Euler's Identity - Explanation

Explanation

The identity is a special case of Euler's formula from complex analysis, which states that

for any real number x. (Note that the arguments to the trigonometric functions sine and cosine are taken to be in radians, and not in degrees.) In particular, with x = π, or one half turn around the circle:

Since

and

it follows that

which gives the identity

The physical explanation of Euler's identity is that it can be viewed as the group-theoretical definition of the number π. The following discussion is at the physical level but can be made mathematically strict. The group is the group of rotations of a plane around 0. In fact, one can write:

with δ being some small angle.

The last equation can be seen as the action of consecutive small shifts along the circle caused by the application of infinitesimal rotations starting at 1 and going for the total length of the arc connecting points 1 and -1 in the complex plane. In fact, each small shift can be written as multiplication by

and the total number of shifts is π/δ. In order to get from 1 to -1 the total transformation would be

Now, taking the limit when δ → 0, denoting iδ = 1/n and using the definition of, we arrive at Euler's identity. The π itself is defined as the total angle which connects 1 to -1 along the arch.

Summarizing, we can say that because the circle can be defined through the action of the group of shifts which preserve the distance between a point and another point, the relation between π and e arises.

This simple argument is the key to understanding other seemingly miraculous relations involving π and e.