Principles of Mathematical Analysis, Rudin, 3th ed, Chapter 1, Problem 1

Problem

If $r$ is rational ($r \neq 0$) and $x$ is irrational, prove that $r+x$ and $rx$ are irrational.

Answer

If $r+x$ is rational then $-r+r+x=x$ is rational, which is contradiction.

If $rx$ is rational then $(1/r)rx=x$ is rational, which is contradiction.

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