/sci/ - Science & Math

>>6547148
Math major here, graduate in the fall.

A lot of e^i identities become very useful to derive things that are too inpractical (for me) to memorize

>>6547149
But what does it mean to you? What sensations does it elicit in you?

It's not as pretty as e^(i*tau)+0=1

>>6547152
When I first saw the proof I was in awe over how neatly everything fit together and how useful the identity was. For a long time I wanted a tattoo of it. Now everyone has it tattooed on their bodies and it just doesn't feel the same way it did when I was fifteen.

a cone has volume 1/3 of a cylinder of similar proportions

among many other geometry shortcuts and ratios

they make me smile

>>6547148
Euler's identity isn't the full story.

Euler's FORMULA is the real thing:
<span class="math">e^{ix} = \cos(x) + i \sin(x)[/spoiler]
it's a very important mathematical result.

>>6547154
>It's not as pretty as e^(i*tau)+0=1
for fuck's sake

OP you are part of the problem with mathematics education and knowledge. Do you really think this pop-math tier formula is some sort of advanced realisation that comes only from deep mathematics?

Go learn complex analysis and you will realise that you are but a mere plebeian in the sea of vast mathematical knowledge. Also, reported for being underage and in high school.

TL;DR

<div class="math">e^{i\pi} = -1</div> follows trivially from babby's first introduction to complex numbers.

>>6547171
As someone without an incredibly deep understanding of math (only went as far as two calculus classes in college), the idea of adding an imaginary number to a real number throws my head into a twist.
And for that matter, raising a number to an imaginary exponent completely boggles me.

I don't understand this stuff, but it is pretty neat, and on some level inspiring, to see how cleanly everything fits into Euler's identity.

>>6547177
Cry more, you fucking hipster.

You're the equivalent of somebody who ditches a band that becomes too popular.

Just because something is widely known, doesn't corrupt its merit. Merit should be assessed on merit.

I have met mathematicians who are far more intelligent and well learned than thou who have expressed admiration at this identity.

>>6547177
And you are the problem with people being (dis)interested in mathematics education and knowledge.
Perhaps instead of being so pretentious about your higher knowledge, stimulate OP's interest and steer him towards equally interesting concepts and fields.

>>6547179
>the idea of adding an imaginary number to a real number throws my head into a twist.
>And for that matter, raising a number to an imaginary exponent completely boggles me.
>I don't understand this stuff, but it is pretty neat, and on some level inspiring, to see how cleanly everything fits into Euler's identity.

I don't agree with this guy's view: >>6547177

but I do agree that if you learn complex analysis, only then will you truly understand this stuff. It's not actually that hard to learn the basics, just look up "introduction to complex analysis" on google.

>>6547180
>you fucking hipster

You are implying that I am the one that is spreading a fucking pop-math formula

>who have expressed admiration at this identity
Your high school teacher doesn't count m8

>>6547182
There is an abundance of interesting topics in complex analysis. Choosing Eulers identity as a "universal truth" shows your lack of knowledge and edge

>>6547186
But my university math lecturers do.

Nowhere have you explained why you think the identity lacks interest or merit, you have only tried to imply an association between popularity and lack of merit.

Perhaps for you that is enough to corrupt something, but not for me,.

>>6547189
Your lecturers are jabronis.

>>6547197
Faggot

>>6547200
>being this mad

>>6547203
I'm not even mad. I'm just wondering how somebody who claims to have a detailed knowledge of mathematics could be so lacking in self-insight and suffer under such a ridiculous complex.

Your mother's vagina had previously been soaked in the semen of many men before you were born into this world. So too has this identity brought pleasure to many. Deal with it.

>>6547197
>jabroni

That's a cool word.

>>6547148
it's a trivial consequence of the definition of the exponential function for complex numbers, which uses an interesting fact about the series for exp, sin, and cos. it doesn't mean anything really, it's just a curiosity. it annoys me that people artificially introduce the +1 so that they can put a 0 in the equation.

>>6547148
It's pretty obvious. Not sure why it took so long to figure it out.

>>6547148

All this because
<span class="math">i^2=-1[/spoiler]

fucking autism convention in here. time to hide.

>>6547210

Because it took a lot of machinery to even parse what these things mean!

Math grad here--and I think it's just reminiscent of how a lot of complex analysis is magic--things like the cauchy integral formula and the argument principle are equally mind boggling, even if they're easy to prove.

Is this a little fact that's brandied about by a buncha math enthusiasts? Certainly. But unlike the math enthusiasts saying "Did you know there are more than one type of infinity?!?!", this one is actually neat.

>>6547219
>more than one type of infinity
Is this something "math enthusiasts" tend to know?

>>6547225
Also, I think this is cooler than euler's identity.

>>6547225

Usually it's accompanied by the phrase "I don't know why, but..."

Of course, there's a bullshit paper on the arxiv that would appear to be disputing that.

(Fun game! Try to find out what his flaw is!)

http://arxiv.org/abs/1404.6447v1

>>6547154
Still not as pretty as
e^(-1*i*pi) + 1 = 0

>>6547148

incomprehensible :-§)

>>6547204
You are not replying to the original heratic

>>6547177

>>6547257
My mistake. My point stands but should be directed towards the dubs heretic >>6547177

>>6547264
Dubs heretic here.

I personally do not value Euler's identity as inferior to others; I appreciate it for it's connection between the most "famous" numbers. I only have a problem when people who are clearly ignorant of the beauties of complex analysis go around spouting off about Euler's identity as if it is the only remarkable thing to come out of complex analysis.

>>6547271
I see where you are coming from. But only as it is part of a more general trend of people trying to better themselves by association with things (maths in this case), rather than their own merits.

>>6547277
I always found De Moivre's formula a lot more interesting. It seemed too good to be true, in my opinion.

>>6547284
I agree.
You spend at least one or two years in highschool being told you can't "distribute" exponents over a binomial, and here you've got something like that happening!
Obviously it's much different than distributing, but it feels very similiar.

>>6547298

I have found, quite ironically, that working with <span class="math">\mathbf{C}[/spoiler] feels very natural, which is intuitively weird. It is a shame that many do not get to learn any complex analysis. Besides solving a quadratic equation with no real roots or babby pre-calculus, I don't think they teach secondary students such.

>>6547186
Richard Feynman called Euler's formula "the most remarkable formula in mathematics."

>>6547320
>Richard Feynman
>Mathematician

>>6547154
... but that evaluates to i.

>>6547345
?
exp(i tau) = exp(i 2 pi) = cos(2pi) +i*sin(2pi)= 1+0i=1

>>6547345
tau is defined is 2*pi.

why? because some math teachers are butthurt or something

>>6547325
Richard Feynman invented the concept of functional integrals before mathematicians did.

>>6547353
what?
Euler, Bernoulli, and Newton before Feynman was born.

>>6547355
...
Do you know what a "functional integral" is?

Take a look:
http://en.wikipedia.org/wiki/Functional_integration
http://en.wikipedia.org/wiki/Path_integral_formulation

>>6547357
>not reading your own fucking source
Functional integration was developed by P. J. Daniell in a paper of 1919[1] and N. Wiener in a series of studies culminating in his papers of 1921 on Brownian motion.

Feynman? Sorry I dont see Feynman in that sentence.

>>6547357
Yeah Hamilton's principle is the variation of a functional integral.

All of calculus of variations was created before Feynman.

>>6547362
Feynman wasn't the first, but he independently developed a different kind of functional integral from Danell and Wiener's original one.

from the wiki article:
>R. Feynman developed another functional integral, the path integral, useful for computing the quantum properties of systems.
>The definitions fall in two different classes: the constructions derived from Wiener's theory yield an integral based on a measure; whereas the constructions following Feynman's path integral do not. Even within these two broad divisions, the integrals are not identical, that is, they are defined for different classes of functions.

>>6547148

e^i times something is a circle having radius 1 in the complex plane.

e^i times pi is half circle. Since you begin at 1 (e^i times 0 = 1 ) half circle means that you are now at -1, and -1+1=0

Check the representation on the complex plane. I think Gauss thesis is related.

>>6547167
>worth

>>6548725
That is so faggy.

>>6547171
>it's a very important mathematical result


yeah it was 250 years ago. Now exponential, sinus and cosinus (in any complexe analysis course) would be defined by their power series so the euler's formula would be completely tautologycal... And pi would be defined to be the first positive real solution of e^ix=-1.


>>6547148
You are outrageously gay, OP.

>>6548725

>Absolutely_disgusting.jpg.png.mp4.mkv.torrent

Fuck you, your mother, and your father, OP.
You're a big flaming faggot.

All that fucking overrated fucking formula says is that the vector (1,0) rotated 180 degrees in the plane is (-1,0). Let the rotation by 180 degrees be R, then what that fucking thing says is R*(1,0)+(1,0)=0, that's all there fucking is to it, no "deep truth".

>>6549033
To be fair, the correspondence between the algebra and its geometric interpretation isn't exactly trivial.

>>6549033
>degrees

>>6549081
If he's making this thread, it's a safe assumption that he's more comfortable with degrees.

>>6549049
I agree, it's just lines like "Yes, I have seen the proofs etc., and I am as atheistic and godless as can be, but it seems to me to represent one of the fundamental truths of this universe." and to a lesser degree, "To me it seems to be an incomprehensible truth.", that really ticked me off.
I should have been more respectful but I hate when people add an element of mysticism to math. No, Euler's formula won't give you a fucking unified theory of physics, nor does it prove that God exists.
Instead of asking stupid questions such as the latter, you should instead try to actually understand. Mystifying elementary concepts (and non elementary) only obscures them further and makes sure that it ends all further advancements in the topic (e.g. mystifying and worshipping nature led to absurd explanations like religion instead of the scientific method in order to actually examine the phenomena. had we instead assumed instead that there is no magic and we should try hard enough to understand it, we would have advanced in science much faster).

Here is how you can start to understand that formula, OP. It's literally in the first chapter of the book and it's very easy to explain: http://libgen.info/view.php?id=608148
The rest of the book requires a bit of knowledge of Analysis (also topology) and of course Calculus, but you could do without Analysis if you try hard enough.

There is one way in which Euler's Identity is great though. It's symbolic of the power of Math to unify what we assume to be completely different concepts, into usually beautiful dualities and simple connections; but that's far as its beauty really goes.

> it seems to me to represent one of the fundamental truths of this universe

e^(ix)=cosx+isinx
say that x=π
e^(iπ )=-1+0i
e^(iπ )+1=0
>that's how its done

>>6548725
That is revolting Why would you not crowbar the 0 in? It looks so much better

>>6550816

Connects the five universal figures neatly and elegantly.

Very nice.

>>6550847
Someone hammer in the Euler-Mascheroni constant gamma and the golden ratio phi.

>>6551367
It's no less valid than the way they get 1 and 0 into the identity...

>>6551738
my fucking sides

>>6547179

> the idea of adding an imaginary number to a real number throws my head into a twist.

Well, the usage isn't really that hard to understand..

Imagine R^2, but with the "normal" numbers as the x-axis and the imaginary numbers as the y-axis.

Then e^ix is a circle with radius 1, moving counter-clockwise as you increase x. In other words, the values of e^ix are repeating periodically.
>picture related

Now since it's a circle you can simply use Sin and Cos as you would do it in R^2:

|cos(x)|^2 + |sin(x)|^2 = 1

The "i" just means they are different dimensions you can not add, like "width" and "heigth"..

> ITT twits parroting trite quotes about this equation they read in popsci books

>>6552991

Oh, I didn't read that one, it has been posted before:
>>6549033


>>6549145
>Here is how you can start to understand that formula, OP.

Uhm...

Well, why don't you just use the infinite series for Sin, Cos and E(ix)? It's really easy to see then..

Autism: Proof of concept
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>>6547180
>I have met mathematicians who are far more intelligent and well learned than thou who have expressed admiration at this identity.
>than thou
>than thou
>than thou
>than thou
/sci/ pls

Not only that, but you're arguing that something is beautiful on the grounds that someone "more intelligent and learned" expressed admiration at it, which makes you even more of a tool.

>>6547148
It just says that -1 is halfway around the unit circle from 1.

>>6553040
Well, he was countering the claim that only people with a shallow understanding like it, I'm guessing. And I have also heard accomplished professors write it up and exclaim it's beauty.

>>6553040
Why are you Sega Genesis?

>>6551738

>>6551738
That's not entirely true since -1 is defined as the number such that 1+(-1)=0, so you're saying e^{ipi} satisfies this so must be -1 which makes it a bit more excusable than those.

e^(i*pi) +1 = 0

d/di e^(i*pi) + d/di 1 = 0

pi e^(i*pi) = 0

e^(i*pi) = 0

proven

>>6554748

same poster here

assuming 1 = i^(4n), where n is an integer

e^(i*pi) + i^(4n) = 0

taking derivatives with respect with i on both sides,

pi* e^(i*pi) + 4n* i^(4n-1) = 0

pi = -4n* i^(4n-1) e^-(i*pi)

if i were to put n = 1

pi = -4

math

works

>>6554759
10/10 would publish

okay You people

Euler's formula is only useful in the context of calculus, where e^kx is king.

d/dx e^ix = ie^ix
d/dx ie^ix = -e^ix
d/dx -e^ix = -ie^ix
d/dx -ie^ix = e^ix

After differentiating 4 times, we arrive back to our original eqn. what sounds familiar? your mind should be screaming SINES AND COSINES.

and the pi-es and -1s and the uses in QM, physics, mathematics just comes naturally.

In fact, Cos x = (e^ix + e^-ix)/2

Says something very deep about trigo and i-es

>>6554769

Normal numbers are only useful to describe +- of the number line dimension.

What about the numbers that are orthogonal to the number line? They don't exist, but we like to imagine an arrow that does not point parallel to the normal number line, but at an angle, into the unknown

The unknown is the realm of the imaginary.

And my friend, it is the greatest invention since the wheel.

Basically its so useful in equations dealing with waves, its hard to imagine how could anyone solve simple pendulum motions without e^(i*pi), let alone solve quantum mechanical wavefunctions.

Am I the only one who thinks the =-1 version is much more elegant?

>>6547186
whatever this post said

it is truth

>>6554786
i^i would have been more elegant than that euler identishit