5.72. Evaluate the proposed proof of the following result. Result If x is an irrational number and y is a rational number, then z = x – y is irrational.

Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
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Chapter2: Second-order Linear Odes
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5.72. Evaluate the proposed proof of the following result.
Result If x is an irrational number and y is a rational number, then z = x – y is irrational.
Proof Assume, to the contrary, that z = x – Y is rational. Then z = a/b, where a, b e Z and b+ 0. Since v2 is irrational, we let x = V2.
Since y is rational, y = c/d, where c, d e Z and d +0. Therefore,
a
ad + bc
V2 = x = y+ % =
%3D
bd
Since ad + bc and bd are integers, where bd + 0, it follows that V2 is rational, producing a contradiction.
Transcribed Image Text:5.72. Evaluate the proposed proof of the following result. Result If x is an irrational number and y is a rational number, then z = x – y is irrational. Proof Assume, to the contrary, that z = x – Y is rational. Then z = a/b, where a, b e Z and b+ 0. Since v2 is irrational, we let x = V2. Since y is rational, y = c/d, where c, d e Z and d +0. Therefore, a ad + bc V2 = x = y+ % = %3D bd Since ad + bc and bd are integers, where bd + 0, it follows that V2 is rational, producing a contradiction.
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