What do e, i and π have in common?

To begin the discussion, as we probably all know,

e is the base of the natural logarithm
i is the number which when squared gives -1
π is the number that is the ratio of the circumference to the diameter of a circle.

What do they have in common? Not really much, but when we put them together, they make -1, as in the following equation:

e^{i π} = -1

It is a surprising result, because the left-hand side is a complex number involving an imaginary component, and the right-hand side is a pure real number. When I was young, a school-friend of mine showed me this equation at a time when I did not even understand what an imaginary number was. Gabriel, now a Ph.D. in physics, works in oncology to help people who cope with cancer. Brilliant people remain brilliant.

Back to mathematics, how is this possible that so many favourite numbers are linked together by this single simple equation? The answer is really quite simple if you read on.

If you have learned about Taylor’s expansions, you will know that the three following mathematical functions can be represented by an infinite series, meaning that if you take enough terms to the point that the remaining terms are small enough, you get the answer to the function.

e^x = 1 + x + x²/2! + x³/3! + x⁴/4! + ...
Sin x = x - x³/3! + x⁵/5! - x⁷/7! + ....
Cos x = 1 - x²/2! + x⁴/4! - x5/5! + ...

For those who have not worked with infinite series before we will show an example:

sin(45 degrees)
=sin(π/4 radians)
=π/4 - (π/4)³/3! + (π/4)⁵/5! - (π/4)⁷/7! + (π/4)⁹/9! - ...
=0.78540 - 0.08075 + 0.00249 - 0.00004 + 0.00000 -....
=0.70710

The accurate answer is 0.7071067812... So now you get the hang of it!

In the exponential equation, let’s put x=i π, then it reduces to:

e^{i π} = 1 + i π + (i π)²/2! + (i π)³/3! + (i π)⁴/4! + (i π)⁵/5! + ...

By regrouping terms, substituting i²=-1, i⁴=1, and factoring out i, we obtain:

e^{i π} = 1 - (π)²/2! + (π)⁴/4! -... _+i ( π - (π)³/3! + (π)⁵/5! - ...)
= cos (π) + i sin(π)
= -1 + i . 0
=-1

Simple, easy and straight forward, as my secondary two English teacher used to say.

Addendum:
Gabriel wrote me some interesting facts on π day on the same subject. Here is what I learned.
He had a suspicion that since e and π are related by the above equation, they should be of equal importance. In fact, π is related to space, namely geometry and dimensions, while e is involved with time, such as growth and decay. These observations are obviously obvious to a physicist!
If I may force a little on the relation between e (as in e=mc², not e=2.71828...) and π, Gabriel also noted another fact: π-day, or March 14, is the birthday of Albert Einstein, author of a piece of intellectual treasure, the theory of General Relativity in 1916, right in the middle of World War I, time-wise (WWI 1914-1918) and space-wise (in the middle of Berlin)!