What's the difference between [eqn]\mathbb{C}[/eqn] and [eqn]\mathbb{R}^2[/eqn]?
>>7961856
The multiplication is completely different.
i
>>7961856
bro they,re letters
they are sounded different
r sounds like errr
c sounds like see
>>7961856
[math] \mathbb{C} \cong {\mathbb{R}^2} [/math]
[math] z \mapsto \left( {\operatorname{Re} z,\operatorname{Im} z} \right)[/math]
[math]\left( {x,y} \right) \mapsto x + \sqrt { - 1} y[/math]
http://www.sjsu.edu/faculty/watkins/complex.htm
The elements of C can be mapped one-to-one with RxR
>>7961856
A base for C is (1,0),(0,i) and a base for R2 is (1,0),(0,1)
That's a huge fucking difference.
>>7961882
No, there's no difference at all. You've shown they are completely indistinguishable as vector spaces, which is how you're considering them by talking about bases.
>>7961887
>You've shown they are completely indistinguishable as vector spaces
C is a field. R^2 is a vector space.
>>7961887
Let the standard basis for C be B and the standard basis for R2 be D.
B1 = D1
but
B2 != D2
Case closed, you are retarded. See in another life.
>>7961890
>showing your ignorance
C is also a real vector space, as you pointed out. In fact, any field F containing a field K is a K-vector space. It's an important point.
>>7961893
Let B = ((1,0), (1,1)) and B' = ((1,0), (0,1)) be (ordered) bases for R^2. Note B_1 = B'_1, but B_2 \ne B'_2. Are these different vector spaces?
>>7961902
Fuck you are actually going to make me generalize? Are you this autistic that you cannot get just simple implications?
What fucking ever.
Proposition #OPISAFAGGOT
Pick any basis from the infinite set of basis for R2 and any basis from the infinite set of basis for C. These will never be the same.
Proof: Take your base for R2. Let V be the first of two vectors and let Vn be that nth component of that vector. For the two bases to be the same then the base for R2 must contain the complex number i so that expands the complex plane but for any V, all Vn will be real numbers. i is not a real number.
QED
WHITE SQUARE
BLACK SQUARE
BLACK LIVES MATTER
OP'S LIFE DOESN'T MATTER
>>7961916
>Pick any basis from the infinite set of basis for R2 and any basis from the infinite set of basis for C. These will never be the same.
This doesn't matter though! Any two vector spaces over the same field of the same (finite) dimension are the same vector space, as you would learn in a freshman linear algebra course. They are completely indistinguishable, as in there is nothing in their behavior as vector spaces that allows us to tell C apart from R^2. As >>7961861 points out, the difference is in the *multiplication*. This is where i makes a difference -- it has nothing to do with bases.
>>7961930
>If V and W are isomorphic then V = W
This is the most elaborate troll I've ever seen, even made me prove a theorem. At least I'll use that as my PhD dissertation.
Nothing to see here, sage and report.
>>7961940
You keep claiming that I'm saying things I'm not. Your entire argument for C and R^2 being different is that they are not literally the same set (and somehow, rather than just pointing out that i is not a real number and being done with it, you've decided to go completely retarded and bring linear algebra into this), and this is a piss poor argument.
R^2 is ordered pairs of real numbers. It is just a set.
C is an algebraically closed field. The two are related, however.
>>7961955
>I just said (0,1) != (0,i)
You said this as means to imply that R^2 and C were different real vector spaces, and in a really fucking smug manner.
>>7961856
>six-point support
>much more than needed
B-cup pls
watching people argue about math makes me feel smart
http://math.stackexchange.com/a/364164
Here's a simple stack exchange answer that you fuckers could have googled in two seconds. Jesus Christ, what is wrong with all of you.
welcome to the fucking rabbit hole
http://math.stackexchange.com/a/710130
this is a good post explaining construction agnosticism and why it rarely matters how an object is constructed, only the properties it exhibits
it is definitely a better post than >>7962560
>>7961965
R^2 is a field as much as C is.
They are isomorphic as vector spaces, though not as fields.
Depends on the context. They are identical as normed real vector spaces but we usually reserve C for the field of complex numbers (that is, we view C as endowed with a multiplication)
>>7962737
thanks for the link, good read
>>7962766
>R^2 is a field as much as C is.
There's only one field structure you can put on R^2 and it is that of C.
Multiplying the factors individually isn't a field because it has zero divisors.
looks like the C = R^2 tards got btfo
>>7962737
THE RABBIT HOLE NEVER ENDS
C is R^2 after you define multiplication · elements.
If a,b,x and y are in R with multiplication *,
and if (a,b) and (x,y) are in R^2, that multiplication · looks like
(a,b)·(x,y) = (a*x-b*y, a*y+b*x)
You may write that as
(a+i b)·(x+i y) = a*x-b*y + i (a*y+b*x)
in which case you dropped the reference to R^2-like pairs. But it's just a representation.
There are literally infinite different representations, any morphism of C into a larger ring gives you a subset of that ring that behaves like C
(or "is" C, if you can make sense of subobjects).
>>7962737
>I've learned this set theoretical construction and [math]\subset[/math] interferes with my intuition of structure
Not even once.
>>7961861
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>>7962966
HS and 1st year undergrad fags
>>7961856
This question cannot be answered without you specifying what category (or more simply: what field of mathematics) we are in. The answer depends on what do you regard R^2 and C as.
Are they vector spaces? They're the same
Are they sets? They're the same
Rings? (where R^2 is the product, in the category of rings, of two copies of R) They're completely different.
>>7963629
samefag
