Here's the answer:
Think about the relationship:
eix = cos x + i sin x
now put in x = . (In geometry we us units of to measure angles instead of using degrees. The reason is that 360 degrees in a circle is a rather arbitrary number why not 518 degrees, for example? Units of are more natural since the circumference of a circle whose radius is one is equal to 2, and units of are much easier to work with than degrees. As an angle, is the same as 180 degrees and 2 is 360 degrees.) And you'll get:
e = cos + i sin
If you remember any high school trig, you'll see that the sine of is zero, so the imaginary term disappears, and the cosine of is -1.
So you can write in one fell swoop:
e = -1
WOW!!! Who would have thought that if you raise the transcendental number e to an imaginary power of another transcendental number, you will get an integer!!!! And a negative one at that!
OK, if you followed this, here's another question: What power of e added to 1 gives zero? Give up? Check here ==> ?