Euclid's Proof that √2 is Irrational

If you know a child in middle school, this is a great way to get them started on "cool mathematical thinking", which I called "Mathematia" when I discussed with my son when he was young (to distinguish from the horrible Math being taught in school):

  1. Introduce ℤ & ℚ - this is easy. Perhaps, fingers and slices of pizza. Now s/he's ready to be as surprised as the members of the Phythagorean cult

  2. Go over the classical proof for √2 given here. We now have a number that's not in ℤ or ℚ!

  3. It's one thing to show a result, a very different thing to *grasp* it. Why is (2) a big deal? It smashes the simple notion Greeks had that *any* two lengths (rational numbers) are commensurable, which is a perfectly simple and obvious (and wrong) thing to believe: "Have one stick for one side of a square and another for the diagonal. You cannot cut both sticks into pieces of the same length, no matter what length you choose." *This is amazing*

  4. We only discovered one such weird number. Are there others? Motivated by the above, how about checking √3. Show that it's weird, too.

  5. √4 is just 2. How about √5? OMG, that's weird, too.

  6. So the square root of an integer is either an integer or one of these weird numbers. It cannot be of the general ℚ form p/q. This is an interesting proof. (While thinking about that with the youngster you can think about another generalization: roots higher than second. Turns out it's true for those, too: https://math.stackexchange.com/questions/4467/how-to-prove-if-a-b-in-mathbb-n-then-a1-b-is-an-integer-or-an-irratio

  7. How do we work these weird numbers? For example, can we add them up, e.g. √2 + √3? How do we do that? Is that another weird number or could it ever be an integer? Some facts about these sums are trivial to prove: https://math.stackexchange.com/questions/157245/is-the-sum-and-difference-of-two-irrationals-always-irrational

  8. Using the wacky notion of adding two numbers as "mating" you can generally outline some higher algebra concepts, e.g. if a lion mates with a lion the result is always a lion. What if it mates with a tiger? (depends, liger or tigon). Can we think of adding a ℚ to one of these weird numbers the same way? Such intuitions may be misleading (remember the Greeks?) but are fun.

Where in Euclid's Elements does this proof appear? I can't seem to find it.

> 1) it's not a proof by contradiction, it's a proof of a negation :grump:

I don't get your complaint. It is a proof of a negation, yes, the conclusion is that √2 ∉ ℚ.

But the proof is done by contradiction; "it's not a proof by contradiction" is flat-out false. "Proof by contradiction" describes the method of the proof, and "proof of a negation" describes its conclusion, which is why one of those phrases uses by and the other one uses of.

Couldn’t you stop the proof at the statement q^2 must equal 2m^2 since it’s obvious there’s no solutions to q^2 = 2m^2.

To explain why it’s obvious, squares always have an even number if factors of two (an even multiple of any prime factor since it’s a square but just focus in on 2 here for now).

A square times two always has an odd number of factors of 2 since it’s the above (an even number of factors of two) plus one more factor.

An odd number of prime factors on one side can’t be equal to an even number of factors on the other side. q^2 can never equal 2m^2 for any integer value of a or m. Therefore it’s irrational.

This ends the proof much earlier right?

https://en.wikipedia.org/wiki/Proof_by_contradiction, https://ncatlab.org/nlab/show/proof+by+contradiction, https://web.stanford.edu/class/cs103/guide_to_proofs#proof-b... all agree (the first three things that came up when I googled for "proof by contradiction"): a proof by contradiction is specifically a proof which shows that P is not false, and concludes that it is true. There is already a perfectly cromulent term for proofs of negations: "refutation by contradiction" (https://ncatlab.org/nlab/show/refutation+by+contradiction), which admittedly nlab says it prefers to its other name "proof of negation".