/sci/ - Science & Math

>>10889046
you're right. you should start explaining that to everyone you know.

>>10889046
you are literally just a shitposter

>>10889046
A cheat at what exactly?

>>10889046
Octonions/sedenions is where shit hits the fan. There is a reason why string theorists masturbate to meme math so much and given that string theory is a dying art/line of study, it makes sense after all.

Not even pythagorean sects where this deluded

>>10889046
the sqrt is a convenient cheat

>>10889046
it's just a natural extension to numbers, the same way subtraction gives negative numbers, division gives fractions, etc

>>10889046
Define a 2-dimensional vector space with the representation of a vector (a,b) where a, b are real numbers that obeys the following multiplication law
[eqn](a, b) \cdot (x, y) = (a\cdot x - b\cdot y, b\cdot x + a\cdot y)[/eqn]
and the addition law
[eqn](a, b) + (x, y) = (a + x, b + y)[/eqn]
now consider
[eqn](0,1)\cdot (0,1) = (0,1)^2 = (-1,0)[/eqn]
according to the multiplication law as defined above. Now clearly we can map every vector (a,b) with b equal to 0 to a. As long as we accept that -1 is really (-1,0), then we have a solution to the equation [math]-1 = x^2 = x \cdot x[/math], in this case it's (0,1). Now let [math]i = (0,1)[math] and let [math]1 = (1,0)[/math] and you have complex numbers.

>>10889359
>As long as we accept that -1 is really (-1,0)

That's not even axiomatic, that's pure dogma. I'm not against the usage of complex numbers, they save a lot of work when doing electrical analysis, but they are literally made up devices to encode maths even more. The problem is when you go full retard with it aka string theory aka 50 years of theory and zero real world tangible production

>>10889367
I wasn't trying to be axiomatic, just hoping to show another way of coming to the result that i^2 = -1 without having the appearance of cheating.

>>10889359
https://en.wikipedia.org/wiki/Complex_number#Matrix_representation_of_complex_numbers

>>10889367
listen you fucking moron, the algebraic closure of a field is a perfectly reasonable construction to make. and guess what? the algebraic closure of R is C. nothing is "made up" about complex numbers, they're literally the correct place to work in almost any context (see: various wonderful properties of complex inner product spaces, complex differentiable (holomorphic) functions, all the obvious algebraic properties of the complex field, complex structures on manifolds, etc etc, the list is fucking endless).
you cannot possibly have any knowledge of what you are talking about if you believe that complex numbers are artificial.

>>10889046
This is just the 12th grader equivalent of negative numbers don't make sense. The problem is that we had to teach math to retards and could only do so by enforcing this idea of countable objects, when math can apply to much more abstract and useful things.

>>10889046
can't solve
x^3-5x+1=0
without imaginary numbers
even though all three solutions are normal real numbers
https://www.google.com/search?q=y=x^3-5x+1

>>10889469
yes my entire point is that you literally just made up an imaginary number to make that equation solvable

>>10889046
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.

>>10889802
2D number
Sucks to be you if 16th century math makes you clutch your pearls
[math] \displaystyle
\boxed{ \mathbb{T} \;
\boxed{ \mathbb{S} \;
\boxed{ \mathbb{O} \;
\boxed{ \mathbb{H} \;
\boxed{ \mathbb{C} \;
\boxed{ \mathbb{R} \;
\boxed{ \mathbb{Q} \;
\boxed{ \mathbb{Z} \;
\boxed{ \mathbb{N}}}}}}}}}}
[/math]

>>10889891
This guy gets it.

I stopped caring about math when I was introduced to the concept of negative numbers. What a crock of shit. If your equation can only be solved by inventing numbers below nothing, like some kind of math debtor, then you are fucking wrong and the math is flawed. Same for arithmetic solutions that basically say "the correct answer is a fraction". 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 exist my ass. Math is just as flawed as any other human construct.

>>10889046
Didn't fermat, or leibniz, or Einstein, or whoever it was, when originally presented with the idea of an imaginary number, called it an imaginary as an insult because they thought the same thing? And then they ended up coining the phrase.

>>10889899
>start gaining properties until you hit H
>suddenly you don't even have a division algebra
literally what's the point? just construct a vector space at this point.

>>10889899
What means t, s, o, and h?

>>10889936
Division algebra goes up to O you retard,
read a book.

>>10889380
Complex numbers are rationalizations designed to take shortcuts and things easier, but it's all made up.

Do they work and help? Yes.

Do they give birth to a bunch of pointless schizophrenic shit aka high math aka octonions / sedenions that literally have no real use other than mental masturbation and comfort for the literal three fags in the entire world that are currently wasting time and resources on them? Lmao yes

>>10889978
32D, 16D, 8D and 4D numbers

>>10889942
32D, 16D, 8D and 4D

>>10889931
This guy gets it.

I stopped caring about math when I was introduced to the concept of decimal numbers. What a crock of shit. If your equation can only be solved by inventing intermediate virtual numbers, like some kind of math accountant, then you are fucking wrong and the math is flawed. Same for elementary operations that basically say "the correct answer is neither 1 nor 0, but in-between". 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 exist my ass. Math is just as flawed as any other human construct.

>>10889996
ha.ha.so original.

>>10889933
https://youtu.be/_qvp9a1x2UM?t=3m

>>10889367
>they are literally made up devices to encode maths even more
All mathematics are "made up." Complex numbers aren't less a part of reality than rational numbers or natural numbers because none of them exist in reality in a literal way. Your hand doesn't have a magic Sesame Street style number 5 floating above its fingers. It's all abstract. Mathematics isn't beholden to anything about our physical world. No part of mathematics is defined by our physical world.

>>10889899
>>10889984
Can you have a complex-dimensional hypercomplex number?
e.g. If Complex are 2D and Quaternions are 4D, can you have a field that's 2+i D?

>>10890044
sort of, tuples of two complex numbers are commonly used in quantum mechanics to describe spin. That's just [math]\mathbb{C}^2[/math] though, so maybe that's not what you mean.

>>10890044
No. The dimension of a vector space is defined as the cardinality of its bases. This means dimension will always be a cardinal number, of which 2+i is not a part.

what does stuff like
[math]
\mathbb{C} \otimes \mathbb{H}
[/math]
mean

>>10890070
>dimension will always be a cardinal number
what about fractional dimension in fractals?
for instance, if you want to specify a position along a koch snowflake line, despite being a line, a single number is not enough to specify anywhere on the line, because there is infinite distance between any two points.

>>10890081
it's an exterior product between the vector space of complex numbers and the vector space of quaternions. basically this just denotes a new vector space where an element of the space is furnished with both a complex number and a quaternion. quite anticlimactic really.

>>10890097
sorry, I meant tensor direct product, not exterior product

>>10889942
H: Quaternions
O: Octonions
S: Sedenions

>>10889046
They are useful cheat because they allow you to reuse certain solution methods, so you don’t have to come up with a separate solution for every case. Take differential equations, you can represent a single differential equation and split into multiple first order differential equations. Then, you can represent differentiation as a linear transformation, change its basis to get a diagonal of eigenvalues, or the generalized version where you put numbers next to the diagonal, and then solve a bunch of easy differential equations, and then switch the basis back to get the solutions. However, not every matrix has real eigenvalues, but they ALL have complex eigenvalues (this is because the characteristic equation will ALWAYS have a complex root). This allows you to reuse the solution method as long as you start using complex numbers.

>>10889942
>>10890126
T: Trigintaduonions

>>10890139
>>10890126
>Sedenions
>Trigintaduonions
do these have any use at all? I mean presumably there are an infinite number of hypercomplex number systems, is there any point in discovering them?

>>10890070
>dimension will always be a cardinal number
Wrong.

>>10889046
>okay i know you cant actually do that... BUT WHAT IF WE PRETEND
lol mathcucks are so dumb lamo

>>10890081
you tell me what you mean by that

>>10890170
You can actually do that. Where are you getting pretend from? It's not any more or less pretend than integers are.

>>10890097
Can you say that [math]\mathbb{C} \otimes \mathbb{H}[/math] and [math]\mathbb{R}^6[/math] are isomorphic?

>>10890207
yes

>>10890148
They're part of the Cayley-Dickson construction which produces a series of R-algebras meant to generalize the jump from R to C and C to Q. It was natural to ask if there were any other division algebras over R aside from the classical ones, and the Cayley-Dickson construction seemed like a natural way to find more. It was later proven through a much different approach that there are no division algebras over R other than R, C, H, and O.

Nowadays there may be some rep theorists or string theory memers who study sedenionic character theory or whatever, but the bulk of mathematics has moved on from them.

>>10890207
As vector spaces, not algebras.

>>10890170
How can you claim we can't do that when we just did it?

>>10890148
Trigintaduonions have a paper showing their use in quantum field theory

>>10889931
>>10889996
Cringe.

>>10890096
>>10890162
He clearly said dimension of a vector space, someone has not passed baby linear algebra

>>10889931
I really hope this is a shit post.... if it's not you don't belong on this board.

>>10890345
It is, look at the replies. People rage against complex numbers rather than checking out why they are useful.

>>10889978
could it go up to S or even H and just needs a solution?

>>10889046

mathematics is really no different from language and there is no such thing as a cheat in either

>>10890345
I really hope this is a shitpost, if you have don't have enough brain cells to understand irony then you don't belong on this board.

>>10890174
not a clue, just saw a lot of that in
https://www.youtube.com/watch?v=3BZyds_KFWM&list=PLNxhIPHaOTRZMO1VjJcs7_3dgyJ2qU1yZ

>>10890025
Best response

Woah math is cool.

>>10889046
Isn't every mathematical construct a cheat of some sort?

It's not complex numbers that are the problem OP...

>>10889046
Yes, it is, literally, just a cheat. But it's not like us mathematicians are trying to pull a fast one over your head.
>imaginary numbers
it's in the name.

>>10889981
Sounds like you havent studied topology very deeply

>>10891309
just wondering, do rational complex numbers exist in his crazyness?

>>10889891
And yet complex numbers are needed to derive models in even classical physics, like springs and waves. It's almost like math isn't an arbitrary invention.

This reply was brought to you by idealism gang.

>>10889933
The considered them useful, but denied that they represented anything real. In other words, they agreed that it's a cheat.

>>10890025
You must be a pretty devoted materialist to argue that math is idealist while restricting what counts as real to the material world.

>>10889329
how mentally ill does someone have to be to think that a 16-dimensional hellfuck of a number is useful? isnt it true that addition doesn't even work at that stage?

All numbers are made up.

Is any number system valid as long as it's logically consistent?

>>10892015
No, by the Incompleteness Theorem that would be far too restrictive a requirement - We'd basically only be safe working with Presburger Arithmetic or Willard's self-verifying systems. Rather, any number system is valid if it can be modeled in ZFC, MK, NBG, PA, Type Theory or whatever formal basis you've chosen to work from that hasn't been proven inconsistent.

Wow, I have literally never seen this thread before

>>10889891

Fucking deja vu, Man.

I remember reading this exact same comment some years ago on this forum. Years ago. Its memorable. The " I stopped caring about math..." and the "the correct answer is whatever the correct answer is..." and the best bit, "ancient math runes.."

Wow, like are you real or are you some sort of bot which is programmed to trawl through math threads and post that comment every time imaginary numbers come up?

Or are you a real human who just happens to keep that comment conveniently located on you computer's notepad so you can copy paste it every time you sniff through a thread like this?

Why would you even bother though, to keep a generic comment on your notepad, just for threads like this?

Nah, you must be a bot. Well fuck you then bot.

>>10891849
They do
https://youtu.be/oybzcvv-ZVo

>>10889367
>but they are literally made up devices to encode maths even more.
you're missing the point, literally all of math is "literally made up device" to encode something more conveniently

>>10890241
What does it even mean for algebras to be isomorphic? How is it different to vector space isomorphism?

>>10892684
wikipedia is your friend

>>10891887
>You must be a pretty devoted materialist to argue that math is idealist while restricting what counts as real to the material world.
Abstraction isn't idealism you brainlet.
Abstraction is the process of behaving around fictional reference points as though they were real.
e.g. We pass around pieces of green paper and pretend there's something underlying this called the 'US Dollar' which has no physical form, can be represented in any number of alternative media (dollar bills, electronic records, checks, money orders, etc).
The physical behavior is what's real. The fictional story is what we pay more attention to though in a way that's somewhat weird / paradoxical seeming at times e.g. with the money example the not literally existing physical concept of currency has more sway over our behavior than visceral biological problems right in our faces, like when someone loses their access to shelter from the elements or their ability to receive food because they no longer have enough money.
None of this requires that you believe in philosophical idealism where these abstractions exist in a literal sense floating around some Platonic realm.

>>10891887
>You must be a pretty devoted materialist to argue that math is idealist while restricting what counts as real to the material world.
Abstraction isn't idealism you brainlet.
Abstraction is the process of behaving around fictional reference points as though they were real.
e.g. We pass around pieces of green paper and pretend there's something underlying this called the 'US Dollar' which has no physical form, can be represented in any number of alternative media (dollar bills, electronic records, checks, money orders, etc).
The physical behavior is what's real. The fictional story is what we pay more attention to though in a way that's somewhat weird / paradoxical seeming at times e.g. with the money example the not literally existing concept of currency has more sway over our behavior than visceral biological problems right in our faces, like when someone loses their access to shelter from the elements or their ability to receive food because they no longer have enough money.
None of this requires that you believe in philosophical idealism where these abstractions exist in a literal sense floating around some Platonic realm.

>>10891887
>>10893049
Other examples of abstraction like this are games, language, your center of gravity, and the eye of a storm.
With games, the real stuff is the behavior e.g. People moving little pieces of wood around on a board in the context of a chess game.
The somewhat weird / paradoxical seeming non-real thing that has even more importance to what they're doing than the real stuff though is the abstraction of the game of chess. The players engage in a fictional concept of the little pieces of wood serving as placeholders for these differently empowered members of fictional armies with rules of engagement and elaborate strategies of what the opponent might do in the future and how you plan to work with that.
The real things of language are mouth noises and scribbles or LED displays comparable to scribbles that people produce or react to other people's productions of. But the non-real stuff that's even more important than the real stuff in language is the "meaning" concept where we behave as though these sounds or drawings stand in for radically unrelated objects or events that don't have much if any resemblance to their representations e.g. the word "bee hive" written out on paper doesn't give you any hints as to what it means if you were looking at it as a non-English speaker.
Center of gravity is a useful fiction too along those lines. There is no literal center of gravity object like there is such an object as a rock or a puddle of water, but we can behave as though this fiction is an object and get benefits from doing so. Eye of a storm is the same deal. It's a fictional reference point for an absence of something in the context where storm activity is present otherwise and makes the absence of storm activity seem like a thing of its own.

>>10889046
there's nothing cheating about taking the free algebra on R and quotienting by the maximal ideal (x^2+1)

>>10889046
The great thing about math is that there is no such thing as cheating, we make up the rules and axioms we choose to work with. All the matters is how useful the axioms we use are when it comes to solving problems and illustrating mathematical concept. As far as that is concerned, the way we've conceptualized complex numbers is actually quite useful, and is a tool which has allowed us to solve problems that were previously unsolvable, and allowed us to express complex problems(pun intended) in a much more elegant way.

>>10892684
vector space isomorphism preserves only linear combinations. algebra isomorphism preserves also the multiplication.

>>10893081
>The great thing about math is that there is no such thing as cheating
Yes there is, and Tooker is an example of it.

>>10893136
How can you cheat in a game with no rules?

>>10893069
Based