Thinking Mathematically (6th Edition)

Published by Pearson
ISBN 10: 0321867327
ISBN 13: 978-0-32186-732-2

Chapter 5 - Number Theory and the Real Number System - 5.3 The Rational Numbers - Exercise Set 5.3 - Page 287: 147


A decimal number with nonterminating decimals can be expressed in the form of integers with the help of a technique thatinvolves solving a one-step equation. The decimal number \[0.\overline{9}\] can be expressed as a quotient of integer as follows: Let \[n\]be equal to the repeating decimal: \[\begin{align} & n=0.9999 \\ & 10n=10(0.9999) \\ & 10n=9.9999 \end{align}\] This is further solved as: \[\begin{align} & 10n-n=9.9999-0.9999 \\ & 9n=9 \\ & n=1 \end{align}\] Thus, \[n=1\] and also \[n=0.\overline{9}\], which gives \[0.\overline{9}=1\].
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