/sci/ - Science & Math

>>10961963
I think this is just your cope for not understanding them

>>10961964
I understand them fine but they seem tacked on. There's probably a way of thinking about math from the foundations that better incorporates the concept of the imaginary dimension.

>>10961963
And you are entitled to your opinion, much like everyone else.

Now try to send a quantum entangled particle to space and we'll see if imaginary numbers are useful or no

https://www.independent.co.uk/life-style/gadgets-and-tech/news/china-quantum-entanglement-satellite-messages-launch-a7794176.html

>>10961964
/thread
>dude ur not supposed to use numbers that confuse me even if they proof ridiculously useful in multiple practical scenarios

>>10961963
Why?

we should make more positive/negative signs to fix math: +,-, @,ß,, etc
like if you agree

>>10961968
It would likely be far more convoluted and fucky to use pracrically

>>10961976
Right but I'm thinking it would be useful for academics. Kind of like how physics diverges for quantum physicists versus the Newtonian model high-school students use.

>>10961969
>>10961972
Once again, not saying the concept isn't without efficacy, just that it should be better integrated.

>>10961974
>like if you agree
Fuck out of here

>>10961977
Academics can rim me. Imaginary numbers will still be taught and used around the more considering how easy they make life. Mathematics is a tool, and it makes sense that we would like our tool to be simple to use.

>>10961968
"Imaginary dimension" - you mean R^2? The imaginary element provides an elegant way to view multiplication. All other ways are convoluted

>>10961963
tonight try to actually take your pills

>>10961980
>, just that it should be better integrated
How could that be accomplished?

>>10961963
"imaginary" is an unfortunate misnomer and ought to be completely stricken out because it gives people the wrong idea.
mathematics involving i actually predict real-world phenomena. the term "complex number" is more accurate.

Dilate your number system OP

>>10961987
By redesigning the entire model of mathematics, down to the most basic arithmetic from the ground up.

>>10961994
Can you even iterate what's so wrong with complex numbers?

Empfindest du sich selber als Teil eines dualistischen Kampfes gegen den Rest der Welt in welchem du in der Defensive verbleibst und empfindest du dich selber als einen natürlichen Antagonisten des Guten?

>>10961999
Because they were discovered and then tacked on to the existing system of mathematics. But if they represent an entire dimensional component of numbers then why shouldn't basic operations account for them?

>>10961963
they are literally just vectors

>>10961963
The right way to see the complex numbers is as the polar coordinates in the plane. Concatenating, scaling and rotating arrows is perfectly intuitive even to kids. You don't even need to recognize i as an special number, but orthogonal bases seem to be natural.

The fact that somehow doing this leads to a solution to [math] x^2 = -1 [/math] is really just a coincidence.

>>10962003
>why shouldn't the basic operations account for them?

What?

What do you mean by "tacked" on? They are a bigger set containing the real numbers with the property that all equations can be solved. They exist because of the axioms of set theory. The real numbers were "tacked" on to the rational numbers in a more contrived way. If anything you should have an issue with that than anything else

>>10962003
>Because they were discovered and then tacked on to the existing system of mathematics
... yeah... Much like the negative numbers, zero, rationals and so many other ones. The main difference is that you don't understand little sense they make either because you grew up using them.

>>10962003
>tacked on
Dude there's an entire field of study called complex analysis, and they reveal more about the regular integers (this is almost all of analytic number theory) you insist are so naturally and "strongly constructed" in a way the complex numbers aren't. It's not that they are useful, but they reveal fundamental truths about the mathematics you seem to have no issue with. There's a reason why a lot of "natural operations" you'd do over a circle or over polygons seem at home with complex numbers. They are in no way "tacked on" and probably feel unnatural to you because of your unfamiliarity with their use.

>>10962013
It's not really a coincidence. It has to do with cycling about the origin i.e. the roots of unity. All the -1 reveals to us is that given complex numbers are way better at sifting out the arbitrary details, as what really matters here is the degree of the polynomial and the magnitude of the constant

>>10962033
I honestly think he probably just accepts the reals as "hey man, it's just natural fact that follows from, like, regular math man. They're like, just a cope because of continuity man. Math already accepts them and doesn't need to be overhauled, man." Hell, I bet he doesn't believe the reals have had more controversy than the complex numbers!
OP is a brainlet who doesn't have "academics' interests" at heart - his posts read like a salty high school student

>>10962041
*they're like, not just a cope
I'm not wildberger

>>10961963
I was helping a primary school girl w maths and she couldn't grasp negative numbers. When I showed her the scale on a C thermometer she understood how it works, or the idea that something can be a few meters above the lake (+) or below it's surface (-). But whenever we diverged from real life examples she was completely lost.

Now you're that girl, but much older and with complex numbers.

>Posted from my phone from a pub
>Cheers,
>Dirty phoneposter

>>10961963
Why do we use complex numbers? Because you can't solve every equation in the real numbers. The complex numbers allows us to do this. In particular, being able to solve every equation allows you do things like diagonal matrices (whatever that means) and whatnot, something physicists like to do (you are constrained otherwise). The complex numbers is the UNIQUE field containing the real numbers that allows you solve every equation. There are natural geometric ways of thinking about complex multiplication (rotation and scaling) so it's not coming from an unintuitive perspective either; it's more than just some abstract generalization of the reals. It's only natural that if you wan want to to do more mathematically, you have to invent more algebraic objects to work with, I. E., the "Imaginary" unit. So, you can dismiss them all you want, but without them physicists and engineers would be crippled mathematically

>>10961963
What's the highest level math course you've taken?

>>10962035
complex analysis is just tacked onto real analysis. you could do analysis without ever diving into complex numbers. in fact a lot of math fields are like that in that they serve no purpose other than to jerk off people who like thinking about numbers and pretend to "get it." It's like one big practical joke. it's the aristocrats of academia.

>>10962003
>and then tacked on
If you don't "tack on" stuff, you end up representing 3 as 1+1+1

>>10961963
Yeah I guess. Just use matrices instead.

But on the other hand. it is easier to just represent the imaginary unit as "i' and follow the algebra from there on.

Würdest du behaupten dass Menschen, die dich nicht mögen und für welche du nicht interessant bist ein Problem für andere Menschen darstellen, die dich nicht mögen und für welche du nicht interessant bist und bist du der Meinung dass nicht du das Problem bist sondern andere Menschen außer du selbst und das ausschließlich?

>>10962071
matrices are just a representation of a vector field, which are an extension of the complex numbers

>>10961968
>the concept of the imaginary dimension.
2D?

>>10962039
>it's not a coincidence, it is [concept that already presupposes our current understanding of complex numbers]

Small brain post. If you were to discover the complex numbers by moving arrows around the ground, you'd see it as a coincidence that by rotating 90 degrees twice you trivially get a solution to [math] x^2 = -1 [/math].

In fact, not even that. Someone rotating sticks on the ground would not even be preocuppied with polynomials. What would happen is that 500 years later, some mathematician would study polynomials and not even see anything special about rotating 90 degrees twice.

>>10961968
r*e^(i*phi)

>>10961974
that's just imaginary numbers

>>10961968
>I understand them fine
Do you, though? Complex numbers are just R^2 equipped with a binary operation that makes them a field. It also happens to perfectly coincide with real multiplicacion for numbers of the form (a,0). How is this "tacked on" and what do you even mean by it? Writing (a,b) as a+bi is just shorthand and the i is there only to identify the first coordinate from the second. The fact that (0,1)*(0,1)=(-1,0) doesn't really have much relevance at all. It's exactly the same thing that physicist do when they write (a,b) as ai+bj to identify what is horizontal and what is vertical.

>>10962048
get off your phone at the pub

We wouldn't have retard-tier opinions like this if the person who named imaginary numbers just called them something else.

>imaginary numbers
>they are no more imaginary than the real numbers are

>>10962033
This.

>>10964260
Well, they do have other names. Complex numbers and lateral numbers are the alternatives. But I agree OP was probably retarded enough to think imaginary numbers actually meant they are imaginary.

>>10964434
>think imaginary numbers actually meant they are imaginary.
Aren't they all?

/sqrt{-1} = i

>>10961963
without them we never would have got the 6,000,000i joke from anon earlier

>>10961963
Okay, well if I called it R^2 with a division algebra structure, would that make you happier?

How is x + R(x^2 + 1) not a well defined element of R[x]/(x^2+1)?
At least if you believe in R.

>>10961963
You should have killed George as soon as he decided heresy was better than the law.

>>10961963

>>10961963
Don't worry. I just overhauled the system for you. Tough stuff.
[math] i = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}, 1=\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}\\
a+bi = \begin{pmatrix} a & -b \\ b & a \end{pmatrix}[/math]

>>10961963
Read on what algebraic closure is

>>10961963
Quite frequently they end up canceling out so they're mostly for bookkeeping until you get to that point.

>>10966697
Great, now integrate that

>>10966932
what ?

>>10961963
This guy gets it.

I stopped caring about math when I was introduced to the concept of imaginary numbers. What a crock of shit. If your equation can only be solved by inventing numbers that can't exist, like some kind of math deity , then you are fucking wrong and the math is flawed. Same for algebra solutions that basically say "the correct answer is whatever the correct answer is". Thats what the math said transcribed to words but god forbid if i wrote in down in english instead of the ancient math runes the teacher word mark me wrong.

Math is logical and numbers never lie my ass. Math is just as flawed as any other human construct.

>>10961963
t. illiterate subhuman

>>10967027
>I stopped caring about math when I was introduced to the concept of imaginary numbers
So you haven't been into math since middle school? Damn that must really suck.

>>10966932
Use ordered pairs of real numbers, damn it. Addition is (a,b)+(c,d)=(a+c,b+d), multiplication is (a,b)*(c,d)=(ac-bd,bc+ad). Real numbers are written as (a,0), and (0,1)*(0,1)=(-1,0) (i^2=-1). It is just more convenient to write a+bi instead of (a,b).

>>10966697
Woah, is [math] \mathbb{C} [/math] just a subalgebra of [math] \mathfrak{su}(2) [/math]? Though it seems you have to fix the phases on your basis elements and then restrict the scalar field to [math] \mathbb{R} [/math].

>>10966932
brainlet

@ll numb3rs a√e |mag|na|ry

>>10967132
missing the point completely. [math]\mathbb{C}[/math] is a subalgebra of the endomorphism ring of [math]\mathbb{R^2}[/math].

>>10967132
You'll find C represented in any C-vector space, so that's moot.
Also, the matrices he wrote down aren't in su(2). If the unity matrix were in it, SU(2) wouldn't even be compact (let alone unitary).

>>10967182
and it cant be subalgebra because the oprations are different

>>10967132
Also, the matrices he wrote down aren't in su(2). If the unity matrix were in it, SU(2) wouldn't even be compact (let alone unitary).
If you pass to sl(2), then you look at a C-vector space and C is trivially represented in it.

>>10967186
If you were replying to me (before I added the sl(2) line), then I don't know why exactly.

>>10967186
>>10967189
Ah, you mean if you'd want it to be a sub-algebra and not just represented as the scalars (given any base vector).

In any case, there's the quaternions and I suppose that's where he'd want to look

>>10964851
That's a different discussion. But the point is, complex numbers are as valid as the others.

>>10967195
I mean that C can't be a subalgebra of any Lie algebra, simply because it's not a Lie algebra.

>>10967211
Yeah, I gotcha.
(The "issue" with that answer is just that he was saying it's a subalgebra of su(2) (which it's not and so that's wrong), but given he was responding to the post constructing a+ib as matrix, it's also clear that he didn't mean that it's a Lie-subalgebra where the algebra operation would be [,]. If x * x isn't 0, then yeah it can't be a Lie-subalgebra of su(2) and that's that, as you say. But if we go back to speaking in terms of the already existing product in a ring over C (of matrices here) and unity I is it, then the c*I represent C. Not very interesting, but just to clarify that there are two notions of "subalgebra" here, as we have two products at hand.)

Incidentally, for people interested in such quotient rings, I did a very light-hearted motivation to universal enveloping algebras yesterday:

https://youtu.be/x_H64Q2GZfw

>>10961968
You mean like this?

>>10967305
if you made this, I suggest actually making a smaller circle within the bigger one and writing just 1, y, y^2, y^3 on its axis intersections.

Moreover, a point could be made about, on the positively line, writing x y^0 instead of x and y^0 instead 1.

>>10967310
>on the positively line, writing x y^0 instead of x
I thought the point of complex analysis was to get the y's the cancel out? In which case, why write y in the real axis at all?
Conversely, how would you differentiate between y^0 and y^4?
> and writing just 1
Scalars aren't real anon

>>10967324
I don't understand the first and last line of your post, and y^0=y^4 for y^2=-1.

>>10967394
>the first line
Every single application if imaginaries I have seen is about getting out purely real values. And scalars aren't real (in the physical sense), since they lack dimension.
> y^0=y^4 for y^2=-1.
Then what does 'y' equal? Remember the goal is to avoid using 'i', and deal purely with reals

>>10967431
What do you mean by "applications". Practical applications are interested in getting rationals.

>And scalars aren't real (in the physical sense), since they lack dimension.
>real
Nothing we have words for "is real" if you ask long enough - certainly not in the sense of materialist realism.

>Then what does 'y' equal? Remember the goal is to avoid using 'i', and deal purely with reals
They were talking about constructing C given R and you can do that by formally* looking at the ring R[X] or finite sequences of polynomials** and taking the quotient w.r.t. the ideal X*X+1.

*in the sense that you can operate with those things in a computably enumerable way
**sequences of X functioning as basis, and sequences of X's being nicely enumerable, i.e. you can write stuff down and compute and even teach a computer how to do it

That is to say, you consider the totality of polynomials with coefficients in R and X*X=-1.
So e.g.
-7 + 2 * X + 5 * X * X + 8 * X * X * X * X * X
and by above ideal rule, this above polynomial is judged to be equal to
-7 + 2 * X + 5 * (-1) + 8 * (-1) * (-1) * X
i.e. this above polynomial is judged to be equal to
-13 + 10 X
Working in the ring of polynomials and with the above ideal rule amounts to just working with what's otherwise called the complex numbers. Executing the rule of replacing all instances of X*X with -1 is formally and thus computationally completely unproblematic.

Sidenote
-7 + 2 * X + 5 * X * X + 8 * X * X * X * X * X
is also
-7 X^0 + 2 * X^1 + 5 * X^2 + 0 X^3 + 0 X^4 + 8 * X^5
so the representation
{-7, 2, 5, 0, 0, 8}
for polynomials suggests itself. This way you're also done with "X" and algebra w.r.t. complex arithmetic is reduced to arithmetic and reduction rules with list:
{-7, 2, 5, 0, 0, 8} becomes the pair (-7+(-1)*5+0, 2-(-1)*0+8), i.e. (-13, 10)

If you already believe in R, then complex arithmetic are just very simple functions on lists of reals (the polynomials)

>>10966657
Based

>>10967475
>Executing the rule of replacing all instances of X*X with -1 is formally and thus computationally completely unproblematic.
Except for when you want to decompose it back to {-7, 2, 5, 0, 0, 8}, but I guess that's a separate issue.

Thank you for explaining what they meant by C

>>10966932
Done.

no number squared equals a negative number. Why wasn't that fact enough to settle on.
>"BUT WHAT IF THERE WAS A NUMBER SQUARED AND IT EQUALED -1"
yeah and what if pigs had wings. eat shit.

>>10967672
What is a number?

>>10967682
>instant reductio ad absurdum

>>10967672
Good thing numbers aren't literal things we have to observe to conclude they exist

>>10967700
what about that post caused two different people into instantly resort to reductio ad absurdum.

>>10967672
What's the highest math course you've taken? Can you construct natural numbers using only set theory? What about reals?

>>10967761
I don't know what you are talking about. I didn't "resort to reductio ad absurdum," I pointed out a fact about numbers.

>>10967672
>what should I do twice to get a point reflection ?
>oh right, a rotation by 90 degrees
complex numbers in a nutshell

>>10961968
Probably, but the patriarchy is preventing us once again from achieving great things for hu(((man)))kind

who is the dildo who decided [math]5^{-1} = \frac{1}{5}[/math] instead of [math]5^{-1} = -5[/math].

>>10967836
These are the poeple, with which I want to discuss mathematics...

>>10967836
Consider the exponential map Z,+ to Q,* and demand it be homo

>>10967796
180 degree rotations are not reflections

>>10967836
What do you suggest we define as the multiplicative inverse then?

>>10967877
not this guy, but I'm sure he just meant the notation. 1/5 is already a (bad) notation for the multiplicative inverse

>>10967860
I said point reflection

>>10967883
You also said two 90 degree rotations, which is not the same.

>>10961963
This board is for math and science, not engineering.

>>10967672
It doesn't matter if it's not intuitive or doesn't follow some axiom you dreamt up. It works and can be used to come up with real word predictions in physical sciences. Maybe you should just appreciate how wondrous math is rather than cling to a simplistic, materialist view of reality.

>>10967908
what is it with self-professed mathematicians and their delusions. it's fucking math, not lord of the rings. just cause you can do math in your head doesn't mean it operates on the same mechanisms of inventing fictional fantasy worlds you mong.

>>10967892
dude what ?

>>10967937
A reflection flips one axis. a 180 degree rotation flips both axes. With a rotation, your i-axis becomes negative too.

mirror reflections and 180 rotations are only degenerate in the reals

>>10967959
planar point reflection = 180 deg rotation

>>10961963
>we should have overhauled our entire system
What do you mean by "we" and "our", Peasant?

>>10967962
Oh wait, is that what you're calling an inversion?

>>10967977
I'm not a native speaker, I don't know what's the most common term
https://en.wikipedia.org/wiki/Point_reflection

>>10967980
>what's the most common term
beats me.

>>10961981
>we should make more positive/negative signs to fix math: +,-, @,ß,, etc
bump the post to agree

>>10967694
>me speak latin
>me big brain