So I've been learning a bit of logic for class and heard that if you begin with a false statement, you can then prove anything to be true, however I don't entirely understand what this means or how to do it.
For example, if $\sqrt{2}$ is rational, can you prove that $1=0$?
just for fun! assume there exist $a,b$ relative prime integers such that $\frac{a}{b}=\sqrt{2}$, we can assume $a$ odd (otherwise we can argue in a similar way with $b$) hence $a \;\text{mod} \;2 =1$. We have $a^2 =2 b^2$ hence $2|a$ and a is even i.e. $0= a \; \text{mod} \; 2 =1$ Q.E.D.
This is known as the principal of explosion. The idea is that as soon as you can prove two contradictory statements from a axiom system (in classical logic), you can prove anything.
For instance, if you have proved that $\sqrt{2}$ is irrational, but also have proved (or perhaps just as an axiom) that $\sqrt{2}$ is rational, then you can argue as follows:
Clearly, it is either the case that $\sqrt{2}$ is irrational or that $1=0$, since we know the former to be true. Since we also know that $\sqrt{2}$ is rational, for the previous statement to be true, it must be that $1=0$.
The trick here is that, you can say "this or that" by knowing "this", but from "this or that" you can show "that" by knowing "not this".
Note that this process requires starting with a contradiction not just with a false statement - but there's no real intrinsic notion of "false" other than "contradictory" within a logical system.
I believe what you are referring to is vacuous truth, and it's for implications.
The statement: "If $\sqrt{2}$ is rational, then $1=0$" is true logically, because the hypothesis (if $\sqrt{2}$ is rational) is false.
Hmmm ... I am not a fan of how that was put ... when it comes to proving things, it is not so much that from a false statement you can infer anything. Logic itself does not care whether things are true or false, and so starting with $P$ does not mean that I can infer anything, even if $P$ turns out to be false.
What is true, however, is that you can infer anything you want from a contradiction.
For example, suppose we have your standard contradiction: we have both $P$ and $\neg P$
Now, from $P$ we can infer $P \lor Q$
But if we have $P \lor Q$, and we also have $\neg P$, then we can infer $Q$
And so yes, since $Q$ can be anything at all, we can infer anything from a contradiction.
To go back to the 'false' though: If you know that $P$ is true, then if you assume that $P$ is false (i.e. you have $\neg P$), then indeed you can infer anything you want. But you can't infer anything you want from a false statement alone.
What this means is simply that the following is considered an allowed proof step:
... and therefore $A$. But we already know that $\neg A$, so therefore we conclude $B$. Q.E.D.
How to do it is just a matter of writing something like the above.
The question you don't ask, but should have, is why people accept this. Here my answer would be:
The purpose of a proof is to learn something like "in every time, place, and world where such-and-such premises hold, this conclusion will also hold". This is the same as saying "it is impossible for the premises to be true and yet the conclusion is false."
When your proof reaches a contradiction what you have shown is that it is impossible for the premises to be true, period. Therefore it is in particular impossible for the premises to be true and at the same time the conclusion is false. The is what it means that the conclusion follows from the premises.
