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 4.6 - Page 206: 10

Answer

True.

Work Step by Step

Suppose there exists some irrational number such that its square root is rational. Then, the square root can be expressed as $\frac{a}{b}$ for positive integers $a$ and $b$. Then, the number is $\frac{a^2}{b^2}$. However, both the numerator and denominator of this number are integers, as they are each the square of an integer, so the number must be rational, but this is a contradiction, so the statement is true.
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