# Proof of the irrationality of sqrt(2)

```         ___
Suppose V 2   is rational, then we can express it as
___
V 2  =  p / q

Where p and q are the smallest positive integers that
satisfy the above equation.

p and q cannot both be even since if they were, we
could divide each by 2 and have a smaller p and q
to work with.

From this we can by squaring both sides get:

2   2                            2
2 = p / q     then by multiplying by q

2     2
2 q  =  p

2              2
From this we can deduce that p is even.  If p is even,
then q must be odd since p and q cannot both be even.

2
Since p  is even, then p must be even.

2
If p is even, then p must be divisible by 4.

Thus,

2
p
-----  is an even integer.
2

But since

2     2
2 q  =  p

Then,

2
p        2
-----  =  q
2

2
Thus q  must be even, and therefore q is even.  We had previously

determined that q must be odd.  q cannot be both even and odd, so

we conclude that no such integer exists, which invalidates our
___
premise that V 2  is rational.

```