Irrational numbers cannot exist

Let's look at the number line. You can see -5,-4,-3,-2,-1,0,1,2,3,4 and so on. If you look between 0 and 1 there is going to be a rational number 1/2 exactly in the middle. There is going to be a rational number 1/4 exactly in the middle of 0 and 1/2. You can do this for any two numbers and make the distance between them smaller and smaller.

So you can fill up the whole number line by cutting it in halves. Therefore, the whole number line is made up of rational numbers and irrational numbers cannot exist.

Assume Sqrt(2) is rational.  Then, by definition, there exists two relatively prime integers a and b such that a/b = sqrt(2).  Implying, a^2/b^2 = 2, and a^2 = 2b^2.

Thus, a must be even, since only the square of an even number is even, and there exists an integer k such that 2k = a.  Thus, (2k)^2 = 2b^2 and 2k^2 = b.  Therefore, b is also even.  But if a and b are both even then they can't be relatively prime integers.
Contradiction.