Geometric geekery

[spudtater]

Non-algebraic proofs of Pythagoras' theorem:
- First an animation. Although it might not be strictly a proof, it's a great demonstration. (Alternate local copy.)
- Then a more thorough proof by tiling. A little less accessible, though.

I want to do geometric proofs! For example,

Is this a proof?

c d a ac ad b bc bd a+b (a+b)(c+d) c+d

Oh, and blah blah Pope blah.

Comments

[sigmonster.livejournal.com]

A defining property of 1 (the identity in an algebraic system) is 1 \times r = r \times 1 = r, for any object r in the system. (Even for non-commutative systems, like matrix multiplication, the identity always commutes.)

So if there are two identities, 1_a and 1_b, we can write

1_a = 1_a \times 1_b = 1_b

and they are the same!

[spudtater.livejournal.com]

That proves that 1_a and 1_b have the same numeric value. It doesn't prove that they are identical.

[sigmonster.livejournal.com]

Hoh yus it does, sunshine. I'm using the strongest possible value of equals (in the given system), *both in the definition and in the equation*, which is identity (in the given system). Ought to be the triple parallel line symbol, really.

Otherwise you'd be saying that 1 \times r and r are not identical. As a matter of typography, that's trivially true - but there is no way at all to distinguish them by the algebraic rules, so we call them identical in that system.

We're getting into some deep waters, here...

[sigmonster.livejournal.com]

Just had another quick thought - this is also the reason 1 isn't a prime; you can tag on 1 \times 1 \times 1 ... and still have the identical integer, so you lose unique prime factorisation. In larger fields, you specify unique prime factorisation *up to units* (where a unit has a multiplicative inverse, basically).

[spudtater.livejournal.com]

Er... oh... okay.
Guess I wasn't taking 'defining property' as literally as I should have been.

[sigmonster.livejournal.com]

It's all consistent, and it all fits the real numbers and their extensions. It doesn't have to make sense.

See also: Proof by Intimidation.