What's so special about this?

it is a succinct and concrete way to express the geometric, algebraic and analytic knowledge that we have accumulated since antiquity

> It has 5 numbers in it, and those numbers are 0, 1, i, pi and e.
e^(0*i*pi) = 1
SORRY THIS IS THE MOST BEAUTIFUL EQUATION IN ALL OF MATHEMATICS IT ALSO HAS 5 NUMBERS IN IT 0,1,i,pi,e

is e^(ix) = cosx + isinx by definition?

Because you are taking OP's image and most of the history of mathematics for granted, that's why it is not intuitive.

What is entailed in the formula?

You have to realize that there is a special number that stands for any one singular thing, by itself, alone. You have to abstract that thing, and use it. (this item is far and away the easiest).

You have to realize that there is a special number which somehow stands for no number at all. Strangely, you then have to give that "number" a name, accept that it /is/ a number, and use it in the context of other numbers (this is a non-trivial, significant cognitive leap, and it betrays an ahistorical modern viewpoint that one may think otherwise).

You have to name the number which stands for the ratio of the circumference of a circle to its diameter,

You have to suppose and name what it might mean to take the square root of an elementary negative number (as late as 1545 A.D., Cardano and his various associates/enemies were struggling with this, discovering it for the first time)

and finally, you have to find and appreciate this other number which is a basis of growth, calculus etc, and has various definitions and properties which are quite nice to calculus. It helps to have later invented calculus two times, to give this number its special motivation toward that subject.

Then on TOP of all of that mathematical development, you have to put all of those special numbers together in a context that makes sense, and realize that a particular formula is true, and has a geometric and pictorial significance which is related to all of them. To get that done, you need coordinate geometry, trigonometry, and complex numbers all as add-ons. Oh yeah, actually prove it too, in whatever fashion is germane to your understanding this decade. All of these lead to the celebrated Barnett identity.

Yes, but yours is trivial

wew lad, careful don't tell normies about the barnett identity

>comes from the e^ix = cosx + isinx case where x = pi

This, and as such it connects the polar and rectangular representations of the complex numbers.

It's also a special case of the matrix/Lie group exponential which gives a self-adjoint map for any 1-parameter unitary subgroup - here the self-adjoints are reals and the unitaries are unit complex numbers, viewed as their left-multiplication maps. This lies at the foundation of quantum mechanics, so I'd say it's pretty important.

proving that e^0 = 1 is just as difficult as e^(pi*i) = -1. Both use the same complex exponential function which is completely different from the standard exponential function.

Yes, but e^(pi*0*i) =1 is still a trivial subcase of a^0=1, a=/=0

But so what? Take per instance the software program "hello world". There's nothing impressive in it.

Unless you want to tell the whole history of how mankind discovered fire, electricity, computers, transistors, capacitors, etc.

not for the complex exponential! a very different thing.

this, humans are stupid and sometimes take centuries to discover simple things that were in plain sight the whole time

>proving that e^0 = 1 is just as difficult as e^(pi*i) = -1

lol, using what definition? The first is trivial if you use the power series. And actually you can *define* pi the second way too. If you define it with cos or sin it takes a tiny bit more work.

Here, crystalized in this post, we have the Nye/Tyson attitude.

The worst sentence of the post: "there's nothing impressive in it", because that sentence perfects the refusal, the utter lack of reflection of its author.

Why is there supposed to be nothing impressive? Because the author cannot get outside his own century, and consider what discovery was like.

Here, there is also a certain petulance. "Oh, Of Course." Let us see what is Oh Of Course two hundred years from now, which has not occured to any of us.

just because the power series for a normal exponential happens to extend nicely to a complex exponential does not mean that the functions are the same thing

You're half right (humans are stupid) and half wrong (in your wrong agreement with the above user, and in utter lack of appreciation for a historical process.).

Except that multiply by zero removes the i

So anything that's built upon layers of trans-generational knowledge is "impressive"? That makes impressive a pretty meaningless word.

They agree on [math]\mathbb{R}[/math], which is sufficient. Literally just plug 0 into a power series, you get the first term, which is 1.

>proving that e^0 = 1

e^1=e
e^0=e/e=1

ok the exponential function is defined in such a way that you are technically correct, but I still see this as a historical accident that the exponential function happened to have this property, even the proof for the pi*i case I see as an accident that it worked out so nicely there is no other way to explain it even mathematicians at the time refused to believe it because it completely changes the definition of the original function when you agree that you can input complex numbers

I have a religious appreciation for simplicity, when humans discover something simple that in no way denigrates the importance of the discovery

Newton had visited from Heaven, and had wanted to have a word with you on your attitude.

But upon re-reading your conceit, he realized that nothing could be done.

Euclid had even entertained visiting from purgatory, but you depressed him as well. You're 0/2 user.

Newton would know, he stole everything from past generations.

if you say that 0 is a complex number, then no you have not removed the i at all, your just orthogonal to it

this sounds more agreeable to me, and you sound to be of a different (better) philosophical attitude to the other user.

Still, I take issue with your same initial agreement with him . It's the same "Oh Of Course" petulance which refuses to appreciate historical discovery on its own terms, which refuses to acknowledge the positive human journey on these points.

Funnily, I am not usually this pro-human. What you are obliged to do, is to appreciate the process on its own terms, and without the ahistorical know-it-all nonsense upon which you (that other guy especially, apparently) have insisted throughout.

That's exactly the point of mentioning him, and that's exactly the reason why I"m right. I'm sure you're a troll by this point, and thank you for having agreed with me.

I don't agree with your definition of "impressive" though. The way humans accumulate and create knowledge is *not* impressive.

The fact that it took mankind until the year 1800 to discover evolution is lame, not impressive.

its a good attitude to have user, i agreed with other user because I personally don't care what other people find "impressive" or not, only what they find "simple" but you are correct that simplicity should not negate its importance in human progress which is what every human should have an appreciation for above all else

You are wrong. If you knew the slightest amount of history you would know the enormous challenges that people had to overcome to come to the knowledge we have today.

But what if people saw species as unchanging through time, due to Plato et al. can culture lead to a predisposition against discovering evolutionary theory?

i leave you with this

"Rule I. We are to admit no more causes of natural things than such as are both true and sufficient to explain their appearances.

To this purpose the philosophers say that Nature does nothing in vain, and more is in vain when less will serve; for Nature is pleased with simplicity, and affects not the pomp of superfluous causes." -Newton

Finally, some god-damn non-autist backup that isn't me. Thanks.

The other guy is onto something, but he's also not onto something. He has a total lack of appreciation of a complex process, and this is what sets my aesthetic, scientific bullshit-meter off about his sensibilities. In an unscientific way, of course.

Da comrad

No, not by definition. You'll see it by comparing the MacLaurin series of all the terms, but also by integrating 1/(1+x^2) = 0.5/(1+ix)+0.5/(1-ix) from 0 to theta (being in the 1st quadrant)

As an amateur mathematician, who didn't knew some of that, I must say that it is indeed an inspirational realization. I can now finally appreciate that formula.

it's used for alternating current to calculate phase angle and some other stuff quickly

I think the definitions more commonly used are
[eqn]\sin{z} = \frac{e^{iz}-e^{-iz}}{2i}[/eqn]
[eqn]\cos{z} = \frac{e^{iz}+e^{-iz}}{2}[/eqn]

From there you can easily work out the identity you cited.

Cyclotomic polynomials etc

ow the pretentiousness

I can guarantee you're not a mathematician, because any course in mathematics doesn't even mention the fact that [math]e^{\pi i} = -1[/math].

Why? because it's a totally obvious result of using polar instead of Cartesian coordinates.

You might as well marvel at the fact that sin^2 + cos^2 = 1, which looks mysterious if you know nothing, and becomes totally obvious when you know something.

I wonder when normies will get their hands on residue theory. They're gonna have a field day with that one.

On the complex plane we can do interesting stuff with complex numbers: we can multiply a complex number by a real number to stretch it, or multiply it by an another complex number to give it a stretch and a twist at the same time.
A complex number (represented by a point M(z) on the complex plane) can be written in the algebraic form as
z = a + bi where a and b are reals (i.e its coordinates)
Or -using some trigonometry on the number's representation in the complex plane- in the trigonometric form as
z = r (cos(θ) + isin(θ))
Here r represents the modulus (or the length) of the complex number
And θ is the argument (meaning the angle M(z) makes with line of reals)
Now here comes the interesting part, if you try to to multiply two complex numbers say z and w in their trigonometric forms you'll find that
z.w = |z|.|w|.(cos(α+β)+isin(α+β)) (De Moivre's formula)
This shows that the length multiply together and the arguments add up.
And what function converts a multiplication into an addition better than the exponential function?
So, for convenience (and some other deeper reasons like the Taylor series of the exponential and trigonometric functions) we can represent a complex number with a complex exponential as follow
z = r.e^iθ
And the e^iπ + 1 = 0 becomes immediately clear since we're multiplying 1 (with length 1 and argument 0) by e^iπ (with length 1 and argument π) and we obtain a number with length 1 and argument π which is nothing but -1 which coincidentally results in a beautiful yet simple formula.

It's more appropriate to define the exponential function as solution to f'(z)=f(z), f(0)=1 and then sin and cos also through differential equations, where it follows

because it was a meme before anoyone thought of concept of a meme

>these brainlets can't into Barnett's Identity

It's an identity.

I majored in math,

your second line is a straight up lie,

And your third line repeats the ahistoricity which has pervaded the thread.