Question

How do I Prove or disprove; For all x, y € R, if x is rational and y is irrational, then xy is irrational

Answer 1

\(x\) is rational, so \(x\) can be expressed as \(\frac{p}{q}\)
Let us assume that \(x\times y\) is rational, then \(xy=\frac{m}{n}\implies y=\frac{mq}{np}\) . This shows that y is rational, which is a contradiction. So \(xy\) is irrational.

Answer 2

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