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True or false: If a is irrational, then a -I is irrational

Chapter 31, Problem 31.23

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QUESTION:

True or false: If a is irrational, then a -I is irrational.

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QUESTION:

True or false: If a is irrational, then a -I is irrational.

ANSWER:

Step 1 of 2

Proof: We use proof of contradiction, let there exists irrational number a, who's reciprocal is rational

Since 1/a is rational, then there exists integers p and q for which 1/a=p/q.

Since a is finite, we know that 1/a is not equal to zero, so p is not equal to zero. So 1/(1/a) exists.

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