The complex number hoax

>R^2 is not a field.
You understood what I meant.
Everyone computer scientist understood what I meant.
Yet you have to bitch about it.

Operations between compex numbers are still not defined though.

Sure, you can go full autism mode and write two paragraphs of R^2 vector operations instead of defining complex numbers and writing 4 symbols, but nobody involved is autistic enough to do it.

>C isn't readily a field either
I think the "official" definition of C is as the algebraic closure of R. So yes, it is readily a field.

Because your whole post is pointless noting that the complex numbers satisfies the convention of a field of ordered pairs. What exactly do you want, a name change?

A completely arbitrary criterion for what constants are physical and that is essentially propaganda for the real numbers. Any complex number can thought to be "physical" as an ordered pair.

OP is right

Literally every time you MAKE UP a """IMAGINARY NUMBER""" or """COMPLEX NUMBER""" you could have instead used a power series simply to get the real part, the part that actually exists.

In my class we defined C to be R^2 with these operations:

(a,b)+(c,d) = (a+c,b+d)
(a,b)*(c,d) = (ac - bd, ad + bc)

Prove to me that this does not form a field, dumbo.

You define C by adding operations to R^2 that are not intristically there by convention. R^k in general is not a field. This is not remotely complicated concept to understand

Furthermore, this is the superior way to define it. Defining it as the algebraic closure of R puts the properties before the set. It is better to construct C out of known elements and then use that definition to prove the fundamental theorem of algebra to establish that indeed, this construction closes R.

This also solves OP's fake problem of realness. In class to simplify operations we obviously defines (0,1) to be 'i' but here 'i' is just a symbol that stands for that pair and we can only say that i^2 = -1 because of you calculate (0,1)*(0,1) you get (-1,0)

That puts the field before the properties. This construction also immediately establish the geometry of complex numbers, as angles and lenghts are well defined properties in the plane that just transfer here. After all, we have the same plane with the same point, just that now we have some nice operations to combine points.

>math
>intrinsically
kek
let me quote Leopold Krocnigger
God created the complex numbers, so that you could kill yourself, virgin aspie
-john von numale

But that is how algebra works!

You realize that R^k is never a field because it is just a set.

Similarly, C = {a + bi, a,b element of R} is also just a set

You still have to define the operations which are 'the operations of the reals and i squared equals -1 lol'

Fields are not sets. They are ordered triples of 3 sets. The first set A can be anything and the other two sets have to be relations from A^2 to A so that they are also laws of composition

great post, memes. I like how you tried to enter the fray and speak english like human beings until it was evident your post made no sense

I don't now what part of convention you and so many others do not understand. When I or anyone else on the planet refers to R^k, we are referring to the euclidean space topology equipped with the bilinear norm and all the other standard properties. This crap you have posted is completely irrelevant because it in no way suggests that C = R^2, which was the initial stupidity proposed in this thread, it is clearly not the same as R^2 because it has different operations, furthermore C is accepted as a convention because it is a field with geometric implications of the plane. It is truly bizarre that a strong rigourist thinks that retreating to axiomatic arguments in someway does a way with the necessity of the geometric language of C when formal axioms were introduced so that previously derived conventions would remain sensible.

>euclidean space topology equipped with the bilinear norm

Sure, you may think it refers to that... but you would be wrong.

I remember back in geometry we defined the arbitrary n-space E and then gave the euclidean axioms it should follow.

Then the professor remarked that R^n was a good candidate for an euclidean space, but it is not necessarily that way.

In particular, if you take any set and find a way to define lines such that they follow the axioms of euclidean geometry the congrats, you have a space!

Your teacher sucked. Do you even study pure math? Your lack of abstraction makes me believe you are some fucking engineer trying to talk about math.

Your argument, or rather the total lackthereof, as well as all of its strawmans falls apart given even the most minute context from the thread. First off you have taken a segment of my post out from which to base your rant when in fact the preceding couple of words in the original post renders everything you said irrelevant, namely that referal to R^k as euclidean k space is a convention. It is obvious that all posters, besides you that is, in this thread are subscribing to this convention. Among many, some have referenced that R^2 is isomorphic to C, that C is constructed as a rotation and scaling of R^2, and that R^2 is a valid replacement for C. Regardless of whether or not any of these things are true, it is clear everyone is talking about euclidean 2-space, not a set of 2-tuples without structure. Finally you questioned my training as a mathematician because I did not subscribe to your arbitrary stupidity that euclidean k-space is not constructed from R^k, which is certainly valid autism, but I never said otherwise. I keep repeating the word convention because the concept of a convention answers every question in this thread, in particular why we use C as opposed to R^2, and more recently that R^2 is most commonly interpreted as a vector space in any disciple.

you got thoroughly owned and raped by

>all these people who don't understand complex numbers

This board really is just a bunch of shitposting freshmen

My mind was definitely boggled but not in a way that flatters the other poster.

Your conventions are bullshit. Conventions only exist inside books where the author says in the introduction that they will refer to something in another way for simplicity.

After you close that book, POOF, that convention does not exist. REST IN PEACE CONVENTION 2016 - 2016

Plus, defining C in any other way is reaaaaaaaaaaaaally dumb. 'C is R plus i' says literally nothing. 'C is the rotation of R' also says nothing in terms of ALGEBRA.

I'd like to see someone prove C is a field from defining it as the rotations of R lol. What a bunch of impractical bullshit.

R^2 = C is the only thing that makes sense.

Plus, you talk about vector spaces you again show your lack of abstraction. Sure, you can make vector spaces with R^k because R is a field. C is also a field so C^k can also be made into a vector space.

Plus, the reason why it is foolish to think of R^k is as THE space is because many other sets can be spaces.

Take the set of points (x,y) such that y = ax + b

If you use R here you get a plane, but also with Q. Q^k is also an euclidean space and given that mathematicians nowadays (like Wildberger) find it useful, it is just as good as R^k, even for topology.

By the way heres the nail in your ridiculous coffin
"Mathematicians denote the n-dimensional Euclidean space by En if they wish to emphasize its Euclidean nature, but Rn is used as well since the latter is assumed to have the standard Euclidean structure, and these two structures are not always distinguished. Euclidean spaces have finite dimension.[2]"

You can now delete your reply if you were typing one because it is thoroughly not important, and your original posts nitpicking shit out of my responses to vomit nonsense everywhere were not relevant to the thread to begin with.

That quote says nothing man.

R^n is euclidean but so is Q^n

So then what on earth is the point of this post: Go ahead and try to explain to me what it had to do with anything

Real numbers are also weird in so many ways you would not consider in everyday life. But they work to build cool stuff with.

The point is conventions are stupid and do not make sense outside of their respective books.

And that C = R^2

I'm glad you think conventions are stupid. You are, by definition, in a minority.

>Approximate me the length of i in real life
1

Now explain to me ho imaginary mass or imaginary time are possible in physics.

>Real numbers are also weird in so many ways you would not consider in everyday life. But they work to build cool stuff with.
Let [math]f[/math] a function [math]\mathbb{C}\mapsto \mathbb{C}[/math].
Then [math]f=\Re f+i\Im f=(f_{1}(x,y),f_{2}(x,y))[/math].
So what you have is a couple of real functions.

Wrong post.
In engineering and physics you don't need anything past a few decimals places.
Maybe ten digits in very special cases.
[math]\sqrt{2}\approx 1.41[/math] deal with it.

>>all these people who don't understand complex numbers
>This board really is just a bunch of shitposting freshmen
Every complex expression is equivalent to a couple of real expressions.
Explain me what's inherently useful about complex numbers.
Being algebraically closed does not respond to any practical or mathematical usefulness.
We decided to make things up. Not cool.

>We decided to make things up. Not cool.
Aw gee, better get rid of all mathematics then

>What exactly do you want, a name change?
STOP using complex numbers altogether.
Unfortunately complex numbers were well established in the mathematical community.
The consequence is that today their are everywhere and present real expressions in compact form.
We need to polish our math from the counter-intuitive complex number madness.

Well now how tf do I describe waves, quantum phases, complex refractive indices, solutions equations like (x^2+2=0) and all the other stuff I use them for?

Real numbers can be defined as equivalence classes of Cauchy sequences.
I'm okay with that because intuitively [math]0.999...[/math] and [math]1.000...[/math] as decimal expansions for [math]1_{\mathbb{R}}[/math] go to the same place.
Topology, measure, ... all of them make sense because they are defined on abstract sets.
Why do they force us to consider complex *things* (in [math]a+ib[/math] mind you that [math]+[/math] in not even a real sum) as number praising their usefulness both in abstract and practical terms?
MUH Schroedinger equation has the imaginary unit.
Then you get imaginary time and imaginary mass.

You surely can express waves and complex numbers using reals:
Then:
>solutions equations like (x^2+2=0) and all the other stuff I use them for.
Why on earth would you need to solve x^2+2=0?

Imaginary numbers can be defined as solutions to equations like x^2 = -1. Nothing in the real numbers can solve it so it only makes sense to invent a new object connected to the reals in such a way that it can solve these equations. Fortunately this produces a consistent system without any ambiguities so everything's fine.

>MUH Schroedinger equation has the imaginary unit.
>Then you get imaginary time and imaginary mass.
Not in the solutions, which are the physical things.

You can but why would you when complex numbers do it better?

OP why don't you like the number i?

Complex numbers were invented as a tool to calculate real solutions of cubic equations.

If you have a cubic equation

[math] a z^3 + b z^2 + c z + d = 0 [/math]
then you can always substitute
[math] x = z + \frac{b}{3a} [/math]
and divide by [math] a [/math] to get a equation of the form
[math] x^3 + p x + q = 0 [/math]
with [math] p = \frac{3 a c - b^2}{3 a^2} [/math] and [math] q = \frac{27 a^2 d - 9 a b c + 2 b^3}{27 a^3} [/math].

Now it's easy to see that one of the solution of the cubic is
[math] x = \sqrt[3]{ -\frac{q}{2} + \sqrt{\frac{q^2}{4} + \frac{p^3}{27}}} + \sqrt[3]{ -\frac{q}{2} - \sqrt{\frac{q^2}{4} + \frac{p^3}{27}}} [/math].

But for some [math] p, q [/math] the terms inside the square roots are negative even tough a cubic equation always has at least one real solution.
Now if you use complex numbers you actually do get those real solutions since the imaginary parts will cancel out at the end.

Complex numbers form a mathematical structure that is extremely common in nature and useful to examine further. It's by far not the only unintuitive thing like that and by far not the weirdest one. The fact that you don't really get it doesn't change anything about it being common and useful. So, I don't know, you can of course continue being stuck in your backwards conspiracy fantasy thinking where everybody else but you is wrong, or you can grow the fuck up and move on. Seriously, you are literally wasting time and thoughts on this that you can spend on learning new things.

Jesus Christ you people are stupid.

>a mathematical structure that is extremely common in nature
>The fact that you don't really get it
Since (You) get it explain how the algebraic properties of [math]\mathbb{C}[/math] are so common and useful.

Than you.
I swear to god I've never seen a proper justification for complex numbers in like 10 analysis books.
Some straight up define them starting from [math]\mathbb{R}^{2}[/math].
Hopefully there are some references from a historical point of view in the wikipedia page about them.
I won't have to read the works of Cardano.
en.wikipedia.org/wiki/Complex_number#Historical_references

The trick is, you've conned yourself into thinking [math]\mathbb{R}[/math] is somehow """real""" because real is in the name. Real numbers, as you'll see if you've seen any real analysis, are completely unrealistic, magical bullshit that we impose onto the world to make sense of it. Same with rational numbers.

Once you accept that, math will be easier to understand. It's all exactly the same amount of "real". The amount of oranges you have is exactly as unreal as the complex numbers in the Schrodinger equation.

(1,0) does not have an inverse element.

>Complex numbers were invented as a tool to calculate real solutions of cubic equations.
It is a common misconception that Cardano introduced complex numbers in solving cubic equations.
en.wikipedia.org/wiki/Ars_Magna_(Gerolamo_Cardano)

But Rafael Bombelli introduced complex numbers in exactly this context.
en.wikipedia.org/wiki/Rafael_Bombelli