Is a non-zero integral multiple of an irrational number guaranteed

madridomot

madridomot

Answered question

2022-05-29

Is a non-zero integral multiple of an irrational number guaranteed to be irrational?

Answer & Explanation

Diego Mathews

Diego Mathews

Beginner2022-05-30Added 6 answers

Yes, it is guaranteed. Suppose for the sake of contradiction that α is irrational, and that n α is rational for some non-zero integer n. In other words, suppose
n α = a b for some non-zero integer n and rational number a b . Then this implies
α = a b n contradicting the definition of α being irrational.

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