1. ## Proving Irrationals

I already posted this on another thread and then found this one! How good!

Here's my problem...

Assume we know that √2 is irrational,

let x,y be any rational number such that y is not 0. Show that x + y√2 is irrational

Any help would be awesome!

2. Originally Posted by lisa21a
I already posted this on another thread and then found this one! How good!

Here's my problem...

Assume we know that √2 is irrational,

let x,y be any rational number such that y is not 0. Show that x + y√2 is irrational

Any help would be awesome!
well, assume it is rational..

then $\displaystyle \frac{p}{q} = x + y\sqrt{2}$ for $\displaystyle p$ and $\displaystyle q$ integers, $\displaystyle q\neq 0$.. solving for $\displaystyle \sqrt{2}$ we have $\displaystyle \sqrt{2} = \frac{\frac{p}{q} - x}{y}$.. since the set rational numbers is a field (i.e. closed under addition and multiplication, and defined for non-zero divisor) the right hand side of the equality is still a rational which is a contradiction to the assumption that $\displaystyle \sqrt{2}$ is irrational.