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: 10


By definition, a number is rational if it is a ratio of two integers, of which the second (the denominator) is not zero. Since $m$ and $n$ are integers, $5m$, $12n$, and $4n$ must all be integers by the closure of the integers under multiplication. From this we conclude that $5m-12n$ is an integer by the closure of the integers under subtraction. Finally, $4n\ne0$ by the zero product property, since $4\ne0$ trivially and we are told that $n\ne0$. Therefore, $\frac{5m-12n}{4n}$ is a ratio of two integers of which the second is not zero, so it is a rational number by definition.

Work Step by Step

The three properties used in this proof are the closure of the integers under multiplication, the closure of the integers under subtraction, and the zero product property. Appendix A of the text provides a review of these and other fundamental algebra concepts.
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