Are there any calculus/complex numbers/etc proofs of the pythagorean theorem? I have been looking for proofs for the pythagorean theorem that don't use area calculation but calculus, complex numbers or any other interesting ways to proof it.

I would love to see any interesting proof,
Shay
Explanation in terms of linear algebra. From  this blog post by Terence Tao


  The statement $a^{2}+b^{2}=c^{2}$ is equivalent to the assertion that the
  matrices $%
\begin{pmatrix}
a &amp; b \\ 
-b &amp; a%
\end{pmatrix}%
$ and $%
\begin{pmatrix}
c &amp; 0 \\ 
0 &amp; c%
\end{pmatrix}%
$ have the same determinant. But it is easy to see geometrically that the
  linear transformations associated to these matrices differ by a rotation,
  and the claim follows.
An elegant proof.  Consider the vector $\mathbf{c}$ in the complex plane: $\mathbf{c}= a + \mathbf{i}b$, where $a$ and $b$ are scalar values, and $a,b &gt; 0$.  Now consider the reflection of $\mathbf{c}$ in the line at an angle of 45 degrees (pi/4 radians) passing through the origin, call that $\mathbf{d}$. $\mathbf{d} = b + \mathbf{i}a$, and $\mathbf{d}$ clearly has the same length as $\mathbf{c}$.

Now take the vector product of c and d.  That has the same length as the square of the length of c, and in terms of polar coordinates, it is at an angle of pi radians (by virtue of the multiplicative properties of complex numbers expressed as polar coordinates).

so $\mathbf{c\cdot d} = i(c^2) = (a+ib)\cdot (b+ia) = ab-ba+i(a^2 + b^2) = i(a^2 + b^2)$.

$c^2 = a^2 + b^2$