I’ve heard this is “the most beautiful and useful equation in all of math. I’ve watched videos, but still don’t understand what it does.
In: 10
People like it for it’s aesthetic value. By most measures, the five most important constants in math are:
– The additive identity, 0
– The multiplicative identity, 1
– The imaginary unit, i
– The base of the exponential function, which is ubiquitous
– pi, which is *weirdly* ubiquitous
The identity e^πi + 1 = 0 brings all those constants together in a shocking way. Why is it shocking? Well…
All these constants are defined independently of one another. π and e are (very) irrational numbers that at first glance are entirely unrelated, and i isn’t even *real*. What does a complex-valued exponent even *mean*?
You’d be forgiven for thinking, then, that it’s absurd for e^πi to equal an integer. Like, what? why? how?
Nowadays mathematicians know that the exponential function e^z is defined in a very pleasing and natural way on imaginary (and therefore complex) inputs, and that π is *intimately* related to that function, which is why π shows up in a lot of places where e^x shows up. Still, you have to imagine how ridiculous this must have seemed to the mathematicians that first made the connection.
By the way, the full Euler’s formula is much more general. It says
– e^ix = cos(x) + i(sin(x))
This is genuinely useful, to the point where nowadays the functions sin and cos are actually *defined* by this relation, rather than their geometric interpretation.
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