You hopefully now understand the concepts that make up Euler’s equation, so let’s move on to how the equation arises. You may have already realised that for eiπ + 1 = 0 to be true, it must also follow that eiπ = -1, simply by rearranging the equation.
Think about that for a second. You’ve got a combination of three extraordinary numbers – e, i and π – numbers that seem to have no relation what so ever, and they combine to make -1. How can this be? Why don’t these numbers create a horrible decimal like 8.23487 or similar? Two of the numbers, e and π, cannot even be written down in full because they are infinity long, and yet shove i into the mix and we get the simple, beautiful result: -1.
It all stems from the exponential function, ex. A function is basically a rule for turning one number into another. You take your independent variable, x, and plug it into your function to get the dependant variable. A simple function might be f, where f(x) = x2. If you stick x = 2 into f, out pops f(2) = 22 = 4. The exponential function works the same way, just replace x with whatever number you are interested in. For Euler’s equation, x = iπ gives us the desired result.
Some functions can be expressed as lots of other functions – an infinite number of functions, in fact. This representation is often of a type known as a Taylor series, and for ex it looks like this (I’ve borrowed a graphic from Wikipedia to make it clearer):
I’ve mentioned factorials like 2! and 3! before, but a quick reminder: n! just means “multiply together all the numbers from 1 to n. The ellipsis at the end of the equation means that the pattern goes on forever. “Well how does that help?” you ask. “Now I’ve got an infinite amount of things to deal with, and we don’t seem to be getting any closer to -1!” Fear not.
You see, ex is actually hiding two other functions you may remember from school – the trigonometric duo, sinx and cosx. These two help you work out the length of a triangle’s sides from its angles, and they crop up everywhere in mathematics. When you stick x = iy (y is just another variable, like x) into our exponential function, it turns out that the Taylor series above becomes equal to two series added together – the series for cosy and isiny.
This means that eiy = cosy + isiny. Like so many things in mathematics, this equation has an alternate, geometric representation – in this case, eiy traces out a circle in the complex plane, a way of representing both real and imaginary numbers. Once more, Wikipedia has an excellent representation. They have used the Greek letter phi rather than the y I use here, but it means the same.
We’re nearly there now. We just need to set y = π, which represents a turn half way around the circle. This also happens to place us on the point where the circle intersects the real numbers at -1; in other words, eiπ = -1. This is because cosπ = -1 and isinπ = 0. This what Euler realised, and although it is thought that he never actually wrote the expression eiπ + 1 = 0, it follows directly from his result.
Euler’s equation summarises addition, multiplication, and exponentiation using five of the most important numbers in mathematics; e, i, π, 1, 0. It’s a truth universal in any language and to any culture. There is quite journey to get the equation, but one I believe is well worth travelling. I hope my explanation here leads you to agree.