/sci/ - Science & Math

>>12505670
I dont even get why it works

>>12505693
it's just a retarded way to write the polar form

>>12505693
e^(i*n) = cos(n) + i*sin(n)
cos(pi) = -1
sin(pi) = 0

>>12505701
You've explained the obvious part while simply asserting the unintuitive part.

>>12505693
It doesn't

>>12505716
why the taylor series exist in the first place?
a non taylor proof of euler's formula?

>>12505716
Think of cos and sin as components of x and y in the imaginary plane. If you start at [1, 0] and you move pi (180 degrees) counterclockwise around the circle, the components end up as [-1, 0].

>>12505670
>Be me, high school algebra, we talk about Euler's identity
>See all of the least attractive people in the room perk up
>The kids with the thickest glasses say things like "this is so beautiful", "it's like poetry"
>Look at formula
>Just looks like any other formula to me
>Doesn't seem to mean anything especially interesting to me
>Fuck it, do the homework and pass the class
I honestly do not understand why permanent virgins cream their pants over this shit. It's mindless. Just a bunch of symbols that don't really actually mean anything or represent anything, just like in every other math question I ever did in school.

[math] \displaystyle
f(x) = e^{-ix}(\cos x + i \sin x)
\\
f^{\prime}(x) = e^{-i x}(i \cos x - \sin x) - i e^{-i x}(\cos x + i \sin x)
\\
f^{\prime}(x) = e^{-i x}(i \cos x - \sin x) - e^{-i x}(i \cos x + i^2 \sin x) \equiv 0
\\
f^{\prime}(x) = 0 \;\;\; \forall \; x \in \mathbb{R}\Rightarrow f(x) \text{ is a constant}
\\
f(0) = e^{0}(\cos 0 + i \sin 0) = 1 \cdot(1+0) = 1 \Rightarrow f(x) = 1 \;\;\; \forall \; x \in \mathbb{R}
\\ \\
1 = e^{-ix}(\cos x + i \sin x) \Rightarrow e^{ix}=\cos x + i \sin x \;\;\; \forall \; x \in \mathbb{R}
[/math]

>>12505737
>monkey sniffs at book, isn't impressed
wanna bananna?

>>12505743
I like this

>>12505737
It's called schizophrenia. People made literal religions over quantities (see pythagoreans).

All of this is as childish as saying "HOLY SHIT, 4 FITS 3 TIMES IN 12 MINDBLOWN!".

>>12505747
The peak human experience is to be found in returning to monke
Even your mathematician king Ted Kaczynski believed so.

Behold:
[eqn]1-1=0[/eqn]

>>12505791
Ook. Ook. Ook. Ook. Ook. Ook. Ook. Ook. Ook. Ook. Ook. Ook. Ook. Ook. Ook. Ook. Ook. Ook. Ook. Ook. Ook! Ook? Ook. Ook? Ook. Ook. Ook. Ook? Ook. Ook. Ook. Ook. Ook. Ook. Ook. Ook? Ook. Ook. Ook. Ook. Ook. Ook. Ook. Ook. Ook. Ook. Ook. Ook. Ook. Ook. Ook. Ook? Ook. Ook. Ook. Ook. Ook. Ook. Ook. Ook. Ook. Ook. Ook. Ook. Ook. Ook. Ook. Ook. Ook. Ook. Ook. Ook. Ook? Ook. Ook? Ook. Ook? Ook. Ook? Ook. Ook! Ook! Ook? Ook! Ook. Ook? Ook. Ook? Ook. Ook? Ook. Ook. Ook. Ook. Ook. Ook. Ook. Ook. Ook. Ook. Ook. Ook. Ook. Ook. Ook. Ook. Ook. Ook. Ook! Ook. Ook! Ook. Ook! Ook! Ook! Ook! Ook! Ook! Ook! Ook! Ook! Ook.

>>12505800
Ooga ooga ooga. Ooga ooga? Ooga.
Ooga ooga ooga ooga. Ooga? Ooga ooga ooga.
Ooga ooga. Ooga? Ooga.
Ooga ooga. Ooga ooga ooga? EE EEE EE

>>12505737
>Just a bunch of symbols that don't really actually mean anything or represent anything
sure

>>12505723
First define natural number exponents as repeated multiplication.
Notice that for small [math]\epsilon[/math]
[eqn](1 + \epsilon)^x \approx 1 + x \epsilon[/eqn]
so that when a is close to 1, we can set [math]a = 1 + \epsilon[/math] in the above obtaining
[eqn]a^x \approx 1 + x (a - 1).[/eqn]
When a is not close to one, [math]\sqrt[n]{a}[/math] is close to 1 for sufficiently large n, and we have:
[eqn]\sqrt[n]{a}^x \approx 1 + x (\sqrt[n]{a} - 1)[/eqn]
[eqn]a^x \approx \left(1 + x (\sqrt[n]{a} - 1)\right)^n[/eqn]
[eqn]a^x = \lim_{n \to \infty} \left(1 + x (\sqrt[n]{a} - 1)\right)^n[/eqn]
We can take the latter as a definition of [math]a^b[/math] for non-natural exponents.

Recall that
[eqn]e = \lim_{n \to \infty} (1 + 1/n)^n[/eqn]
so
[eqn]e^x = \lim_{n \to \infty} \left(1 + x \left(\sqrt[n]{(1 + 1/n)^n} - 1\right)\right)^n[/eqn]
[eqn]= \lim_{n \to \infty} (1 + x/n)^n.[/eqn]

We have
[eqn]e^{ix} = \lim_{n \to \infty} (1 + ix/n)^n.[/eqn]
The power of n in this expression is just repeated multiplication, and we know that multiplication of two complex numbers multiplies their magnitudes and adds their angles. By repeatedly multiplying [math]1 + ix/n[/math], we arrive at a complex number with magnitude
[eqn]\left(1 + x^2/n^2\right)^{n/2}[/eqn]
and angle
[eqn]n \tan^{-1}(x/n).[/eqn]
The first converges to 1 as [math]n \to \infty[/math], and the second converges to x.

>>12505670
Meanwhile I don't even know what this nerd shit is lmao

>>12505834
The symbols only mean something if you impose your own meaning on them.
We tend to share imposed meanings.
But math isn't about meaningful statements, it's about formulae.
Math can be applied to real-life events and objects, and that can be meaningful.
But e^(pi*i)? That's meaningless on its own. I always regarded those 100-question homework assignments as pointless busywork because they were always just boring pure math stuff that didn't apply to anything I was interested in at all. Now I have a master's degree and run my own business and I still haven't found a single thing I can apply e^(pi*i) to in my own life. Maybe it's relevant to some mathematicians out there, but to me it has never served any purpose and has always been meaningless.

>>12505701
Holy shit, the universe is two dimensional sine and cosine.

>>12505844
>The symbols only mean something if you impose your own meaning on them.
Meaning is intrinsic to information, not something you'd see on a regular day in the jungle.

>>12505853
It depends on what you mean by "meaning".
I follow a combination of subjectivism and a pragmatic definition of truth (read William James). If it isn't useful to me it isn't true to me. Likewise, if I find no meaning in something, then it has no meaning (to me). It may have meaning for you, as does a doodle from a childhood notebook of yours, but if no real attempt is made to communicate that meaning then it really is just your own meaning, and is still meaningless to me. Since I have never encountered a situation where e^(pi*i) has provided me with anything of value outside of a classroom while discussing pure mathematics, it is functionally meaningless to me regardless of what Euler intended.

>>12505844
It's pretty useful for electrical engineering. Or analysis of signals with Fourier transforms. Or dealing with waves (including quantum mechanics). Not everyone will use it of course. Also the thing on the right hand side of the equation is useful if you want the coordinates of something after rotating it, for example to plot it on the screen, but you don't really need the e part for that.

>>12505844
>math isn't about meaningful statements, it's about formulae
bait or beyond retarded

>>12505855
>if I find no meaning in something, then it has no meaning (to me).
You never developed object permanence?

>>12505737
>>12505844
>>12505855
just because you've never bothered to understand the symbols and the equation (or rather nobody has ever spoonfed it to you) doesn't mean they're meaningless. it means you're a brainlet incapable of doing his own research.

It's astonishing after precalculus and before complex analysis.
But then it's like discovering that a baffling magical illusion has some shitty gimmick.

>>12505848
That is phase, not dimensional, sine and cosine occupy the same space out of phase by pi.
Dimensions each occupy different orthogonal space.

>>12505737
I'll take things that didn't happen for 300 Alex.

>>12507678
I could see it happen. I honestly nerded out about that identity when seeing it.

People will say that euler's formula is boring but then claim that Riemann-Roch is beautiful

>>12507777
This. I hate pseuds who only allow themselves to appreciate bloated math.

>>12505670
>This shit impressed me in freshman year of high school

If this impressed you in highschool, then you are a popsci faggot and you don't actually understand the formula, what it means, or why it is "beautiful". Of course it is over-rated and over-memed, but it's still kind of cool because of the whole Taylor series connection. If you were a freshman in high-school, then I can almost guarantee that you would not have known what a Taylor series is, and if you were one of the few (maybe a few hundred or a few thousand) highschool freshman in the world who did, then you would not be a scientifically illiterate popsci faggot as an adult.

>>12507687
But you are attractive anon.

>>12507862
y-you too

The most impressive math shit is that 0 seem to exists

>>12505670
How about this?

>>12507868
So does infinity and the infinitesimal if you use non standard analysis.

>>12505693
It's because of shitty notation.
We're not raising something to a power, like how we do 2^5 = 2 * 2 * 2 * 2 * 2.
It's better to write it down as exp(iπ) + 1 = 0.

If you use the definition of exp as (exp(x))' = exp(x) and exp(0) = 1, and generalize it to the complex numbers you'll get (exp(ix))' = (ix)' exp(ix) = i exp(x)
Since multiplying by i is just clockwise rotation by 90 degrees, we have that the derivative (velocity) of exp is always perpendicular to the position and pointing to the left.

Since:
* the velocity is always perpendicular to the position
* the velocity has length equal to the position's length
* we start at 1+0i (meaning that initial velocity is 1)

we have that the length velocity is always 1, it's always perpendicular to the left of the position, and we start from 1+0i.

So, you're just tracing the circle. If you trace it by π (i.e. half the circle, clockwise), you'll end up at (-1+0i)

>>12507870
>tau
Holy cringe

>>12507870
reddit tier

>>12507922
>>12507923
Alright hotshots what's the most beautiful theorem in math then?

>>12507930
I like the first isomorphism theorem, just not the reddit tier τ shit that you posted.
I don't know which theorem is the most beautiful desu. They always lose their beauty when you properly understand them.

>>12507930
Galois correspondence.

>>12507930
sin^2 + cos^2 = 1

>>12507947
>>12507956
Nice these are great choices. But how does a theorem lose its beauty when you understand it? I find they become more beautiful.

>>12507930
for me, it's Sneed's Lemma

>>12507930
1=.999...

>>12507930
Residue theorem.
Fight me.

>>12507845
Wow dude. How insecure are you?

>>12507930
>>12507967
a^2+b^2=c^2

>>12508079
What makes you think I'm insecure? You're not even using that word correctly. Maybe you mean 'frustrated' or 'offended' or something? If you're mad at someone for taking a different perspective from yourself (e.g. getting mad about a difference of opinions regarding Euler's identity), that's not "insecurity". Like I said it could be described as "frustration", "offense", or you could even argued that I'm emotionally unstable or prone to anger, but the word "insecure" doesn't really apply there, so your comment makes no sense.

You also seem to have poor reading comprehension, and you have completely misunderstood my post. I'm not criticizing OP, nor am I in any way mad or bothered by his post. My point was just that OP is LARPing and this thread is bait. I'm not saying that in a negative or disparaging manner. I say that simply as an observation. Yes, the tone I took in my post is slightly aggressive, but I don't really care if OP is a "popsci faggot" or not. It was mostly banter.

>>12505723
There is one using a sequence and basic trig identities
https://hsm.stackexchange.com/questions/4907/eulers-first-proof-of-eix-cosxi-sinx

>>12507930
(LW + G)/Yaw = TMI

>>12508153
If this isn't copypasta, then you're definitely insecure.

>>12508153
>it was just banter
Keep telling yourself that kek

How does all that relate to
i^n = 1 if n mod 4 eq 0
...
-i if n mod 4 eq 3

>>12508582
Multiplying by i rotates numbers in the complex plane by 90 degrees counterclockwise.
i^n = cos(n * 90 degrees) + i * sin(n * 90 degrees)
Multiplying by e^i rotates by 1 radian counterclockwise.
(e^i)^n = cos(n radians) + i * sin(n radians)
The number e^i is equal to i^(2/pi), and the number i is equal to (e^i)^(pi/2).
People like to write things with a real number base like e because it gets rid of annoying technical complications due to needing branch cuts, plus e has nice properties in calculus.

So what's the point of imaginary numbers, now that we have linear algebra? What does [math]i[/math] do that a vector doesn't do better?

>>12505670
e^(i * Tau/2) = -1

>>12508801
e^(i * Tau) = 0

e^(i * Tau) = 1

>>12508780
complex numbers can be multiplied, vectors can't

>>12505882
Tell me what Euler's identity tells you about the world without making any reference to mathematical terminology.
Tell me how this impacted your ability to live a good life.
Tell me how it brought you closer to objective, absolute truth.
Tell me whether it is something you would teach your children in a SHTF situation.

>>12508892
For what purpose?

>>12508978
To scale and rotate stuff.

>>12508984

You can scale and rotate vectors via a linear transformation, which seems to me to be a lot more flexible and intuitive.

>>12507845
>>12508153
Cringe

>>12509035
In the general case, linear transformations do other stuff besides scaling and rotating. If what you want is to scale and rotate 2D vectors, you need the subset of linear transformations which correspond to complex numbers.

Another important feature of complex numbers, of course, is the fundamental theorem of algebra, which is useful in linear algebra when we want to find eigenvalues.

>>12509035
C is isomorphic to a subalgebra of 2x2 matrices so your question doesn't have a satisfactory answer, because you can always say "but matrices can also do that"

>>12508904
>walk 1 meter right
>walk x metres up
>walk x^2/2 metres left
>walk x^3/6 metres down
>walk x^4/24 metres right
>walk x^5/120 metres up
>etc.
>you're approaching the spot which is exactly 1m away from your starting position and whose angle with the right direction is exactly x
mindblowing? no. meaningful? yes.

>>12509101
Well that's what I'm saying. Linear algebra does everything complex numbers do, and more, all without asking you to believe that -1 has a square root. Seeing people care about complex numbers today is a bit like seeing people care about the luminiferous aether.

>>12509288
But that's wrong.
(-1 0
0 -1)
has lots of square roots.

>>12509288
Let a(n+2) = a(n+1) - a(n) with initial conditions a(0) = 0, a(1) = 1. How would you go about finding the closed formula for a(n)? Your linear algebra tools for solving this sort of problem are broken if you don't use complex numbers.

>>12508892
>multiplied, vectors can't
dot product
cross product

>>12509288
no, you're missing the point completely. complex numbers are not "replaced" by linear algebra, they are included in linear algebra. when you say we shouldn't care about complex numbers, you're saying we shouldn't care about some distinguished subset of matrices, because they're the SAME thing.

>>12508213
>insecure
You're still not using that word correctly though. Insecure doesn't mean someone who is irritable or easily offended. Insecure has more to do with anxiety, and self-doubt.

>>12508153
>>12509752
>>12507845
>You're still not using that word correctly though.
>What makes you think I'm insecure? You're not even using that word correctly. Maybe you mean 'frustrated' or 'offended' or something? If you're mad at someone for taking a different perspective from yourself (e.g. getting mad about a difference of opinions regarding Euler's identity), that's not "insecurity". Like I said it could be described as "frustration", "offense", or you could even argued that I'm emotionally unstable or prone to anger, but the word "insecure" doesn't really apply there, so your comment makes no sense.

This is bullshit. You are insecure . That world accurately describes you. You're a whiney loser who got pissed of because some random person on a message board doesn't like Euler's identity as much as you. You're self-obsessed with your own intellect and you are personally offended that someone on the internet doesn't share you tastes. You are a low IQ loser and you are completely enraged about reddit-tier topics for pseuds, namely Euler's identity. You also type like a literal autist and you clearly take yourself very seriously.

>>12505670
So you just decided to make a post about something no one asked about, just to prove how over it you are?

>>12507930
quadratic reciprocity

[math]e^{it}[/math] represents a point on a circle where t is an angle. Once you know this fact, [math]e^{i\pi}[/math] is not that interesting.

>>12512333
>eiteit represents a point on a circle where t is an angle
it's this fact which is interesting

>>12510392
no one ever asks for a thread.
threads are inflicted upon lurkers.
If you think 4chan is a demand/supply system you are dead wrong.

>>12505737
>I honestly do not understand why permanent virgins cream their pants over this shit
you wouldn't get it

>>12505670

This shit is so fucking cringe. Totally superficial

Still, I admire it

>>12505855
>William James

u mean this?

>>12515424
>William James
this one?

>>12505670
Bro I'm just impressed that complex exponential are eigenfunctions of lti systems. Also true Chads Only use the complex Fourier series

>>12505800
>>12505826
based monkeys

>>12505855
>if I find no meaning in something, then it has no meaning (to me)
>if no meaning, has no meaning
obvious fucking shit for 5¢, Alex

>>12505670
anyone who gets pogchamp about this after high school is a moron

>>12515716
https://www.dcode.fr/ook-language

> e^i*pi = -1

> If you rotate something 180 degrees it'll be upside down

>>12505693
see this
https://www.youtube.com/watch?v=-dhHrg-KbJ0

>>12509814
>>12508213
>>12508079
>You are insecure . That world accurately describes you. You're a whiney loser who got pissed of because some random person on a message board doesn't like Euler's identity as much as you.

That's not what insecure means. Getting mad over a post on an internet message board is not 'insecurity'. It could point to immaturity, autism, instability, or aggression, but not insecurity. Someone who is insecure is the opposite of combative. They are shy, emotionally withdrawn, and often display flattened affect, meaning they appear less emotional and less reactive to events. Someone who is insecure would generally tend to avoid confrontation and disagreements.

https://en.wikipedia.org/wiki/Emotional_security

You're claiming that I am pissed off, but the irony is that you genuinely seem to be getting pissed off by the fact that you don't know what 'insecurity' means. You sound like your understanding of human behavior comes from a mixture of internet personality quizzes, and daytime TV like Dr Phil and Oprah.