Let p and q be two integers where q is nonzero. Any rational number is of the form (ratio of two integers)
If was rational, then for a certain (p,q) value.
Multiplying both sides by q leads us to which becomes
The right side of is an integer (p is an integer)
The left side of should be an integer as well; however, it is not an integer. The expression is surely an integer because 2 times any integer is an integer, but the expression is NOT an integer. It is only an integer if q was of the form , but it is not. Again, q is an integer with no irrational parts.
In short, on the left side is NOT an integer while on the right side is an integer.
So this is where the contradiction lies. This contradiction makes the claim to be false, therefore is not rational. The only thing left is that must be irrational.
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