Originally Posted by

**iknowone** This is a classic example of "assuming what you are trying to prove."

In algabraic proofs, especially when you are first being introduced to proofs, it is tempting to write down an equation then manipulate it until you get something that is "obviously" true. It cannot be stressed enough that this is not a proof, since what you assume (namely that the equality is true) may not be!

Here is a simple but enlightening example:

I claim I will prove -1=1.

**Proof:**

$\displaystyle -1 = 1$

$\displaystyle (-1)^2 = (1)^2$

$\displaystyle 1=1$ QED!

Now did I just rock the very foundation of mathematics by proving -1 = 1? Why not? If I have an equality I am allowed to square it as long as I do it to both sides, right?

The problem is that I started by assuming what I was trying to prove. I wanted to prove -1 = 1 and what I did was I wrote that down and did some math to both sides of the equation and ended up with an equality I know to be true. So I really didn't prove anything.

Start with one side of the equation (usually the more complicated side) and through the use of proven identities and simplification rearrange it into the other side. This constitutes a valid proof.