Question #324393

Prove that a real number 𝑥 is irrational if and only if 5𝑥 is irrational


1
Expert's answer
2022-04-06T17:22:35-0400

LetxQ.Suppose5xQ,thenx=5x5Q,whichisacontradictory.Let5xQ.SupposexQ,then5x=5xQ,whichisacontradictory.ThusxQ5xQLet\,\,x\notin \mathbb{Q} . Suppose\,\,5x\in \mathbb{Q} ,then\,\,x=\frac{5x}{5}\in \mathbb{Q} , which\,\,is\,\,a\,\,contradictory.\\Let\,\,5x\notin \mathbb{Q} . Suppose\,\,x\in \mathbb{Q} , then\,\,5x=5\cdot x\in \mathbb{Q} , which\,\,is\,\,a\,\,contradictory.\\Thus\\x\notin \mathbb{Q} \Leftrightarrow 5x\notin \mathbb{Q}


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