What is so great about Euler’s identity?

Answer by Yasha Berchenko-Kogan:

Euler's identity [math]e^{i\pi}+1=0[/math] gets a lot of hype because it contains a lot of important constants, but the thing that's actually interesting about it is that raising a number to an imaginary power makes any sort of sense.
Raising a number to a whole power makes sense: It's just repeated multiplication. Raising a positive number to a rational power p/q makes sense: First take the qth root of the number, and then raise it to the pth power. Even raising a positive number to an irrational real power can be defined by approximating the irrational exponent with rationals. But what could raising a number to an imaginary power possibly mean?
Next time someone tells you that Euler's identity is beautiful, see if they can answer that question. They can't possibly understand why the formula is beautiful without first understanding what it even means!
The answer to this question comes from an incredibly important formula. For any real number x,
[math]e^x=1+x+\frac{x^2}2+\frac{x^3}{3!}+\frac{x^4}{4!}+\frac{x^5}{5!}+\dotsb[/math]
This formula gives a new way of raising e to a power. Instead of taking roots and repeated multiplication of e, we compute an infinite sum involving the exponent.
What makes this formula especially interesting is that the sum still makes sense if we plug in an imaginary number. For example, if we plug in [math]x = i[/math], then using [math]i^2=-1[/math], [math]i^3=-i[/math], and so forth, the above sum is
[math]1+i-\frac12-\frac i{3!}+\frac1{4!}+\frac i{5!}+\dotsb[/math]
If we were to start adding the terms of this summation, we'd eventually get around 0.5403 + 0.8415 i. So it makes sense to say that [math]e^i[/math] is approximately 0.5403 + 0.8415 i. More generally, we can use that formula to define what it means to raise e to a complex power in a way that's consistent with what it means to raise e to a real power.
The last piece of the puzzle comes from analogous formulas, called Taylor series, for sine and cosine. The formulas are
[math]\cos y = 1-\frac{y^2}2+\frac{y^4}{4!}+\dotsb[/math]
[math]\sin y = y-\frac{y^3}{3!}+\frac{y^5}{5!}+\dotsb[/math]
If you plug in [math]x = iy[/math] in the formula for [math]e^x[/math], you can see that [math]e^{iy}=\cos y+i\sin y[/math], which is Euler's formula and is actually beautiful: It shows us that exponential functions and sinusoids, which seem completely different, are actually two sides of the same coin if we know how to use complex numbers. Euler's identity is just the special case where [math]y=\pi[/math].

What is so great about Euler's identity?

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