help on irrational numbers

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• March 9th 2009, 09:23 PM
b0mb3rz
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
• March 10th 2009, 08:21 AM
red_dog
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.