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Pythagoras theorem

Holy shit. I just had a test and I needed to use the fact that [math] \sqrt{ a^2 + b^2 } \leq |x|+|y| [/math] but I wasn't 100% sure that it was true so I analytically proved it and it wasn't that hard but holy crap, all I needed to do was look at a goddamn triangle. Never forget your geometric intuition, children.

>>11046749

>>11046749Pythagoras theorem... Home...

>>11047019But that isn't true.

>>11047088i dont believe you

>>11047092a=1, b=2, x=3, y=4.

>>11047088I now see that I replaced a and b with x and y for no reason, but you get what I'm saying. If you mean that [math] \sqrt{ a^2 + b^2 } \leq |a|+|b| [/math] is not true, then I don't know what to tell you, man. You're literally looking at it.

>>11047096Sorry, other way around.x=1, y=2, a=3, b=4

>>11047099>>11047096Holy fuck you are stupid

>>11047077|1+i|=sqrt2

>>11047077Well you can by definition not have complex length since [eqn]i \notin \mathbb{R}[/eqn] and the reals is the set of all lengths in euclidean space

>>11047110>>11047099The mistake was mine for fucking up the notation in the first place. There's no need to be an asshole, anon. On the other side, there's also no need to be a smart-ass either, other anon.

>>11047129What if you extend the complex numbers by adding a "hallucinatory number" h, where h is the solution to [math]|h|=-1[/math]?

>>11047149>What if you extend the complex numbers by adding a "hallucinatory number" h, where h is the solution to [math]|h|=-1[/math]?You could but you'd have to redefine norm in a way that allows for it to be negative but it's more useful and coincides with its current definition. And it would still make no sense talking about imaginary lengths, because lengths are real numbers greater than or equal to zero and imaginary numbers are none of those.

>>11047149How would it work for |x+hy|, x and y real numbers?

>>11047149https://en.wikipedia.org/wiki/Minkowski_space

>>11047077The illustration in this is a bit disengenuous. Since the imaginary numbers are orthogonal to the reals, the side with length i would now point back along the side with length 1 (in order to keep the 90° angle) so the image would make sense.

>>11047110They're correct w.r.t the original equation.

>>11047588I assume |x| - h|y|

>>11046749Its not doing anything

>>11046958Once again, like with the circle, the infinite staircase has infinitely many deviations from the line

>>11046749tan^-1(b/a)sin(tan^-1(b/a))=b/ccsin(tan^-1(b/a))=bc=b/sin(tan^-1(b/a))

>>11046958fractals will never cease to be a mindfuck