I have two views when it comes to this unique identity. My first is to hide my face as people go on and on about its beauty. Seriously, some people are waaaay too fanatic. To have these 5 constants in one equation is, no doubt, unexpected, but it’s not magic. It’s maths!
To understand where this funky equation comes from, we start off by considering complex numbers. Most people are familiar with the real numbers: we can draw a number line stretching from negative to positive infinity and any point on this line is real as shown below.
Now imagine we add a new line from negative to positive infinity running up the page. Instead of using 1, 2, 3, etc. we express units on this line in terms of imaginary units i, 2i, 3i, etc. We end up with something known as an Argand diagram:
We can express complex numbers as being a combination of these two lines as if we were reading off a graph such as 3 + 2i as shown by point P above. Complex numbers were originally created to solve equations which otherwise had no solutions. Since then, the field of complex analysis has developed by leaps and bounds and we have a lot of interesting equations associated with it. The equation we’re interested in is known as Euler’s formula (which, sadly, I won’t be proving):
If we set our value for x as being pi (a.k.a 180 degrees) we end up with:
By considering the values for cos and sin when the angle is pi, we can reduce this equation to:
Personally, I prefer this form but by rearranging you can end up with the more common form that’s the header image for this post.
As I said at the start, I have two views on this equation. I gave my slightly pessimistic opinion first, but I still think that this equation is really cool. The fact that all these seemingly unrelated constants can end up in one equation honestly astounds me. I just feel that we have to be wary of drowning in our own hyperbole and losing sight of the mathematics behind this!
<courtesy to Wikipedia for the equations and images>