I'm a sucker for mathematical ingenuity and neatness. Now, I can't really stand mathematical formalism and rigour myself, but what I mean by ingenuity and neatness is being able to arrive at something profound using simple, yet clever mathematical manipulation. 

The derivation of Euler's formula was like art to me. Upon seeing it, I was rendered absolutely speechless. In my head all I could think of was simply, "Wow".

When dealing with expressions with complex roots, we see in it's solution, a real part and an imaginary part, which comes in the form a+ib. As we are now concerned with an ordered pair (real and imaginary), we can express the solution graphically. Using polar coordinates, we arrive at the expression,

z = r( cos x + i sin x )

Using Maclaurin's expansion, we find that,

Therefore, we find that our complex solution may be expressed as the following,

As

Where n = 2, 6, 10, 14, ...

When n = 4, 8, 12, 16, ... however, the nth power of i would be equal to 1.

Therefore, in attempting to get rid of all the minueses in our complex solution, we have,

Using Maclaurin's Expansion again, we know that,

Hence, we arrive at the following conclusion,

Pure beauty....