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/sci/ - Science & Math

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File: 21 KB, 400x205, pythag.png [View same] [iqdb] [saucenao] [google]
11338059 No.11338059 [Reply] [Original]

I can intuit the truth of this by looking at it visually, but I'm unsure if this technically counts as a rigorous proof. In what sense are these kinds of visual "proofs" appreciated or taken seriously by mathematicians?

>> No.11338065

They are appreciated but not rigorous.

>> No.11338258

>>11338065
That's what I suspect, but what makes you say that? You do math?

>> No.11338488

Why isn't it rigorous enough? Because it assumes shapes don't change their areas when moved?

>> No.11338516

>>11338059
when accompanied by the corresponding mathematical formulas
the drawing image is a graphical representation for conceptualization sake

>> No.11339233

It's a bit disconcerting how easy it can be to make invalid proofs using pictures. Having formulas is necessary.
I would even suggest throwing away the picture when double checking the proof to make sure you don't make any false assumptions.
For instance you may have three points ABC, so you draw a little triangle. This triangle looking very triangly and all, it may cause you to forget to check whether ABC lie on the same line.

>> No.11339249
File: 5 KB, 640x400, bh2.png [View same] [iqdb] [saucenao] [google]
11339249

Here is a "proof" that the area of a triangle is base * height / 2.
The triangle can be divided into a left and right part, each of which is half the area of a corresponding rectangle. The two rectangles make up a bigger rectangle with area base * height.

Which implicit assumption have we made?

>> No.11339259

>>11338488
because that's a proof for only one particular triangle, the one in the picture. are you sure that the argument will work for every triangle ? are you sure you haven't made other assumptions which aren't part of the theorem ?

>> No.11339264
File: 90 KB, 792x785, 2eb3d1fde391571be6e22a07380a0101.jpg [View same] [iqdb] [saucenao] [google]
11339264

"Mathematical proofs" are just arguments of a sort. And like any other argument, they can be very rock solid, totally indisputable, or they can be somewhat dubious or weak, even if the the statement to be proven is known to be correct.
The pic in the OP is certainly an argument for the Pythagorean theorem, therefore it counts as a proof.
>>11339249
Only applies to right triangles. I can think of a way to sketch up a visual proof for any kind of triangle, but I am lazy.

>> No.11339266

>>11339259
>that's a proof for only one particular triangle
No it isn't. a, b, and c are totally unspecified. The only restriction is a+b>c. It counts for all right triangles.

>> No.11339297

>>11339266
the picture is literally a proof only for the triangle that is on the picture

>> No.11339305

>>11339297
Good thing the triangle in the picture is equivalent to all right triangles

>> No.11339316

>>11339305
>equivalent
don't use that word if you don't know what it means

>> No.11339323

>>11339316
Okay. I know what it means, though.

>> No.11339343

>>11339264
> Only applies to right triangles.
No. That's not it.

>> No.11339354

>>11339343
What is it, then?

>> No.11339358

>>11338059
Pictures are not proofs, period. Pictures are only useful for explaining a concept to someone.

>> No.11339371
File: 2 KB, 640x400, bh2_2.png [View same] [iqdb] [saucenao] [google]
11339371

>>11339354
We assumed the triangle is contained in the box that shares the triangles base, and has the height of the triangle.

>> No.11339404

>>11339358
Why not?

>> No.11339409

>>11339404
Because they can be drawn to be correct.
In the same amount of time that you can discern truth to that image in OP, I want you to assign values for a, b, and c that make the image true. It isn't possible.

>> No.11339459

>>11339371
>not using the largest side as the base
Shiggy diggy.

>> No.11339497
File: 22 KB, 1288x305, 1578519124213.jpg [View same] [iqdb] [saucenao] [google]
11339497

>> No.11339521

>>11339249
This proof is fine since all triangles have at least one side (the largest one) that can be connected to its opposing vertex by a perpendicular line.

>> No.11339624

>>11339459
The formula works even if you don't. Just the proof that doesn't.

>> No.11339630

>>11339316
It doesn’t seem like you understand the proof.

>> No.11339671

>>11339624
The proof does work though. Can you not read what you responded to?
>>Not using the largest side as a base
If you do use the largest side as the base (which your drawing didn't) the triangle is contained inside the rectangle.

>> No.11339687

>>11339671
According to the theorem, you can use any side you prefer as base. So you need to prove that it works for any side. Not just the biggest.

>> No.11339709

>>11338059

it is not rigorous because this proof depends on Euclidean space to work, whereas the Pythagorean Theorem works in all kinds of other spaces as well that are not Euclidean, and whose graph would not prove the theorem at all.

>> No.11339732
File: 4 KB, 640x400, bh2_3.png [View same] [iqdb] [saucenao] [google]
11339732

For reference here is how to do it when it's outside.

Yellow triangle is half of (b+x)h
Red triangle is half of xh

(b+x)h / 2 - xh/2 = bh / 2

>> No.11339774

>>11338059
Just imagine a and be sliding around. You can create absolutely any right angled triangle doing that, so I is a proper proof. If you have that kind of intuition making a more formal proof is trivial, but I you wanted to publish something, thats usually what you'd have to do. Thinking about stuff like this is more what you do when you're incubating an idea in your head.

>> No.11339786

>>11339709
>the Pythagorean Theorem works in all kinds of other spaces as well that are not Euclidean
Name one(1) non-euclidean finite dimensional space where Pythagoras holds.
And don't hit me with some bullshit like "it holds on the manifold's tangent space."

>> No.11339847

>>11339709
The diagram can be the basis of a proof, but also one must prove certain assumptions for it to work. For example, prove that putting together two right triangles of the same size and shape but one is flipped on the long longest side results in a square external perimeter. This is true in Euclidean space and can be proved.

>> No.11339855

>>11339847
actually the two triangles are not flipped relative each other. Opps.

>> No.11340292

>>11338059
the only reason the diagram isn't rigorous is because you haven't specified the geometric axioms you are using. if you assume Euclid's axioms, then yes, one could argue from this diagram a proof of pythag's theorem. however, this hinges on parallel lines never meeting axiom. the situation is different in hyperbolic and elliptic geometries.