What's so special about this?

>>8254507

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

links e, i, pi and contains 0 and 1 (sinpi= 0)

i can't think of any practical applications, but it is a pretty equation

>>8254507
Should use [math]\tau[/math] obviously.

>>8254507
You have to appreciate that you got an exponential to be a negative number without a negative base.

>>8254572
with a complex number... not impressive

The exponential function is the most important function in mathematics. [math] e^{a+b} = e^a e^b [/math] is a bit suggestive of this, but to understand what makes it very very good you need to know about the fourier transform which it defines. The fourier transform turns integral equations into algebraic equations and is ideally suited for the physics of our world.

>>8254507

The formula is a previously-quite unexpected, and certainly not immediately obvious relationship of five of the most important numbers in math. That is the significance and appeal of the formula, anon, and any "nuh uh I knew better already this stuff is old as hell" posing to the contrary, is both disingenuous and misses the point entirely. There is both a historical context as well as a personal experiential context to the formula, which both start out and advance quite far without realizing that the formula is true.

>>8254654
How is it not intuitive when you look at the circle?

>>8254507
the complex exponential has interesting mathematical properties im sure, but as an engineer its completely abused by every pretentious shitty professor who whants to feel like he is smart for using the complex exponential to add vectors rather than vector coordinates, which is gay because to evaluate these additions you have to convert back to sine and cosign representation which is the same as the original vector formulation. The interesting stuff starts happening when you multiply complex numbers but this is only interesting because you are now performing 2d matrix transformations but obfuscating the linear algebra with shitty exponential notation and it gets even more obsurd when mathfags start using quaternions when everything they are doing can be written with matrix equations, or vector algebra ( see Euler–Rodrigues formula )

so basically all engineers are smarter than mathematicians

>>8254507
It's one of the most beautiful formulae in mathematics. It has 5 numbers in it, and those numbers are 0, 1, i, pi and e. If you can't appreciate that, you have no soul

>>8254507
it is a succinct and concrete way to express the geometric, algebraic and analytic knowledge that we have accumulated since antiquity

>>8254711
> 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

>>8254507
>>8254525
is e^(ix) = cosx + isinx by definition?

>>8254670

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.

>>8254753
Yes, but yours is trivial

>>8254761
wew lad, careful don't tell normies about the barnett identity

>>8254507
>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.

>>8254771
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.

>>8254800
Yes, but e^(pi*0*i) =1 is still a trivial subcase of a^0=1, a=/=0

>>8254761
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.

>>8254805
not for the complex exponential! a very different thing.

>>8254806
this, humans are stupid and sometimes take centuries to discover simple things that were in plain sight the whole time

>>8254800
>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.

>>8254806

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.

>>8254819
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

>>8254815

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

>>8254815
Except that multiply by zero removes the i

>>8254820
So anything that's built upon layers of trans-generational knowledge is "impressive"? That makes impressive a pretty meaningless word.

>>8254822
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.

>>8254819
>proving that e^0 = 1

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

>>8254824
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

>>8254823
I have a religious appreciation for simplicity, when humans discover something simple that in no way denigrates the importance of the discovery

>>8254826

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 anon.

>>8254842
Newton would know, he stole everything from past generations.

>>8254824
if you say that 0 is a complex number, then no you have not removed the i at all, your just orthogonal to it

>>8254840

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

Still, I take issue with your same initial agreement with him >>8254815 . 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.

>>8254843

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.

>>8254854
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.

>>8254851
its a good attitude to have anon, i agreed with other anon 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

>>8254856
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.

>>8254856
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?

>>8254851
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

>>8254872

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.

>>8254757
Da comrad

>>8254757
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)

>>8254761
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.

>>8254525
it's used for alternating current to calculate phase angle and some other stuff quickly

>>8254757
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.

>>8254525
Cyclotomic polynomials etc

>>8254761
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.

>>8254757
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

>>8254507
because it was a meme before anoyone thought of concept of a meme

>>8255037
>>8254806
>these brainlets can't into Barnett's Identity

>>8254507
It's an identity.

>>8255037

I majored in math,

your second line is a straight up lie,

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