prove to me that complex numbers aren't arbitrary circlejerking

It's just a vector dumb dumb

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This is you.

>>8343354
>taking numbers out of your ass to be able to solve equations
>someone actually defends this

>>8343328
physics

>>8343354
What does "balancing a checkbook" actually entail, other than adding and subtracting numbers from others, thereby creating a sequence of numbers.
What does the guy ask for, what do you need to learn other than writing down what you spend and doing addition and substraction??

>>8343328
Quantum mechanics

>>8343409
/thread

>>8343385
>>8343354
what can complex numbers do that reals can't, besides dragging more the calculation of non existent numbers?

>>8343328
Prove to me that negative numbers aren't arbitrary circlejerking bullshit

>protip, you can't

Is this a bait thread? A simple google search will find enough applications for complex numbers to shut you up.

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>>8343416
Can your real numbers do this?

>>8343386

most brain dead faggots live on credit. balancing a check book becomes taking your $500 paycheck and decided who to pay (whos interest charges to pay) with the money.

people are retarded, do you honestly think "rent to own furniture" stores and Paydays stores make a profit based on the small amount of people who need new furniture because of a flood? go look at /g/, most are burger flippers yet have a $3000 battlestation

>buy $1500 couch with $50 down
>no payments for 1 year
>1 year later they cant pay it off
>35% interest rate and principle payment with back interest
>"oh hai repo men!"

>>8343416
Can real numbers derive solutions to 2nd order differentials?

>>8343428
Ah, k, well I live in a social democratic country (Germany) people don't do loans unless they buy a house

>>8343416
provide roots to all integer polynomials

also try looking at differential equations, you know those things that are everywhere.

The complex numbers are the smallest algebraic closure of the reals. Also, complex cobordisms are intimately tied to chromatic homotopy theory and algebraic K-theory. Plus, the complex numbers are an algebraic artifact from the "complex" Hopf fibration, so they are very natural and show up a ton when reasoning about why certain patterns arise in the homotopy theory of the spheres.

>>8343437
This. Underdamped responses would be a hassle without Euler's formula.

Even with people pushing for a sort of unification of homotopy theory and logic itself, I have a hard time putting it of strange topologies over field theory.

>>8343417
Prove to me that numbers aren't arbitrary circlejerking bullshit

>protip, you can't

>>8343328
>the matrix ([1,1],[-1,1]) doesn't exist
ok

>HUR HUR REAL NUMBERS ARE REAL
>IMAGINARY NUMBERS ARE FAKE

All numbers are equally unreal you retard.

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>>8343328
What the fuck did you just fucking say about complex numbers, you little bitch? I’ll have you know I stopped caring about math when I was introduced to the concept of imaginary numbers, and I’ve been involved in numerous secret raids on Al-Gebra, and I have over 300 crocks of shit. I am trained in equations that can only be solved by inventing numbers that can't exist and I’m the top math deity in the entire US academic forces. You are nothing to me but fucking wrong. I will wipe you the fuck out with math the flaws of which have never been seen before on this Earth, mark my fucking words. You think you can get away with saying that shit to me over the Internet? Think again, fucker. As we speak I am contacting my secret network of algebra solutions across the USA and your IP is being traced right now so you better say "the correct answer is whatever the correct answer is", maggot. The math that says the pathetic little thing transcribed to words. You’re fucking dead, kid. I can be anywhere, anytime, and I can mark you wrong in over seven hundred ways, and that’s just if you write it down in english instead of ancient math runes. Not only am I extensively trained in unarmed combat, but I have access to the entire arsenal of the United States Logical Math Corps and I will use numbers that never lie to their full extent to wipe your miserable ass off the face of the continent, you little shit. If only you could have known what unholy flaws your little “clever” human construct was about to bring down upon you, maybe you would have held your fucking tongue. But you couldn’t, you didn’t, and now you’re paying the price, you goddamn idiot. I will shit complex numbers all over you and you will drown in it. You’re fucking dead, kiddo.

>>8343328
because thanks to "invention" of complex numbers you can shitpost on your phone

>>8343515
This picture is a great representation of intuitionist logic. Saved.

>>8343328
they sure help with the description of underdamped circuits but you're probably too much of a fag to know about that

>>8343425
Why aren't there any responses to this? I'm an EE, and complex numbers literally take up most of our calculations which have physically verifiable effects in the real world...

>>8343416
You're exactly right. Complex numbers can be realized as pairs of real numbers (a,b) with certain properties like
(a,b)=(c,d) iff a=c and b=d
(a,b)+(c,d)=(a+c,b+d)
(a,b)*(c*d)=(ac-bd,ad+bc)
from the last property:(0,1)*(0,1)=(-1,0)
if we choose to define (0,1)=i for notational convenience then i*i=-1 and (a,b) is equivalent to a+bi.

>>8343556
they can also be realized by matrices ([a, b],[-b, a]) <=> a+bi

which is a specific instance of the more general form [eqn]a+b\sqrt{N}[/eqn] being ([a,b],[bN,a]) where N=-1

>>8343456
>>8343456
>I have a hard time putting it of strange topologies over field theory.

literally what

>>8343409
/////////////thread

>>8343416
The original historic reason for the complex numbers was not to solve the equation [math] X^2+1=0 [/math] despite what you hear in classes. The reason for them was to solve degree three polinomials, look here
>https://en.wikipedia.org/wiki/Cubic_function#General_formula

Even if you have real solutions, the intermediate steps involve manipulation the "fake" numbers, and yet you end up with the correct solution. That is exactly what you're asking for.

>>8343586
>link goes to calculating square roots
>literally the necessary step in solving general quadratics
>somehow this is magically different

>>8343590
The solutions for the quadratic were understood since antiquity, you don't have to invent the complex numbers when you can just say when they have no solutions.
And the necessary step involving quadratics is exactly what I said, the point is that the imaginary part coming from two different square roots can cancel out, and you get a legitimate real solution.

you can't be posting without complex numbers. checkm8 faggot.

>>8343378
>can't solve x + 2 = 0
>make up "negative numbers" like -2
>pulling numbers out of your ass to be able to solve equations
>someone actually defends this
I knew math was a joke once some autists decided there must be more than the natural numbers

>>8343328
if complex numbers aren't real then explain this
[math]\frac{\sqrt{2}+\sqrt{2}*i}{2}[/math]

Some people say "imaginary numbers" is unfortunate naming. I insist it's a good thing since it keeps the plebs out.

>>8343328
define:
[math]\mathbf{1} = \begin{bmatrix}
1 & 0 \\
0 & 1

\end{bmatrix}[/math]

also define:
[math]\mathbf{i} = \begin{bmatrix}
0 & 1 \\
-1 & 0

\end{bmatrix}[/math]
you will find that these two definitions will satisfy the identity:
[math]\mathbf{i}^2 = -\mathbf{1}
[/math]
Consider complex numbers as a linear combination of these two matrices:
[math]z = a\mathbf{1} + b\mathbf{i} [/math]
We have now defined complex numbers solely in terms of real numbers. These can be used to solve many problems in engineering and physics. A popular shorthand is to drop the boldface:
[math]z = a + bi[/math]

what now OP?

>>8343425
This

>>8343328

they are found in nature

for example impedance calculations

>>8343700
>he thinks 'are found in nature' is the same as 'are used by humans to model nature'

w e w

e

w

l a d

a

d

>>8343354
>balancing a checkbook
>useful

automatized already

>>8343705
>he thinks there's a difference
w e w

e

w

l a d

a

d

>>8343328
prove to me that "real" numbers aren't arbitrary circlejerking bullshit

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>>8343785
fucking this
everyone who doesn't understand that all mathematics is a set of rules which can be used to model reality needs to pull their no math head out of their asshole

>>8343785
no

>>8343641
you're late

Prove to me that negative numbers exist.

Give me one real, physical example of a negative number.

And don't give me some shit like "Oh I have -$2000 in my bank account". No, you have $0 in your bank account and you owe $2000 (a positive value).

Negative numbers do not exist in reality. Mathematicians just made them up.

>>8343416
No numbers "exist". Complex numbers are imaginary in the same way that negative numbers are imaginary.

It was probably a bad idea to call i the "imaginary number".

God i love this thread!

Most people will live rich full lives and never have a need to apply imaginary numbers. That is also true for irrational and basically negative numbers. But the need for all of these types of numbers is real. In the case of imaginary numbers the need is very subtle and complex. Which makes their need, and therfore existence, very hard to explain to non-math folk

This video series explains why we use them very well.

https://m.youtube.com/watch?v=T647CGsuOVU

>>8343554
>responses
>to real sci/math
you seemed to have mistaken this board for a non-shitposting one

>>8343328

If we just taught students that [math]\mathbb{C}^n \cong \mathbb{R}^{2n}[/math] from the get-go I bet we wouldn't have threads like these.

>>8343961
Not a "math person" but know a fair bit, tell me if this is flawed thinking in some way:

While I can't think of any non-mathematical or computational use for numbers like i, it seems to me that there doesn't need to be a "why" as in a purpose for them to exist, their existence is self evident in the logic of numbers. Square roots and negative numbers both exist insofar as any mathematical concept exists, it stands to reason a negative number has a square root and we should be able to mathematically express it. I was never really confused by them conceptually for this reason, it seemed very natural that -1 should have an expressible square root.

And maybe this is only because I'm a little younger than most here (19), but no one in school ever had a problem with negatives. I think it was explained to us that it could represent debt, which I guess was concrete enough that no one complained that "lol never gon use this something something balance checkbook :DD" and zone out.

>>8343969
the problem is introducing square roots, square roots are generally as unbelievable as complex numbers and should be introduced at the same time in the same way

The common that square root of 2 is not rational still assumes it exists, which in my opinion remains to be proven. Square root of 2 is taken on faith for many years, perhaps forever, before any kind of retarded construction of reals is ever taught. But finite field extensions of rationals can be introduced basically at the exact same time as matrices, and in fact are probably a really good motivation for introducing them in the first place.

>people are still answering my thread almost 6 hours later
didn't expect this

>>8343462
this is the only correct answer
Am mathematician

>>8343386
This is why I have no furniture except for a mattress on the floor at my 1br apartment.

>>8343615
>can't solve equations like x-1=1
>invent "natural" numbers like 2 to have vacuous answers to problems with no solutions
>math autists literally have no clue what they are talking about

Look mom, I'm shitposting with the cool kids!

>>8343409
The exact reason, the kernel of it
https://youtu.be/gCAxGTt7nLg?t=11m25s

>>8344021
Baisically yes.

Math people say pretty much what you said but like this:
>>8343443

>>8344021
It doesn't "make sense" that you can take the square root of a negative.

>>8343328
it does have meaning.
for example, imaginary power is power that was transmitted but never used (reactive power).
so it has a real world application and a real world meaning, meaning it's not worthless.

>>8344306
You can't. Square root is a simple redneck function that doesn't even claim to be able to do that.

[math] \displaystyle
\sqrt {x^2} \ne \pm x, \quad \sqrt {x^2} = \left | x \right |
\\
\left | x \right | =
\left \{
\begin{align}
x, & \hspace{1em} x \geq 0 \\
-x, & \hspace{1em} x < 0
\end{align}
\right .
[/math]

>>8343437
Only if you get lucky.

>>8343354
I gotta be real, I'm 23 and I have no idea what "balancing a checkbook" means. Is this an American thing?

>paying taxes?
shit comes out of my income automatically.
>paying off credit cards
a matter of transferring enough money from one account to another

what is it?

All numbers are imaginary and intangible.
Do any of you guys think they are real?

>>8343354
wow realy maeks u think

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>>8344350
Here's my two (2) apples.
Now show me your i (i) apples.

>>8343575
I means putting math involving strange topologies conceptually and logically before something "pure" as fields.
To say complex numbers are natural because they arise in some relatively complicated settings is not a convincing point

>>8343969
But that's wrong unless you set up a quite strange structure on R^2n (the one isomosphic to C^n as a vector space) where pairs of numbers from different rows get involved with each other upon scalar multiplication.
Uf you don't mean isomorphism by that sign but just bijection as sets, then
C^n = R
is true just as much.

>>8344355
strawman

>>8343328
If complex numbers were bullshit then most of our models of electronic circuitry, signal processing and so on would be totally incorrect which does not seem to be the case. They are one of the more heavily used things at least in electrical engineering and without them the work they do would be extremely difficult

>>8344375
you're a complex guy

>>8344379
for you

All math is arbitrary circlejerking bullshit made up by humans to represent relations found in nature and elsewhere. However, as it is consitent in a good description of nature and is usefull to us, we do not render it arbitrary circlejerking bullshit.

>>8343554
>EE
>doesn't understand that your complex number calculations are just glorified trigonometry
Leave the conversations about complex numbers to the people that actually understand them :^)

>>8344355
show me your -2 apples.

>>8344402
show me the logic behind i^2 = -1

>>8344355
You would first have to identify what an apple is.
By following logic, you will find there is no apple.
There is no boundary as such. It is only your imagination creates that. Thus, your "2" apples are imaginary.

>>8344405
I define i to be the number which when squared = -1

how unfortunate that some guys thought it clever to call the numbers "complex", "real", "imaginary", "rational", "irrtational", "natural", etc.

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>>8344402

As another mathamatican
All of math is circle jerking. You make up sets, usually of numbers, and decide on the rules that apply to operations within that set. Bla bla

Some times the problem requires only gives meaning to natural numbers like savages trading beads.

Sometimes you need to divide land and you use the the real numbers. Allowing you to figuring the area of a circle.

And sometimes you need to work with the complex numbers.

There is no logic behind why i^2=-1. There is no logic to why in calculus we get to pretend infinity is a thing. Whenever mathmagians are presented with a problem you get to decide what set of numbers you want to work with and what properties you want them to have. It just so happens that when you are working on some problems using complex numbers just works.


so yall nailed it. we mathmagicians circle jerked them int exsitance. just like we did with other insane shit that everyone swallows. My favorites are -1x-1=1 the existance of 0 and the irrational numbers.

So when a problem arises that we know has a solution, and no tools work. We sit around circle jerking and and reading math papers until we abitraraly will numbers to behave a certain way. then if our new load of the good stuff magically solved the problem; we unload it all over the world like the intilectual bukkake the plebs deserve!

Describe the relationship between current and voltage in an oscillating circuit without using imaginary numbers.

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>>8344512
>Everyone swallowing the reals
lurk more

>>8344544

Oh yeah swallow more!
those irrational numbers. They totally make sense.
*glug*
I bet you like 1+2+3+4+...=-1/12 too.
*Glug*
Oh god quit talking and swallow more of my circle jerked math.
Ooo
Hyoerbolic geometry!!!!
Oooh god
1 isnt prime!!
Oooo Swallow swallow.

....
....
Ahh that was medium good.

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>>8344584
idiot

>>8344306
Well, it literally does make sense to me. But thanks for letting me know that intuitively understanding a relatively simple math concept is wrong somehow.

>>8344405
i = -1^1/2
literally highschool level math, not sure why you're on this board if you don't even understand that

>>8344430
>nihilism
;^)

>>8344511
Stupid negger

>>8344698
Is that nihilism?
I don't know. I havn't read much into it. maybe I should.
Im danish, so I could easily read Soeren Kierkegaard.

I just know that what I say is true in the sense that it is coherent. At least according to how I percieve it.

>>8343869
at least im right

>>8344727
It's often called Nihilism. Basically it's the thought that nothing is truly distinct from anything else, at it's most extreme everything is just a system of energy and particle reactions and any further definition is imposed by humans. Basically anti-platonism.

It's different than Nietzsche's nihilism, which is his description for the anti-value system between when the "old order" collapsed and when a new order would rise up, almost totally unrelated.

>>8344355
yes, there are two apples in the picture, but how that that prove the existance of a number two? there being TWO apples is a property (a quantitative property, in this case). in a similar manner, complex numbers are used to represent a property of something, be it a physical or an abstract object

>>8343328
You can use them to prove shit like trig identities desu.

>>8345067
>It's often called Nihilism. Basically it's the thought that nothing is truly distinct from anything else, at it's most extreme everything is just a system of energy and particle reactions and any further definition is imposed by humans.

I'm uncertain that particles can exist. Wouldn't that be impossible to determine?
I mean, they would have no real boundary. They would just consist of smaller particles, that themselves would consist of ever smaller particles into infinity.
Following logic, it would mean that everything amounts to nothing, and nothing amounts to everything.
That is what the unified field is all about, and why they can't come up with a theory.
They are quantifying that which is unquantifiable.

>>8344512
>there is no logic to why in calculus we get to pretend infinity is a thing
>mathamatician [sic]

>>8344355
What the fuck is an apple? They're just concepts made up by humans. The only things that actually "exist" are quantum fields. All else is abstraction done by humans to make our models simpler.

>>8344349
>balancing a checkbook
"Balancing a checkbook means you've recorded all additions (deposits) made to your account and subtractions (withdrawals). Each deposit and withdrawal is called a transaction. The purpose for balancing a checkbook is to know how much actual money you have in your checking account at any given time."

Seems worthless with the advent of online banking.

>>8345837
You have no idea how hard it is to balance a checkbook if you are a medium sized business that, instead of hiring 1 CPA at 60k per year to do all of the accounting for the business, decides to hire 4 accountants who are not certified in any way, at 40k per year each.

imaginary numbers are very important placeholders, they need to be there to draw conclusions about things.

>>8343328
Everything made by humans is cirlejerking. Complex numbers work, like it or not.

>>8343883
most straightforward simple response in the thread.

complex numbers aren't more or less real than the Reals. They are a natural algebraic system when you want to have certain properties.

The reals are only as real as your need for completion in some topological sense.

The integers are only as real as your need to be able to invert addition.

The naturals are only as real as you need to be able to iterate addition.

The complex numbers are as real as you need to be able to solve arbitrary polynomial equations.

Once you know what the solutions HAVE to be, it is always possible to restrict your attention to situations where it gives the answers you WANT.

This last situation (having solutions for arbitrary polynomial relations) shows up a lot, and the complex numbers are uniformly the best way to deal with that, from diff eq to class field results.

btw, you don't need to go full C to bump into complex numbers. The same reasoning I expressed introduces many intermediate rings, such as Z[i] or Q[i], depending on what you need.

>>8344355
Well can you show me pi (pi) apples? I guess you can't, too bad now pi doesnt exist anymore lol

>>8344512
0/10, go back to >>>/b/

>>8343328
I like complex numbers. Really fun what you can do with them when you have sinusoidal circuits with caps and inductors.

>>8343428
And this happens exactly because primary schools don't focus on teaching life skills. I'm not saying don't teach complex numbers. I am saying teach kids about applied finance. Or just hate the poor. Enjoy your oranges, asshole.

>>8348714
But the Democrats want to keep the poor incompetent and dependent on government handouts so they can keep generation upon generation of faithful voters.

>>8343328
C is the smallest algebraically closed field that contains R.

>>8343354
I'm so glad I deleted facebook. I'd see images like this posted unironically on a daily basis.

>>8348720
I think that neither the dems nor the republicans truly have the interest of the poor at heart. The poor don't fund their campaigns. I think math has lots of joy but if you graduate high school without knowing how to avoid predatory lenders, what's the point?

>>8343713
Right? Like do kids not understand that balancing a checkbook was only considered annoying/difficult before calculators were common place.

>>8348812
Dem trees

>>8343515
>Al-Gebra

fuckign lost

>>8343428
Anyone who thinks debt is retarded is retarded. The modern system of debt is one of the most brilliant fucking inventions of all time. Without it, consumption is only possible after saving. While moralists and ascetics might appreciate this, the rest of us would like to be able to drive to work while we work to afford a car. In reality there is no reason to not have a car while you pay for it: the manufacturer works on marginal cash flow, you work on marginal cash flow, so redirecting flows makes much more sense than putting up "money dams" everywhere to accumulate flows into one big release.

We COULD put massive battery packs in every residence and business in the world then only generate electricity once a month, or we could have a continuous flow of electricity which is only subtly adjusted at the margins on demand as need requires.

>>8349239
debt is not retarded. But interest rates are.

>>8349307
interest rates are just the price of debt, not that scary really.

>>8343378
complex numbers are actually needed to solve some cubic equations even for real roots; that's why they were originally invented

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>>8349392
Yes! Good goyim. Interest rates are good yes!

hate this board now fucking high schoolers can we make a /sci/ allowed only for those who pass a test.

>>8349444
/sci/ has been high school algebra shit from day 1 newfag get fucked back to normiebook

>>8349392
https://www.youtube.com/watch?v=7HdO5cO7dPw

>>8344033
Perelman go back to feeding your cockroaches

>>8349239
This nigga gets it.

>>8349239
A capacitor (originally known as a condenser) is a passive two-terminal electrical component used to temporarily store electrical energy in an electric field.
Capacitors are widely used in electronic circuits for blocking direct current while allowing alternating current to pass. In analog filter networks, they smooth the output of power supplies. In resonant circuits they tune radios to particular frequencies. In electric power transmission systems, they stabilize voltage and power flow.

This is a perfect analogy for why it is useful to make savings.

>>8349392
interest rates is a ponzi scheme.
it is fraudulent. The collective debt can never be paid off.

>>8343875
I am almost certain that greedy elite bankers and similar organizations were involved in shaping science as well to benefit their agenda.

>>8343416
Almost every useful calculation in Electronic Engineering.

Without Laplace and Fourier transforms (both complex functions) there can be no practical signals processing.

>>8349720
kys u retarded fuck

>>8349706
Yes. Why would they not.

>>8349687
>bank loans you money to buy a car
>no interest so there is no reason for them to give the loan
The interest is just a fee for doing the loan based on the risk involved, the total amount of the interest over the loan can be calculated and shown to you as if it was a lump sum fee. Also many loans let you pay ahead of interest.

In short you know nothing about finance.

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>>8349967
>In short you know nothing about finance.
In short you know nothing about how money is created today, and how the bankers run the whole fucking show.
In short, you pretend to know how it all works, while having done no research.
In short you know not of what you speak.

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>>8349967
>bank loans you money to buy a car
>no interest so there is no reason for them to give the loan
Thats not the banks job.
thath is the job of the car manufacturer and the salesman - to make it affordable.

>>8343354
I never understood this meme.

isnt balancing a checkbook just addition and subtraction of a column of numbers?

>>8350012
If the right way to live is without credit, then how could bankers run the show for you?

>>8350105
it's tuff for 'muricans cuz you run out of fingers real quick

>>8350112
Credit =/= Usury

>>8343328

OPfaggot hasn't understood yet that numbers profoundly are arbitrary

OPfaggot needs to go back to school

>>8350243
This.

All numbers reside in the imagination. They are all imaginary

all numbers are imaginary in the purest sense retard. are you going to say negative numbers are stupid too?

They're just numbers that obey different rules that make certain things possible that would otherwise not be.

All numbers, and math, are just a human invented concept that is arbitrary. You could define anything as anything.

>>8349687
>>8349392
look up 'time value of money'

time is money

time without your money is time that could be spent using that money to accumulate more money for yourself

interest is the price of that.

>>8343328
They're just the exponential equivalent of negative numbers or fractions.

>>8349664
Neat

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>>8350500
>time = money
wat?

>Interest = The price of time without your money, that could be spent using that money to accumulate more money for yourself.
Wat?

Please clarify what you mean.

>>8350533
not that guy, but, think of loaning money with interest as making an investment. you wouldn't invest in a stock that had a 0% return. i think the time=money bit is just saying that it doesn't make sense to put your money in an investment with no return when there are more lucrative opportunities available. the more time you keep your money in an investment, the more money you get

>>8350500
>time value of money'
That concept is only true within a ponzi scheme structure like the banking system.
In an usury-free system based on gold and silver for example, moneys value rise over time, and you are better off waiting to recieve the money if you don't have anything to use it for right away.

>>8350569
>ponzi scheme
>muh banks
>muh usury
>commodity currency
top
kek

Back to >>/pol/, kiddo

>>8350586
>muh funny post
>muh irrelevant to the discussion post
>muh not unnersdant engyfing ob wad u sey
>muh muh muh
>muh like to be goyim sheep BAAAAAAAAAAH!!

>>8350628
>being jealous of G-D's chosen
Oy vey

>>8350655
Not being jealous. I pity them.

>>8350670
>implying we need your pity
>implying you're not the pitiable one for not understanding economics

>>8350675
>we
they*

>>8350675
>not understanding economics
You knoweth not of what you speaketh.

>>8344512

Yeah man! Fucking High Five!

I just love fucking those plebs over. What crap should I lay on them today? Holy Fuck yes! Pi! Oh yeah. Then multiple infinities! Haha! Watch their brains bubble over. Oh fuck yes. Oh God yes. Haha! oh! Oh!. I think I'm going to jizz.

>>8343641
Just curious but can anything interesting be done with the matrices?

>>8350837
Yes.

>>8350837
Google eigenvalues and eigenvectors, it's from basic Linear Algebra but their applications are many

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>>8344699

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>>8344430

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>>8351076
I vill stant for se trus. Se trus vill prevail no matter hav muts juv try to ignorre it.
Se trus alvays vins in se ent.
So vy not embotty se trus?
Sat vets es tru, vill last in se lengs of se illusion of time.
He hu novs se trus, vil nov vot evrising es, ant he vil not fear to stant in se fase of se nossingness of reality.

>>8343328
Ok, sit down OP and learn.
Complex numbers are more of a vector space (a matrix) than a "number".
Due to this, they are fairly useful for:
>Solving equations otherwise unsolvable.
>Pretty much everything in electrical engineering

The important thing is
>Google it before asking stupid questions

>>8352208

Ok, let's be not so inaccurate.

(i) a matrix is a countable, finite object (basically it is an element of K^n x K^p). a vector space is infinite and has a structure (operation laws, topology).

(ii) a complex number is a sometimes useful representation for a 2-dimensional vector : instead of writing :
- vector(A) = x. vector(i) + y. vector(j), we can write
- A = IAI exp(i. phi) as a complex number (in the complex plan)

(iii) it is this way of formatting vectors which is used in electricity (and many other fields)

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>>8344355

>>8344405
>he doesn't know polar coordinate system

>>8344400
Already aware of that...
The point was to say that complex numbers were "real" and not "imaginary". Trigonometry shows "real" things. Complex numbers are just a pretty compact way to show that.

Which is cool.

1=(-1)2
√1=√(-1)2=-1
1=-1

Hey low IQ OP just take 2x2 matrices and treat the anti diagonal identity matrix as i. So now matrix multiplication and every single science that depends on matrices is also circle jerking bullshit? \thread

>>8352281
> 1 = (-1)2

that's wrong

>>8352281
√1 = +1 or -1
multiplying two negative numbers give you a positive number anon

>>8352292
>>8352292
Omg sqrt is non injective on R, so you're going to declare a solution you have to restrict to a domain in which it is injective. Once you have done so there is no ambiguity. You're solving for its pre image but are pretending misunderstanding inverting a function is a form of cleverness.

1=√1=√(-1)^2=√-1•√-1 = i•i=i^2=-1
1=-1

>>8344511
Cheeky cunt

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Your move.

>>8352431
There is no number you can write which, when squared, is 2. The proof for this is very common. Usually the claim is that this proves the square root of two is not rational; but, all it proves is that the square root of two does not exist. The same motivation for creating integers from naturals, the same motivation from creating rationals from integers, applies to square roots. Unfortunately for the mystery school religion that is modern mathematics, there is no way to extend the rationals to contain square roots—it can contain an arbitrary but finite number of extensions, limited only by the patience of the practitioner, but no one can do arithmetic with an infinite number of extensions.

The situation is worse for so-called "algebraic" numbers since there's actually no extension at all to cover fifth degree solutions to polynomials, as demonstrated by the amazing Galois theory. Math has gone from an expert craft to sleight of hand in trying to extend the field of rationals any further than quartics and specific higher orders.

Put simply, there is no field of "algebraic numbers." Such a thing is never constructed, and never can be constructed, in the same way that integers, rationals, and other extensions can be constructed. And anyone claiming to do arithmetic with them is lying.

>>8343354
Imaginary is really an awful name for them.

>>8352469
Yes. All numbers are imaginary!

>>8352454
Are you talking about the incompleteness theorem?

>>8352454
No rational number, pretty sure [math]\sqrt {2}[/math] describes exactly such a number.

>>8352289
>1 = (-1)2
no, that's correct

the second equality is correct too

the third one isn't, because sqrt(a2) = a if a > 0, and -a if a < 0. in our case a=-1 < 0, so sqrt(a2) = - a = - (-1) = 1.

>>8344405
The logic is this: i represents a 90-degree rotation of the plane. If you rotate by 90 degrees twice you get a reflection, and multiplication by -1 represents a reflection, so i^2 = -1.

>>8352454

N (integers) is included in Z (relative numbers), Z is included in Q (rational numbers).

All of these sets contain the same "amount" of numbers, which is the cardinal of integers N.

However, the next step (defining irrationals), and the following steps (algebraic, transcendental, etc.) makes a gap that anon describes.

The gap is: even the irrational numbers can't be "linked" to N. They are "uncountable". More than that : they are as numerous as the whole set of real numbers.

This gap says that there is a difference between the amount of numbers in N and the amount of numbers in the set of real numbers R.

It also says that there isn't any "size" inbetween. This is quite unexplained so it has been accepted, as the "continuous hypothesis".

As for C (complex numbers), it has the same "amount" of numbers as R (which is the same "amount" as there is in [0,1], but that's another story).

>>8343328
Actually negative numbers are weird as well. I mean you could actually define them the same way you define complex numbers. Like, I just call -1 = h. It has the property that h^2 = 1. Then you get stuff like

[eqn] e^{hx} = \cosh(x) + h \sinh(x)[/eqn]

They only become the negative numbers as we know it if we additionally introduce an equivalence relation

[eqn]a_1 + h b_1 = a_2 + h b_2 \ \mathrm{if} \ a_1 + b_2 = a_2 + b_1[/eqn]

The generated equivalence classes are then the negative and positive numbers.

>>8352584
You can also have a unit which when squared is 1, but this unit isn't equal to 1. You can also have a unit which when squared is 0, but this unit isn't equal to 0. All of these are the matrices [[a, b],[bN, a]]. N=-1 is complex, N=1 is split-complex, N=0 is dual, N=2 is square root of 2, etc. etc.

But they cannot operate together. If you want to use both complex and dual numbers you need a 4x4 matrix (use a 2x2 matrix to represent one, then use this to make a block matrix like the other), to use three of them you need an 8x8 matrix, etc.

>>8352584
you missed the sqrt in your definition of h didn't you ?
why to do introduce hyperbolic cosine and sine ?
you define your equivalence relation on which set ? RxR ?

>>8352237
Do matrices have to be finite?

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>>8352266
nice dubdubs

>>8352631
>you missed the sqrt in your definition of h didn't you ?
No.

>why to do introduce hyperbolic cosine and sine ?
For the same reason why the cosine and sine is introduced for the exponential of complex numbers.

>you define your equivalence relation on which set ? RxR ?
You can define it on R, but also on N, Q or Z I guess. It makes the most sense on R+0 and N0.

>>8352673
As far as I know, yes.

To define a matrix, you need to set two integers (generally called n and p), which are the sizes of the matrix. Then, the matrix is just a collection of n x p elements. A 2 x 2 matrix (on R let's say) is defined by 4 elements (elements of R).

So yes, a matrix is "finite".

>>8352728
Wikipedia says otherwise my guy

>>8352732
well could you be more precise please then? so that we discuss fruitfully?

>>8352736
Not sure what you mean. I'm just saying that matrices with one or both dimensions being infinite can exist and Wikipedia agrees with me..

>>8352726
>For the same reason why the cosine and sine is introduced for the exponential of complex numbers.
Thing is: it is the exponential notation which is introduced. Steps are : function f(t) = cos(t) + i sin(t) verifies exponential-like properties ( f(t+t')=f(t)f(t'), f' = i f, f(0)=1), so that this function is written like exponential.

>You can define it on R, but also
I was talking about the very one you mentioned. It is apparently defined on a cartesian product, so I'm surprised you say R or whatever, and not R x R.

>R+0 and N0
I don't understand what you mean by R+0 and N0, sorry, could you explain that ?

>>8352745

well you have to teach me because I don't know about that

for me, a matrix is a representation of a linear application between to vector spaces of finite dimensions.

As soon as on of the spaces is of infinite dimension, I never used the notion of matrix.

But if there is a definition for matrices in such cases, and if you can teach me what this definition is, I'm glad to learn something. Thank you in advance.

>>8352758
Sorry senpai but I know nothing about this subject. Just didn't think there was any specific reason to think infinite matrices didn't exist. You're better off going to Wikipedia (https://en.wikipedia.org/wiki/Matrix_(mathematics)#Infinite_matrices) or MathSE

>>8352777
Ok, I had a look on that wikipedia page, thanks. You've just expanded my knowledge.

What is presented in this paragraph of the wikipedia article is a generalization of the notion of matrix.

If you don't know what a "basic" matrix is, of course you couldn't be "limited" by this definition, so you could think about infinite matrices, without seeing the technical problems this "infinite" causes (very definition, representation, how to define multiplication ? determinant ? eigenvalues ? ... and so on).

One aspect of mathematics is to be candid. It is an important aspect.

The technical work needs then to be performed, to secure the visions.

>>8352777
nothing infinite exists

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>>8352924

are complex number deniers just memeing or do they actually believe the shit they're spouting?

>>8343328
Any math can be defined as true as long as long a's it doesn't contradict any previously defined axiom.

Imaginary and complex numbers contradict 0 axioms. Therefore they can be defined as part of real mathematics.

It has as much validity as aleph null. It exist because we say it exists, and it violates 0 axioms.

>>8343409
is there anything that isnt proven with quantum mechanics?