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I'm of the opinion that anyone who spends a non-trivial amount of time
talking about complex numbers
needs to at least mention the Euler's identity.
So the Euler's identity states the following:
e to the power i pi plus 1
is equal to 0.
My first encounter with this was through a book that I read when I
was in high school.
At that time, I was totally blown away by this identity.
The remarkable thing is, it involves five
very celebrated constants in mathematics.
0 of course is a very important number. 1 is the building block of all
counting numbers
and eventually all rational numbers. pi is a constant that has been known for a
long time
and it gives the area of a unit circle.
And e, known as the Euler's number, is the base of the natural logarithm.
And of course we have i,
our imaginary number.
Even though I was amazed by this identity,
there were some doubts that I had. What does it really mean to take
e to the power i pi.
But that question was never really settled in my mind
until I took a complex analysis course in university.
If you search for Euler's identity, you'll see many videos and resources
giving a proof of this.
And what you might not notice is
many of them do not acknowledge or point out the subtlety
involved in proving this identity.
So let me show you
a very common way of arriving at this identity
and this is known to be the way that Euler arrived at this identity himself.
And the key is the following.
So the power series expansion of e to the x is given by
1 plus x plus x squared over 2 factorial
plus x 3 over 3 factorial
and so on. So the generic term is
x to the n, n factorial. So if you take
e to the i x, what you end up with
is 1 plus i x minus x squared over two factorial
minus i x cubed over three factorial
plus x to the fourth over four factorial
and so on. And now you can write this as
one minus x squared divided by two factorial
plus and so on, and then plus
i times x minus
x cubed divided by three factorial
plus x to the fifth divided by five factorial
minus x to the seventh divided by seven factorial
and so on.
And this thing here is the power series expansion of cos x.
And here is the power series expansion of sine x.
This might look convincing at first sight.
But there are a few questions that need to be answered.
So first of all, what is a power series expansion.
Right? How do you really add infinitely many terms?
That's a question that is answered
in an introductory
analysis course. And once you have established the meaning
of a sum that involves infinitely many terms, you can ask the same thing.
Okay, what if this x is a complex number?
Does this infinite sum still make sense? And if so in what sense?
The other thing is, can you really do this?
I've splitted this infinite sum as the sum of two infinite sums.
Now if there are only finitely many terms here, this is not a problem
But again, because we have infinitely many terms, the precise meaning
of this operation need to be carefully defined.
And once you have taken care of
all those details, then you can conclude that e to the i x
is equal to cos x plus i sine x. And now if you plug in
x equal to pi, you get Euler's identity here.
This of course is not the only proof.
As I mentioned there are many different proofs.
But to write down a completely satisfactory proof,
one really needs the machinery of modern calculus.