDEAS

1. Every irrational number has a decimal representation. What is true of that decimal?

2. Find an interval of the given width containing √ _____

10.01 . a. One b. One tenth c. One thousandth d. One millionth

3. Find an interval of the given width containing √ ___

65 - 1. a. One b. One tenth c. One hundred-thousandth d. One ten-millionth

4. Use a calculator to estimate the result to the nearest thousandth. a. √

___ 32 + √

__ 2 b. √

______ 32 + 2

c. √ ___

32 - √ __

2 d. √ ______

32 - 2 e. √

___ 32 · √

__ 2 f. √

_____ 32 · 2

g. √

___ 32 _

√ __

2 h. √

___

32 _ 2

5. Consider the irrational number j = 1.24681012… where the consecutive digits are the digits of the consecutive even positive integers.

a. j is between what two consecutive whole numbers? b. Place j in an interval one tenth long. c. Place j in an interval one thousandth long. d. Place j in an interval one trillionth long.

3 Using Algebra to Prove

Lesson 13-7B

APPLYING THE MATHEMATICS

6. Explain why π + 50 must be an irrational number.

7. Explain why ( √ __

2 _ 3 ) 4 is a rational number.

8. Is √ ____

5.29 rational or irrational? Explain.

9. Without a calculator, fi nd the two consecutive integers between which √

_______ 123.456 lies.

10. Copy one of the number lines in this lesson and show the graphs of π, π - 0.1, and π - 0.2 on it.

11. Draw a number line and graph the following numbers on it: √

__ 3 , 1 + √

__ 3 , 1 + √

__ 3 _ 2 .

Fill in the Blank In 12–14, approximate the value of each expression and then write <, =, or > in the blank to make the statement true.

12. √ __

3 + √ __

2 ? √ _____

3 + 2

13. √ __

8 - √ __

6 ? √ ___

10 - √ __

8

14. √ __

2 ? √ __

8 _ 2

15. Here are three approximations of π. For each, give the decimal equivalent correct to the number of places that match the actual value of π.

a. 22 _ 7

b. 333 _ 106

c. 355 _ 113

In 16 and 17, use the following information. The golden ratio,

ϕ = 1 + √ � 5

_ 2 , is an irrational number. The ancient Greeks thought

rectangles with width w and height h in the proportion w _ h = ϕ

_ 1

were the most pleasing to the eye.

16. Write an interval with a width of at most one thousandth containing the golden ratio by using the decimal representation √

__ 5 = 2.236067977… .

17. An architect is drawing plans for a rectangular building with sides in the ratio

ϕ _ 1 . If the shorter side is 15 meters, fi nd the approximate

length of the longer side accurate to the nearest hundredth.

18. Recall that the volume of a cylinder is V = Bh. Refer to the cylinder at the right.

a. Find its exact volume. b. Find its volume correct to the nearest hundredth of a

cubic centimeter.

9 cm

4 cm

9 cm

4 cm

Decimal Approximations of Irrational Numbers 4