Discrete Mathematics with Applications 4th Edition

Published by Cengage Learning
ISBN 10: 0-49539-132-8
ISBN 13: 978-0-49539-132-6

Chapter 4 - Elementary Number Theory and Methods of Proof - Exercise Set 4.2 - Page 168: 11


Let $n$ be some integer. Then $n=\frac{n}{1}$, because any number divided by $1$ is equal to itself. But $\frac{n}{1}$ is a ratio of integers, of which the denominator is not zero, so by definition, it is a rational number. Since $n=\frac{n}{1}$, it must be that $n$ is also a rational number. Because our choice of $n$ was arbitrary, we conclude that every integer is a rational number.

Work Step by Step

Note that we call $n$ a ratio of two integers because it is equal to a ratio of two integers, namely, $n$ and $1$. This follows from our understanding of the symbol $=$, which implies that the object on the left side of the equality has all the same mathematical properties as the object on the right side.
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