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> It becomes a theorem when you prove (i.e. explain why) this relationship always holds, based on more evident things. [...] whereas the Greeks definitely did (
by orra 3y ago
> It becomes a theorem when you prove (i.e. explain why) this relationship always holds, based on more evident things. [...] whereas the Greeks definitely did (we don't know whether Pythagoras himself did it
Wait, really? I thought the proofs back then were geometric, so only proved specific instances. And so it wasn't until algebra, maybe trig, was discovered it was properly proven?
- n4r9 3y agoThere are fully general and valid geometric proofs, eg http://mathandmultimedia.com/wp-content/uploads/2010/02/pythagrean41.png http://mathandmultimedia.com/wp-content/uploads/2010/02/pyth...
- orra 3y agoDespite labels $a$, $b$, $c$, I don't see how it's fully general. At best you could say it works for similar triangles, if you have that concept. But I don't see how it proves anything if $a$ were say doubled, and $b$ kept the same.
- n4r9 3y agoI'm not sure what you mean, sorry. It's definitely fully general. It demonstrates that: * For any real positive values $a$ and $b$ * let $T$ be the (unique) right-angled triangle with non-hypotenuse sides of length $a$ and $b$ * then $a$^2 = $b$^2 = $c$^2 * where $c$ is the (unique) length of the hypotenuse of $T$. You could give me the instance $a$ = 2, $b$ = 3 and it will prove it for this case. Then you could later double $a$ and give me the instance $a$ = 4, $b$ = 3 and it will prove it just as easily for this case.