Why is the square root of two irrational?

If √2 is rational, then there exist mutually prime integers p and q such that

p/q = √2

and so

(p/q)^2 = 2 [using ^ to indicate an exponent, as we usually do in the Math section].

Then

p^2 = 2 q^2 by elementary algebraic operations

This means that p^2 is even,

so p must be even

and p^2 is divisible by 4.

Thus, q^2 is divisible by 2,

so q^2 is even

and therefore so is q.

But having p and q both even means they have a common factor,

which contradicts the assumption that they were mutually prime.

Thus, by contradiction, we have shown there ARE no such p and q,

and √2 is not rational.

I suppose God could have made it some other way, if God hadn't been more interested in making mathematics consistent with rationality than in making all numbers rational. ("Rational" has two different meanings in the different contexts.)

I count it among the blessings we have received by the grace of God that mathematics works.

How does God determine mathematics? We use mathematics to analyze and characterize the world through our own, human eyes. A foot is 12 inches because we say so. A unit is = 1 because we say so. This has been an evolutionary processes for many thousands of years, and although the way people wrote the numbers changed, the concepts stayed the same.

Well he did make the value of pi a nice even 3.

Yeah, god sucked at math.