TITLE: Is there a direct proof for there is no rational number whose square is 2?
QUESTION [11 upvotes]: One of my math teacher did not like proof by contradiction. He mentioned that it is a indirect way to prove something and not a genuine way to prove a statement.
In elementary math class, there is a following statement: "there is no rational number whose square is 2." I have only seen a proof by contradiction so far.
Is there a direct proof for there is no rational number whose square is 2?
Also I want to hear your opinion about proof method: proof by contradiction. Is this really bad? Should I avoid using this method whenever possible?

REPLY [4 votes]: I'm going to agree with others that a proof by contradiction is completely valid, assuming mainstream, classical logic.
That said, if either option is available, a direct proof can often be more explanatory, easier to read, more understandable, and possibly more useful in pointing to other constructed facts later on. Many of us find that students tend to go to proofs by contradiction almost fetishistically, when a direct proof is available and shorter. So some of us try to advise students about that blind spot. It's possible that the OP's instructor was trying to say something similar, and was misunderstood.
Now, here's the thing I really want to add. Consider the standard definitions of the terms "rational" and "irrational".

Rational: A real number $r$ is rational if $r = p/q$, for some integers $p$ and $q$, with $q \ne 0$.
Irrational: A real number is irrational if it is not rational.

The thing I'd like to draw attention to is that the definition of "rational" has a formula for it, whereas "irrational" does not. So proofs about rationality can be done directly, by manipulating that formula algebraically (or manipulating some other formula into a matching form).
With proofs about a number being irrational, that's not an option. It doesn't have a formula; it only has a proposition containing the "not" operator. So really you have no choice but to use a proof by contradiction:  The initial assumption of a negation of your goal statement cancels the other negation, and then you're left with a positive statement about a rational number, for which you have a formula for algebraic manipulation.
So based on the way "irrational" is defined, not-some-other-property, you inherently need to use a proof by contradiction. Again: this assumes standard, classical logic (in which proof by contradiction and "negation" are equivalent).