The following is taken from Keith Devlin's book, Mathematics: the Science of Patterns (Scientific American Library: New York, 1994), p. 101.

I hope it blows your mind as much as it did mine!

Euler's Formula

Complex numbers (numbers involving the "imaginary" number "i" which is the square root of -1) have connections to many other parts of mathematics. A particularly striking example comes from the work of Euler. In 1748 he discovered the amazing identity

This is true for any real number x.

Such a close connection between trigonometric functions, the mathematical constant "e", and the square root of -1 is already quite startling. Surely, such an identity cannot be a mere accident; rather, we must be catching a glimpse of a rich, complicated, and highly abstract mathematical pattern that for the most part lies hidden from our view.

In fact, Euler's formula has other surprises in store. If you substitute the value for x in Euler's formula, then, since cos = -1 and sin = 0, you get the identity

Rewriting this as

you obtain a simple equation that connects the five most common constants of mathematics: e, , i , 0 , and 1.

Not the least surprising aspect of the last equation is that the result of raising an irrational number to a power that is an irrational imaginary number can turn out be a natural number. Indeed, raising an imaginary number to an imaginary power can also give a real-number answer. Setting x =/ 2 in the equation at the top of the page, and noting that cos / 2 = 0 and sin / 2 =1 , you get

and, if you raise both sides of this identity to the power i, you obtain (since = -1)

Thus, using a calculator to compute the value of , you find that

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