TITLE: Non-geometric Proof of Pythagorean Theorem
QUESTION [11 upvotes]: Is there a purely algebraic proof for the Pythagorean theorem that doesn't rely on a geometric representation? Just algebra/calculus. I want to TRULY understand the WHY of how it is true. I know it works and I know the geometric proofs.

REPLY [2 votes]: A "proof" of the Pythagorean Theorem depends on some kind of definitions of:

right angle
length/area
stright line

The axioms of Euclid are not completely formalized but we have other formal axiomatic systems that mimic euclidean axioms and definitions of these notions (for example the Hilbert's axioms) so that we can derive Pythagorean theorem there. A formal proof with these axiomatic systems wouldn't require any reference to pictures in principle.
Proving Pytagorean Theorem in completely different context such as analytic geometry (or"calculus") could be possibly trivial or meaningless depending on what definition of "right angle" we are going to consider. For example it would be trivial if you define a right angle with the scalar product and the distance with $\sqrt{(x_1-x_2)^2+(y_1-y_2)^2}$, but you could try with different ones and the proof of the theorem could get more and more complicated depending on which definition you want to take (you could want to define areas with Peano Jordan's measure for example).