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10504625 No.10504625 [Reply] [Original]

How does one prove the Euclidean norm (or Pythagorean theorem) is the obvious choice?

I was thinking something like this. Let us have a right angled triangle with 45 degree other angles and side length 1. The hypotenuse can be approximated as a sequence of ladders made from small x and y vectors. When the vectors are infitesimally small, the sum of their length should equal the sum of the hypotenuse.

How to make this formula now?

>> No.10504629

>>10504625
>Let us have a right angled triangle with 45 degree other angles and side length 1.
thats way too specific to be useful
>The hypotenuse can be approximated as a sequence of ladders made from small x and y vectors.
wrong

>> No.10504643
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10504643

>>10504625
this is how you prove it
both squares have side length a + b
so they have the same area
both squares contain 4 identical triangles
so after removing them, whats left must have the same area still
the remaining area on the left is a^2 + b^2
the remaining area on the right is c^2

>> No.10504650

>>10504629
wait, you are right
it's always going to be longer
oops

>> No.10504654

>>10504625
Sin^2 + cos^2 = 1

Sin = a/c
Cos = b/c

Sin^2 + cos^2 = a^2/c^2 + b^2/c^2 = 1

(a^2 + b^2 ) = c^2

Then c = sqrt(a^2 + b^2) which is the hypotenuse aka a line.

Btw, that norm isnt the obvious choice. It's just useful because flat things are easy to measure.

>> No.10504681

>>10504654
This is circular reasoning (pun intended). That identity is solved by way of the Pythagorean Theorem. It is a very helpful mnemonic to memorize identities, I suppose.

>> No.10504683

>>10504681
>solved
*proved

>> No.10504779

>>10504625
the euclidean norm is the obvious choice if you accept the postulates of euclidean geometry.
also your attempted proof yields c = a + b. I don't know a rigorous way of explaining why it's wrong, but the general idea is: at infinitesimal scale, most things look linear, but the rules of Euclidean geometry still apply. your method describes a fractal hypotenuse composed of only horizontal and vertical lines.

>> No.10504794
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10504794

>>10504625

>> No.10504806
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10504806

>>10504625

>> No.10504823

>>10504625
https://matheducators.stackexchange.com/questions/10751/should-my-8th-graders-see-a-proof-of-the-pythagorean-theorem

>> No.10504930

>>10504625

Because if you consider an euclidian space with an inner product <. , .> , you can always orthonormalize a basis with the Schmidt algorithm to obtain an orthonormal basis for < . , . > , and WITH this new basis, the inner product become the canonical inner product thus the euclidian norm is kind of "natural".

>> No.10504961

>>10504930
You can do this with any inner product.

>> No.10506568
File: 7 KB, 367x368, abc.png [View same] [iqdb] [saucenao] [google]
10506568

>>10504806
(a+b)^2 = c^2 + 4*(ab/2)
a^2 + 2ab + b^2 = c^2 +2ab
a^2 + b^2 = c^2

>> No.10507055

>>10504625
You prove the pythagorean theorem whenever your geometry is eucledian. That is, if your geometry has the basic geometry axioms, (two circles in the same plane, where their centers are on each others circumferences, have exactly intersection points for their circumferences, could be one example of an axiom), in addition to a fifth postulate axiom, or equivalent.

>> No.10509026

>>10504643
>this is how you prove it
Pythagoras' theorem has like 100 valid proofs I think.