# Thread: Need help proving by contraposition

1. ## Need help proving by contraposition

Hello,

I have a home work question that asks me to prove by contraposition that for all positive real numbers that if x is irrational, then x^1/2 is also irrational.

So far I have the Contrapositive of for all positive real numbers, x^1/2 is rational then x is rational. Which means (x^1/2)/1. But this is where I am stuck. I don't know what to do with it beyond that.

Can anyone give me even just a hint? Thank you in advance.

2. Originally Posted by Dko
Hello,

I have a home work question that asks me to prove by contraposition that for all positive real numbers that if x is irrational, then x^1/2 is also irrational.

So far I have the Contrapositive of for all positive real numbers, x^1/2 is rational then x is rational. Which means (x^1/2)/1. But this is where I am stuck. I don't know what to do with it beyond that.

Can anyone give me even just a hint? Thank you in advance.
The contrapositive of the statement is:

"If $\displaystyle \sqrt{x}$ is rational then $\displaystyle x$ is rational."

Now set up your arguments.

If $\displaystyle \sqrt{x}$ is rational, it can be written as a fraction of integers.

$\displaystyle \sqrt{x} = \frac{a}{b}$

$\displaystyle (\sqrt{x})^2 = \left(\frac{a}{b}\right)^2$

$\displaystyle x = \frac{a^2}{b^2}$, another rational number.