31 Days – Euler’s Identity

When something is true by coincidence, we can just enjoy it…

HE: What brings you to a grocery store in the middle of the night?

SHE: I’m looking for a pie.

HE: Wow! Me too!

When the truth is serendipitous, we start to wonder whether Something More might be going on:

SHE: What kind?

HE: Strawberry rhubarb with macaroons on top.

SHE: That’s incredible! Strawberry rhubarb with macaroons on top is my favorite! And why are you up in the middle of the night?

When it’s true by definition, it’s no big deal…

HE: I can’t sleep. I’m a mathematician and there’s a problem I can’t solve. It has to do with pi.

SHE: Strawberry rhubarb pie?

HE: No, pi — defined as the ratio of a circle’s circumference to its diameter.

…but when it must be true and nobody knows why, our hearts start to pound.

SHE: Oh! Pi! As in Euler’s Identity, which <batting her eyelashes> mysteriously relates the five most fundamental numbers in mathematics: pi, e, i, 1 and 0, using only multiplication, addition, exponentiation and equality <bat, bat> exactly once each?

HE: I love you.

And here it is: Euler’s Identity. No list of 31 Wonders would be complete without it.

Euler's Formula

In case you’re not as familiar with math as our two protagonists, I’ll explain the three letters there. The wonder is how they can possibly be related so simply.

You can think of “e” as pertaining to continuously compounded interest. If you were to invest a dollar at a 50% interest rate, you’d have (1+1/2)2 dollars after two years, which is $2.25. At a 33% interest rate for three years, you’d have (1+1/3)3, or $2.37. Invest a dollar at an infinitely tiny interest rate for an infinite number of years, and you’d end up with (1 + 1/∞), or “e“, dollars. That’s about $2.72. (How amazing is it that we can even compute that answer!?) More on “e” here.

As an aside, “e” also happens to be expressible as an infinite series:


…which is pretty cool.

i” is the square root of -1. Can you think of a number that, squared, equals negative 1? Neither can anyone else, which is why “i” is called an imaginary number. In fact, in math it is the prototypical imaginary number. All other imaginary numbers are multiples of it.

You already know what pi () is, but here’s an infinite series for it:

Pi Series

…which is maybe even cooler than the one for “e“.

Now what five numbers could be less related to each other than e, i, pi, 0 and 1? How amazing is it that they do, in fact, combine into such a compact equation as Euler’s Identity? Many mathematicians consider Euler’s Identity to be the most beautiful result in all of mathematics.

If 1 + 1 = 2 is for kids and a2 + b2 = c2 is for grown-ups, then Euler’s Identity is for the gods.

And Euler’s Identity is only a special case of the more general Euler’s Formula, which brings trigonometry into the picture.

Tomorrow, I promise I’ll bring you something less geeky. For today, I hope you’ve enjoyed this excursion into the mysterious and beautiful world of math.

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