# Thread: help on irrational numbers

1. ## help on irrational numbers

suppose x is an irrational number
-show that 1/x is irrational
-are there any integers p/q both greater than 0 such that (p/q)x is rational?

suppose x is an irrational number and y is any real number bigger than zero. show that at least one of x-y and x+y is an irrational number. .

thank you

2. Suppose that $\frac{1}{x}$ is rational.

Then $\frac{1}{x}=\frac{p}{q}\Rightarrow x=\frac{q}{p}\Rightarrow$ x is rational, contradiction

Suppose $\frac{p}{q}\cdot x=\frac{m}{n}\Rightarrow x=\frac{mq}{np}$, contradiction.