# Math Help - proof by contradiction

how would you prove by contradiction that if n is even then n^2 is even??

2. To prove by contradiction you assume the opposite of what is stated, ie that $n^2$ is odd.

As you are starting with an even number, then instead of $n$, lets use $2n$.

$(2n)^2 = 4N$, therefore if $n \in Z$(intergers), $n^2$ is always even.

This is how I would do it anyway, I am sure there are different methods out there.

Craig

Edit: solved

3. Originally Posted by craig
To prove by contradiction you assume the opposite of what is stated, ie that $n^2$ is odd.

As you are starting with an even number, then instead of $n$, lets use $2n$.

$(2n)^2 = 4N$, therefore if $n \in Z$(intergers), $n^2$ is always even.

This is how I would do it anyway, I am sure there are different methods out there.

Craig

And of course, once you assume that $n^2$ is odd, the only way that is true is if $n$ itself is also odd. But that contradicts the original assumption that you made about it being even. So that completes the proof.
By law of excluded middle: n is odd or n is even. Assume n is even, then $n = 2m$ where $m \in \mathbb{Z}$. But $n^2 = 4m^2 = 2(2m^2)$, which is a contradiction.