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Prove each of the statements in , 24, G in two ways: (a) by contraposition and (b) by contradiction. 24. The reciprocal of any irrational number is irrational. (The reciprocal of a nonzero real number a is 1/æ.)

Prove each of the statements in , 24, G in two ways: (a) by contraposition and (b) by contradiction. 24. The reciprocal of any irrational number is irrational. (The reciprocal of a nonzero real number a is 1/æ.)

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Prove each of the statements in 22, 24, and 26 in two ways: (a) by contraposition and (b) by contradiction. 24. The reciprocal of any irrational number is irrational. (The reciprocal of a nonzero real number \( x \) is \( 1/x \).)
Transcribed Image Text:Prove each of the statements in 22, 24, and 26 in two ways: (a) by contraposition and (b) by contradiction. 24. The reciprocal of any irrational number is irrational. (The reciprocal of a nonzero real number \( x \) is \( 1/x \).)
Expert Solution
Step 1

(24) We have to show that the reciprocal of any irrational number is irrational.

(a) Now, prove the given statement by contrapositive.

The contrapositive statement is "If 1x is rational then x is a rational, for x∈R, x≠0

Assume that 1x is a rational number.

Clearly, 1x is nonzero.

By the definition of rational number, 1x=ab for a,b∈Z, a≠0,b≠0

⇒ba=x

Hence, by definition of rational, x is a rational number.

Hence, proved.

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