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Q75E

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Discrete Mathematics and its Applications

Found in: Page 155

Book edition 7th

Author(s) Kenneth H. Rosen

Pages 808 pages

ISBN 9780073383095

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Short Answer

Prove that if x is a positive real number, then
$\begin{aligned} (a) ⌊ \sqrt{⌊ x ⌋} ⌋ = ⌊ \sqrt{x} ⌋ \\ (b) [\sqrt{⌊ x ⌋}] = ⌈ \sqrt{x} ⌉ \end{aligned}$

$\begin{aligned} (a) ⌊ \sqrt{⌊ x ⌋} ⌋ = ⌊ \sqrt{x} ⌋ \\ (b) [\sqrt{⌊ x ⌋}] = ⌈ \sqrt{x} ⌉ \end{aligned}$

See the step by step solution

Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step: 1

If x is a positive integer, then the two sides are equal. So suppose that $x = n^{2} + m + ε$ , where is the largest perfect square less than x, m is a nonnegative integer, and $0 < ε \leq 1$ . then both $\sqrt{x} and \sqrt{⌊ x ⌋} = \sqrt{n^{2} + m}$ between n and n+1, so both sides equal n.

Step: 2

If x is a positive integer, then the two sides are equal. So suppose that $x = n^{2} - m - ε,$ , where $n^{2}$ is the smallest perfect square less than x, m is a nonnegative integer, and $0 < ε \leq 1$ . then both $\sqrt{x} and \sqrt{⌊ x ⌋} = \sqrt{n^{2} - m}$ between n-1 and n-1, so both sides equal n

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