7 Chapter Rational Numbers as Decimals and Percent Copyright © 2016, 2013, and 2010, Pearson Education, Inc.

7-3 Repeating Decimals • Why repeating decimals occur and how to tell if a decimal will repeat. • How a fraction can be converted to a repeating decimal and vice versa. • Ordering repeating decimals efficiently. • That and investigate various ways to show this is true.

Repeating Decimals We have examined decimal numbers such as 0.475, which stop, and are called terminating decimals. Not all rational numbers can be represented by terminating decimals. For example, converting 2/11 into a decimal using the long division process indicates that we will actually get 0.1818…, with the digits 18 repeating over and over indefinitely.

Repeating Decimals A decimal of this type is a repeating decimal, and the repeating block of digits is the repetend. The repeating decimal is written 0.18, where the bar indicates that the block of digits underneath is repeated continuously.

Example Convert the following to decimals: 0.142857 0.153846

Writing a Repeating Decimal in the Form Write 0.5 as a rational number in the form Step 1: Let x = 0.5, so that x = 0.5555… Step 2: Multiply both sides of the equation x = 0.5555… by 10. (Use 10 since there is one digit that repeats.)

Writing a Repeating Decimal in the Form Step 3: Subtract the expression in step 1 from the final expression in step 2. Step 4: Solve the equation 9x = 5 for x.

Writing a Repeating Decimal in the Form

Writing a Repeating Decimal in the Form In general, if the repetend is immediately to the right of the decimal point, first multiply by 10m where m is the number of digits in the repetend, and then continue as in the preceding cases. Suppose the repeating block does not occur immediately after the decimal point. A strategy for solving this problem is to change it to a related problem we know how to solve; that is, change it to a problem where the repeating block immediately follows the decimal point.

Writing a Repeating Decimal in the Form Write 2.345 as a rational number in the form

A Surprising Result Write 0.9 as a rational number in the form

Ordering Repeating Decimals Order repeating decimals in the same manner as for terminating decimals. Compare 1.3478 with 1.347821. Line up the decimals, then starting at the left, find the first place where the face values are different. Since 3 > 2, 1.3478 > 1.347821.

Example Find a rational number in decimal form between 0.35 and 0.351. Starting from the left, the first place at which the two numbers differ is the thousandths place. One decimal between these two is 0.352. Others include 0.3514, 0.3515, and 0.35136. There are infinitely many others.