TITLE: If the square root of two is irrational, why can it be created by dividing two numbers?
QUESTION [16 upvotes]: I'm doing pre-calculus and got a bit caught on this question… I looked online and it said that a number is rational if it can be the quotient of two integers. So I did this:

I know that it is irrational because if this was actually rational, mathematicians would have figured out. So my question is, why is the square root of two irrational?

REPLY [11 votes]: There's nothing so mysterious:
$$
\left(\frac{141421356237}{10^{11}}\right)^{\!2}=
\frac{19999999999912458800169}{10^{22}}\ne 2
$$
Thus $\dfrac{141421356237}{10^{11}}$ is not $\sqrt{2}$.
Without computations: $141421356237$ is odd, so its square is odd too. Hence the fraction cannot simplify to an even integer.