Question

If x is not equal to 0 is rational and y irrational, prove that x + y, x − y, xy, x/y and y/x are all irrational. 

Answer 1

x is rational. That means, that x can be represented as a/b, where a and b are integers.

y is irrational. That means, that it cannot be represented like that.

1) x+y

Let x+y be rational. x+y=c/d, c and d are integers. Then:

$y=\frac{c}{d}-x=\frac{c}{d}-\frac{a}{b}=\frac{cb-ad}{bd}$

cb-ad and bd are integers. So, y can be represented as fraction of two integers, but y is irrational.

If x+y is rational, we get contradiction. That means, x+y is irrational.

2) x-y

Let x-y be rational. x-y=c/d, c and d are integers. Then:

$y=x-\frac{c}{d}=\frac{a}{b}-\frac{c}{d}=\frac{ad-cb}{bd}$

ad-cb and bd are integers. Contradiction. So x-y is irrational.

3) xy

Let xy be rational. xy=c/d, c and d are integers. Then:

$y=\frac{c}{d}\frac{1}{x}=\frac{c}{d}\frac{b}{a}=\frac{bc}{ad}$

bc and ad are integers. Contradiction. So xy is irrational.

4) x/y

Let x/y be rational. x/y=c/d, c and d are integers. Then:

$y=x\frac{d}{c}=\frac{a}{b}\frac{d}{c}=\frac{ad}{bc}$

ad and bc are integers. Contradiction. So x/y is irrational.

5) y/x

Let x/y be rational. y/x=c/d, c and d are integers. Then:

$y=x\frac{c}{d}=\frac{a}{b}\frac{c}{d}=\frac{ac}{bd}$

ac and bd are integers. Contradiction. So y/x is irrational.