Why does the Pythagorean theorem work? I know the proof representing the squares geometrically...

...

An 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.

Not a proof and not intuitive.

it is a proof and it only involves simple shapes.

Then you won't have trouble accepting pic related which shows that 32.5 = 31.5.

>which shows that 32.5 = 31.5.
how so?

It is both a proof and intuitive!

...

This.

Also pls ban to highschool fas that cant find a proof for Pythagorean theorem.

by shaving off a very tiny portion of each square

It 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

this is the best proof

There's always a relationship when you compare lengths. It's like asking why 1m = 3.3 ft
You 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.

>just like earth being a ball
So the pythagorean theorem is propaganda?

The 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 = 0
To find the length of a vector, you dot it with itself then take a square root.
||a||^2 = a • a = length(a)^2

Lets 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•b

Since a and b are orthogonal, a•b=0

a•a + a•b + b•a + b•b = a•a + b•b = ||a||^2 + ||b||^2

Thus for 2 orthogonal vectors, you find the combined length like so...

||a+b||^2 = ||a||^2 + ||b||^2

install allah it is

Btw here are the axioms you are allowed to work with in order to prove pythagoreans theorem

en.m.wikipedia.org/wiki/Inner_product_space

Two vectors are perpendicular/orthogonal if their dot product evaluates to 0

Look right above for proof

I 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.

thats just the way right angled triangles evolved to their surroundings. isoceles evolved in a different way for a different purpose

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.

>32.5 = 31.5.

Made me kek. Have a you.

besides the area of the two shapes is 32 and 33 but our brainlet friend was too dumb to notice

mathematical intuition is not something anyone
is born with, it is developed by years of study

OP here, thank you all, I'm starting to understand it. Geometry is fuckin crazy.

It's basically a projection of the area of the smaller squares by the parallelogram rule.

Nobody claimed it was.

Pear shaped.

It's related to the sacred portion and the very nature of reality itself.

Because then it would be the Pythagorean Law instead of just something that works so far on every triangle we've tried so far.

Because 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.

Coast guard says oblate spheroid so I say oblate spheroid. You want to call it a pear, issue me an Oceans license.

>unicode
[math]\sf \color{red}{Git\; gud,\; brainlet!!!}[/math]

more upset by the lack of inner products
for two orthogonal vectors [math] a [/math], [math] b [/math]
[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]

>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.

Is it possible to write the equation for a 60 degree triangle as

c^2 = (ax)^2 + (bx)^2

[math]c^2 = a^2+b^2-2(a)(b)cos(C)[/math]

...

Circular logic at its finest.

Did it in phone, rip

>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)

Are you retarded?

yes

Thats what I expected...

that 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.

Read Euclid's Elements.

Kill yourself you idiot. What a shit thread.

No, you are. The dot product is so defined so as to construct classical geometrical results in Euclidean geometry.

>The Pythagorean theorem works because of inner product spaces.
This is not even wrong.
Nice 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

My bad. You use Pythagoras theorem to define the length of the vector, that's the reason why its a circular argument.

This is the only good reply in this thread.

>You use Pythagoras theorem to define the length of the vector
> length(a)^2 is derived somewhat from Pythagoras theorem
This 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.

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

>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)

>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.

>thread is full of brainlets talking about metric spaces
For anyone actually interested in why the Pythagorean theorem holds: en.wikipedia.org/wiki/Hilbert's_axioms

What?
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.

This is a relationship that applies to more than squares

youtu.be/ItiFO5y36kw?t=2m50s

>what does the triangle INEQUITY have to do with Pythagorean's theorem?
Everything. Also
>inequity

I think he's saying, "stfu and study".

are you, like, only pretending to be retarded?

Guys triangles are sentient 90 dimensional polyhedrons were just looking one side aaaaaaaaahbbbbbbb

I can be explained simply by rearranging areas.

You'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.

>dude Pythagorean theorem lmao

if this isn't enough of a proof for some, they're just too lazy to fill in the formalities of the proof anyway.

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.

*tile a plane

I 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.
google.ca/amp/s/strathmaths.wordpress.com/2011/10/15/pythagoras-two-favourite-proofs/amp/

>far more complicated
Whoops, didn't mean to say that, I meant it's just a little more complicated but far more insightful.

Damn, why so bitter? Do you actually go to Veeky Forums looking for quality content?

There 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.

His 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.

>Irrational numbers are the reason I believe in God.
wat

>Irrational numbers are the reason I believe in God.

jwilson.coe.uga.edu/EMT668/EMT668.Student.Folders/HeadAngela/essay1/Pythagorean.html

Pythagoras' proof is pretty intuitive. Just start from "The area of the first square is given by (a+b)^2..."

My IQ is 142, according to WAIS-IV. Don't be mad just because intelligent people like to question things.

Why is Pi Pi?

You're eventually going to realize you're not nearly as clever as you think you are.

I don't follow, how does that imply a god?

I disagree
Thanks for this image I like it
Surprised this wasn't posted sooner. This is how I learned it.

It really isn't that hard OP

>being a brainlet

it's actually the very first link if you fucking google "proof of the pythagorean theorem"

jwilson.coe.uga.edu/EMT668/EMT668.Student.Folders/HeadAngela/essay1/Pythagorean.html

my man there is but 1 thing you should do

i just like drawing these triangles