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File: 2 KB, 347x263, PythagoreanTheoremFigure_1000.gif [View same] [iqdb] [saucenao] [google] [report]

Why does the Pythagorean theorem work? I know the proof representing the squares geometrically, but why do the squares of the two legs equal the area of the square of the hypotenuse? It just seems like magic to me. I can't register any intuition for it.

 >> Anonymous Fri Jun 9 12:52:20 2017 No.8966238 >>8966217it's just the way it is, don't question it. just like earth being a ball
 >> Anonymous Fri Jun 9 12:58:54 2017 No.8966243 >>8966238oblate spheroid
 >> Anonymous Fri Jun 9 12:59:14 2017 No.8966245 >>8966217It's just the formalism we've chosen. We are lucky to have so many nice properties from a handful of axioms and definitions.
 >> Anonymous Fri Jun 9 13:05:21 2017 No.8966258 >>8966245I mean, isn't there a step by step, logical sequence following the axioms//properties that explains why it works?
 >> Anonymous Fri Jun 9 13:13:57 2017 No.8966275 >>8966238but there is a very detailed and explainable reason for why earth is a sphere(ish), also physics =/= math
 >> Anonymous Fri Jun 9 13:34:12 2017 No.8966300 File: 1.34 MB, 360x263, giphy.gif [View same] [iqdb] [saucenao] [google] [report]
 >> Anonymous Fri Jun 9 13:36:02 2017 No.8966305 >>8966300That doesn't explain shit, that just shows that it's true for this example
 >> Anonymous Fri Jun 9 13:37:57 2017 No.8966308 >>8966305It explains everything. It's true for any example.
 >> Anonymous Fri Jun 9 13:39:45 2017 No.8966314 >>8966308Like I said, I understand the proof, I just don't understand why the legs have the relationship with the hypotenuse, besides "it just does"
 >> Anonymous Fri Jun 9 13:39:58 2017 No.8966315 File: 39 KB, 736x818, 736px-Pythagorean_proof_(1).svg.png [View same] [iqdb] [saucenao] [google] [report]
 >> Anonymous Fri Jun 9 13:40:31 2017 No.8966316 >>8966308An example doesn't prove anything, and showing that the two areas are the same by a process as hard to follow as a flowing fluid doesn't provide any intuition either.
 >> Anonymous Fri Jun 9 13:41:33 2017 No.8966318 >>8966315Not a proof and not intuitive.
 >> Anonymous Fri Jun 9 13:49:19 2017 No.8966333 >>8966318it is a proof and it only involves simple shapes.
 >> Anonymous Fri Jun 9 13:55:47 2017 No.8966343 File: 27 KB, 800x694, 1.png [View same] [iqdb] [saucenao] [google] [report] >>8966333Then you won't have trouble accepting pic related which shows that 32.5 = 31.5.
 >> Anonymous Fri Jun 9 14:01:37 2017 No.8966357 >>8966343>which shows that 32.5 = 31.5.how so?
 >> Anonymous Fri Jun 9 14:02:01 2017 No.8966358 >>8966318It is both a proof and intuitive!
 >> Anonymous Fri Jun 9 14:03:47 2017 No.8966362 File: 19 KB, 949x433, fag.png [View same] [iqdb] [saucenao] [google] [report]
 >> Anonymous Fri Jun 9 14:05:10 2017 No.8966365 >>8966362This.Also pls ban to highschool fas that cant find a proof for Pythagorean theorem.
 >> Anonymous Fri Jun 9 14:05:17 2017 No.8966367 >>8966357by shaving off a very tiny portion of each square
 >> Anonymous Fri Jun 9 14:42:29 2017 No.8966436 >>8966217It has to be something. Same thing with pi. Why is it ~3.14 and not 60? Just philosophy. It is a very specific relationship with how space exists between 2 dimensions
 >> Anonymous Fri Jun 9 14:43:31 2017 No.8966438 >>8966362this is the best proof
 >> Anonymous Fri Jun 9 15:43:47 2017 No.8966563 >>8966314There's always a relationship when you compare lengths. It's like asking why 1m = 3.3 ftYou start with definitions and theorems and end up with results. It just happens that in our construction, this is the relation. There's an infinite number of other ways to construct it as well. There's nothing special about it except that it's convenient.
 >> Anonymous Fri Jun 9 15:48:56 2017 No.8966573 >>8966238>just like earth being a ballSo the pythagorean theorem is propaganda?
 >> Anonymous Fri Jun 9 15:55:18 2017 No.8966589 >>8966217The Pythagorean theorem works because of inner product spaces.We commonly use the dot product to determine orthogonality.For example, lets say that a and b are n-dimensional vectors, and that they are orthogonal to eachother - aka a • b = 0To find the length of a vector, you dot it with itself then take a square root. ||a||^2 = a • a = length(a)^2Lets try this out!Once again, let a,b be orthogonal vectors...||a+b||^2 = (a+b)•(a+b) Since dot products are linear in the real numbers, we can split this up.(a+b)•(a+b) = a•a + a•b + b•a + b•bSince a and b are orthogonal, a•b=0a•a + a•b + b•a + b•b = a•a + b•b = ||a||^2 + ||b||^2Thus for 2 orthogonal vectors, you find the combined length like so...||a+b||^2 = ||a||^2 + ||b||^2
 >> Anonymous Fri Jun 9 15:58:26 2017 No.8966600 >>8966573install allah it is
 >> Anonymous Fri Jun 9 16:00:56 2017 No.8966604 >>8966258 Btw here are the axioms you are allowed to work with in order to prove pythagoreans theoremhttps://en.m.wikipedia.org/wiki/Inner_product_spaceTwo vectors are perpendicular/orthogonal if their dot product evaluates to 0Look right above for proof
 >> Anonymous Fri Jun 9 16:03:46 2017 No.8966610 >>8966314I mean why wouldn't they? Just imagine a right triangle. And you make one leg longer or shorter, you see the hypotenuse get longer and shorter accordingly. Same with the other leg.It clear that there is SOME kind of relation between the legs ans hypotenuse.As for why it is sqrt(b^2 + c^2). Who knows, why is PI the number it is, Why is the speed of light the speed it is. It's a stupid question.Just be glad it's something easy unlike other trig functions like sin or cos.
 >> Anonymous Fri Jun 9 16:07:36 2017 No.8966619 thats just the way right angled triangles evolved to their surroundings. isoceles evolved in a different way for a different purpose
 >> Anonymous Fri Jun 9 16:13:18 2017 No.8966630 File: 64 KB, 661x652, pythagoras.png [View same] [iqdb] [saucenao] [google] [report] It's a thing stemming from the property of Corresponding Angles. Note that the colored pieces fit to each other only because of those angles. The squares are the key for a reason.
 >> Anonymous Fri Jun 9 17:06:27 2017 No.8966760 File: 638 KB, 624x588, bs9gG.gif [View same] [iqdb] [saucenao] [google] [report] >>8966343>32.5 = 31.5.
 >> Anonymous Fri Jun 9 18:40:49 2017 No.8966984 >>8966619Made me kek. Have a you.
 >> Anonymous Fri Jun 9 18:58:00 2017 No.8967010 >>8966760besides the area of the two shapes is 32 and 33 but our brainlet friend was too dumb to notice
 >> Anonymous Fri Jun 9 19:26:38 2017 No.8967064 >>8966217>>8966316>>8966318mathematical intuition is not something anyone is born with, it is developed by years of study
 >> Anonymous Fri Jun 9 21:25:09 2017 No.8967223 >>8966630>>8966589>>8966362>>8966604OP here, thank you all, I'm starting to understand it. Geometry is fuckin crazy.
 >> Anonymous Fri Jun 9 21:26:22 2017 No.8967225   File: 10 KB, 668x681, pyth.png [View same] [iqdb] [saucenao] [google] [report] It's basically a projection of the area of the smaller squares by the parallelogram rule.
 >> Anonymous Fri Jun 9 21:36:03 2017 No.8967240 >>8967064Nobody claimed it was.
 >> Anonymous Sat Jun 10 15:34:59 2017 No.8968575 >>8966243Pear shaped.
 >> Anonymous Sat Jun 10 15:36:03 2017 No.8968576 >>8966305It's related to the sacred portion and the very nature of reality itself.
 >> Anonymous Sat Jun 10 18:55:15 2017 No.8968873 >>8966258Because then it would be the Pythagorean Law instead of just something that works so far on every triangle we've tried so far.
 >> Anonymous Sat Jun 10 19:15:25 2017 No.8968910 >>8966217Because the inner square in the square diagram can slide completely from one side of the outer square to the other, changing size appropriately. This animation generates a set of right triangles to which all right triangles in general are similar, so that PT, which is clearly true in any one case, is true for all such cases, and by similarity applies to all right triangles in general.
 >> Anonymous Sat Jun 10 21:12:42 2017 No.8969052 >>8968575Coast guard says oblate spheroid so I say oblate spheroid. You want to call it a pear, issue me an Oceans license.
 >> Anonymous Sat Jun 10 21:18:59 2017 No.8969059 >>8966589>unicode$\sf \color{red}{Git\; gud,\; brainlet!!!}$
 >> Anonymous Sat Jun 10 21:35:28 2017 No.8969084 >>8969059more upset by the lack of inner productsfor two orthogonal vectors $a$, $b$[eqn] \|a+b\|^2 = \langle a+b , a+ b\rangle = \langle a , a \rangle + \langle a , b \rangle + \langle b,a \rangle + \langle b,b \rangle =\|a\|^2 + \|b\|^2 [/eqn]
 >> Anonymous Sat Jun 10 21:44:26 2017 No.8969096 >>8966217>Tesla theorized that the application of electricity to the brain enhanced intelligence. In 1912, he crafted "a plan to make dull students bright by saturating them unconsciously with electricity," wiring the walls of a schoolroom and, "saturating [the schoolroom] with infinitesimal electric waves vibrating at high frequency. The whole room will thus, Mr. Tesla claims, be converted into a health-giving and stimulating electromagnetic field or 'bath.'"[180] The plan was at least provisionally approved by then superintendent of New York City schools, William H. Maxwell.
 >> Anonymous Sat Jun 10 22:01:10 2017 No.8969120 Is it possible to write the equation for a 60 degree triangle as c^2 = (ax)^2 + (bx)^2
 >> Anonymous Sat Jun 10 22:07:47 2017 No.8969127 >>8969120$c^2 = a^2+b^2-2(a)(b)cos(C)$
 >> Anonymous Sat Jun 10 22:21:56 2017 No.8969141 File: 11 KB, 793x423, 60.png [View same] [iqdb] [saucenao] [google] [report]
 >> Anonymous Sat Jun 10 22:55:20 2017 No.8969174 >>8966589Circular logic at its finest.
 >> Anonymous Sun Jun 11 04:08:00 2017 No.8969508 >>8969059Did it in phone, rip
 >> Anonymous Sun Jun 11 04:55:35 2017 No.8969553 >>8966258>I mean, isn't there a step by step, logical sequence following the axioms//properties that explains why it works?Yes, there is and it is over 2000 years old.Euclid himself proved the Pythagorean theorem.You can do the same in "modern" linear algebra within 1 line, even for arbitrary scalar products on arbitrary vector spaces.(It is even true for functions in L^2)
 >> Anonymous Sun Jun 11 04:56:39 2017 No.8969556 >>8969174Are you retarded?
 >> Anonymous Sun Jun 11 05:30:39 2017 No.8969604 >>8969556yes
 >> Anonymous Sun Jun 11 05:31:53 2017 No.8969608 >>8969604Thats what I expected...
 >> Anonymous Sun Jun 11 06:23:52 2017 No.8969674 >>8969608that wasn't me. ||a||^2 = a • a = length(a)^2 is derived somewhat from Pythagoras theorem. At least that's what my lecturer told us.
 >> Anonymous Sun Jun 11 06:26:11 2017 No.8969677 >>8966217Read Euclid's Elements.
 >> Anonymous Sun Jun 11 06:30:14 2017 No.8969680 >>8967223Kill yourself you idiot. What a shit thread.
 >> Anonymous Sun Jun 11 06:31:54 2017 No.8969681 >>8969556No, you are. The dot product is so defined so as to construct classical geometrical results in Euclidean geometry.
 >> Anonymous Sun Jun 11 06:35:40 2017 No.8969685 >>8966589>The Pythagorean theorem works because of inner product spaces.This is not even wrong.>>8969084Nice circular logica there dumb-ass.>still not getting that inner product spaces are DEFINED as they are defined so that the Pythagorean theorem holds in them
 >> Anonymous Sun Jun 11 06:35:53 2017 No.8969686 >>8969608>>8969674My bad. You use Pythagoras theorem to define the length of the vector, that's the reason why its a circular argument.
 >> Anonymous Sun Jun 11 06:36:42 2017 No.8969688 >>8969553This is the only good reply in this thread.
 >> Anonymous Sun Jun 11 06:50:48 2017 No.8969706 >>8969686>>8969674>You use Pythagoras theorem to define the length of the vector> length(a)^2 is derived somewhat from Pythagoras theoremThis is only the case for the 2 norm on R^n, but as it turns out any norm you choose (on a space with a scalar product which induces that norm) will lead you to Pythagorean theorem.Pythagoras theorem is also true when you consider much more abstract function spaces, something like L^2, where function can have a "length" and be "orthogonal" to each other.
 >> Anonymous Sun Jun 11 06:58:24 2017 No.8969711 I don't understand why euclidian distance is e=sqrt(a^2+b^2+c^2), based on pythagoras' theorem i'd figure it to be e=sqrt(c^2+d^2) where d=sqrt(a^2+b^2) so e=sqrt(sqrt(a^2+b^2)+c^2)but supposedly it's the former and i cannae see why
 >> Anonymous Sun Jun 11 07:04:04 2017 No.8969718 >>8969711>I don't understand why euclidian distance is e=sqrt(a^2+b^2+c^2), based on pythagoras' theorem i'd figure it to be e=sqrt(c^2+d^2) where d=sqrt(a^2+b^2)e=sqrt(c^2+d^2)d=sqrt(a^2+b^2)=>e=sqrt(c^2+(sqrt(a^2+b^2))^2)=sqrt(a^2+b^2+c^2)=/=sqrt(sqrt(a^2+b^2)+c^2)
 >> Anonymous Sun Jun 11 07:09:01 2017 No.8969723 >>8969706>as it turns out any norm you choose (on a space with a scalar product which induces that norm) will lead you to Pythagorean theorem.Gee, you think? Does the triangle inequality ring any bells? You know, one of the defining properties of a norm?Keep using circular arguments dumb-ass. > something like L^2, where function can have a "length" and be "orthogonal" to each other.More circular shit. Nice. You're an idiot.
 >> Anonymous Sun Jun 11 07:14:47 2017 No.8969728 >thread is full of brainlets talking about metric spacesFor anyone actually interested in why the Pythagorean theorem holds: https://en.wikipedia.org/wiki/Hilbert%27s_axioms
 >> Anonymous Sun Jun 11 07:16:57 2017 No.8969732 >>8969723What?what does the triangle INEQUITY have to do with Pythagorean's theorem? Aside from the fact that the EQULITY holds if the 2 vectors are orthogonal?Do you have any clue what you are talking about.Orthogonality in R^2 is equivalent to a 90° degree angle between vectors and if you accept that the usual distance in our world is ||.||_2 then Pythagorean's theorem will go from:"in a triangle with a right angle a^2+b^2=c^2" to "||x||^2+||y||^2=||x+y||^2 if x is orthogonal to y"If you want a logical deduction NOT based on linear algebra go read euclid, but the assertion that any logic is "circular" is nonsensical.
 >> Anonymous Sun Jun 11 07:24:43 2017 No.8969739 >>8966217This is a relationship that applies to more than squareshttps://youtu.be/ItiFO5y36kw?t=2m50s
 >> Anonymous Sun Jun 11 07:45:30 2017 No.8969758 >>8969732>what does the triangle INEQUITY have to do with Pythagorean's theorem? Everything. Also>inequity
 >> Anonymous Sun Jun 11 08:12:30 2017 No.8969779 >>8967240I think he's saying, "stfu and study".
 >> Anonymous Sun Jun 11 10:25:21 2017 No.8969962 >>8969732are you, like, only pretending to be retarded?
 >> Anonymous Sun Jun 11 10:37:49 2017 No.8969987 Guys triangles are sentient 90 dimensional polyhedrons were just looking one side aaaaaaaaahbbbbbbb
 >> Anonymous Sun Jun 11 14:51:21 2017 No.8970335 File: 633 KB, 480x312, ezgif-2-e78ed736d9.gif [View same] [iqdb] [saucenao] [google] [report] >>8966217I can be explained simply by rearranging areas.
 >> Anonymous Sun Jun 11 15:37:44 2017 No.8970412 >>8967010You're not very bright, are you? The evidently false "proof" argued that the two shapes are the same except for the missing square, which only works if the diagonal is straight (which it is not), under which assumption the areas are 13*5/2 and 13*5/2 - 1.
 >> Anonymous Sun Jun 11 15:39:36 2017 No.8970416 >dude Pythagorean theorem lmao
 >> Anonymous Mon Jun 12 12:15:10 2017 No.8971744 >>8970335if this isn't enough of a proof for some, they're just too lazy to fill in the formalities of the proof anyway.
 >> Anonymous Mon Jun 12 14:57:15 2017 No.8971936 File: 12 KB, 1000x588, pythagorean-tiling.png [View same] [iqdb] [saucenao] [google] [report] This is how I make sense of the Pythagorean theorem, and I think it's the most intuitive. If you have tiles of any two different sizes, you can tile a space with them like in pic related. Then if you connect recurring points, you get a new grid of squares larger than the tiles you started with and tilted at an angle. No matter what recurring point you pick (could be the centers of the smaller squares, could be the centers of the larger squares, for example); when you connect them and make a grid, the pieces inside the grid can always be rearranged to make the one of each of the original tiles. This just makes sense. Then when you pick a corner for the recurring point, you form a right triangle with the tiles as the legs and the grid square as the hypotenuse.
 >> Anonymous Mon Jun 12 15:04:14 2017 No.8971946 >>8971936*tile a plane
 >> Anonymous Mon Jun 12 17:47:34 2017 No.8972201 >>8966217I think what will satisfy you is the proof by dimensional analysis (in essence, the analysis of units). It is far more complicated than the geometric proof, but I think it is far more insightful in terms of asking "why" rather than "how do you know". It demonstrates that the result of the theorem is necessary rather than just being "just the way it is". Here's a good explanation of the proof.https://www.google.ca/amp/s/strathmaths.wordpress.com/2011/10/15/pythagoras-two-favourite-proofs/amp/
 >> Anonymous Mon Jun 12 17:51:03 2017 No.8972207 >>8972201>far more complicatedWhoops, didn't mean to say that, I meant it's just a little more complicated but far more insightful.
 >> Anonymous Mon Jun 12 19:12:36 2017 No.8972331 >>8969680Damn, why so bitter? Do you actually go to 4chan looking for quality content?
 >> Anonymous Mon Jun 12 21:18:16 2017 No.8972497 >>8966217There are different geometric proofs. You need to say which one you're talking about. By the wording of your question I assume that you actually don't understand the proof.
 >> Anonymous Mon Jun 12 23:51:29 2017 No.8972706 >>8972497His question is more related to certain mathematical relationships like Pi and e being irrational numbers, although the Pythagorean theorem applies as well. Irrational numbers are the reason I believe in God.
 >> Anonymous Tue Jun 13 22:03:47 2017 No.8974409 >>8972706>Irrational numbers are the reason I believe in God.wat
 >> Anonymous Tue Jun 13 22:09:31 2017 No.8974414 File: 681 KB, 1098x550, Screen Shot 2017-05-25 at 11.23.15 PM.png [View same] [iqdb] [saucenao] [google] [report] >>8972706>Irrational numbers are the reason I believe in God.
 >> Anon !!3wHWqEMsmPf Tue Jun 13 22:12:57 2017 No.8974416 >>8966217http://jwilson.coe.uga.edu/EMT668/EMT668.Student.Folders/HeadAngela/essay1/Pythagorean.htmlPythagoras' proof is pretty intuitive. Just start from "The area of the first square is given by (a+b)^2..."
 >> Anonymous Tue Jun 13 22:14:39 2017 No.8974418 >>8969680My IQ is 142, according to WAIS-IV. Don't be mad just because intelligent people like to question things.
 >> Anonymous Tue Jun 13 22:39:38 2017 No.8974453 >>8974409Why is Pi Pi?You're eventually going to realize you're not nearly as clever as you think you are.
 >> Anonymous Wed Jun 14 19:43:17 2017 No.8975964 >>8974453I don't follow, how does that imply a god?
 >> Anonymous Thu Jun 15 01:50:46 2017 No.8976415 >>8966318I disagree>>8966315Thanks for this image I like it>>8966362Surprised this wasn't posted sooner. This is how I learned it.
 >> Anonymous Thu Jun 15 02:12:07 2017 No.8976433 File: 65 KB, 800x436, pythag_proof_china.jpg [View same] [iqdb] [saucenao] [google] [report] >>8966217>>8966362It really isn't that hard OP
 >> Anonymous Thu Jun 15 02:27:42 2017 No.8976449 >>8969174>being a brainlet
 >> Anonymous Thu Jun 15 02:33:32 2017 No.8976454 >>8966217it's actually the very first link if you fucking google "proof of the pythagorean theorem"http://jwilson.coe.uga.edu/EMT668/EMT668.Student.Folders/HeadAngela/essay1/Pythagorean.htmlmy man there is but 1 thing you should do
 >> Anonymous Thu Jun 15 04:51:41 2017 No.8976600 File: 13 KB, 394x407, pthag_proof.png [View same] [iqdb] [saucenao] [google] [report] i just like drawing these triangles
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