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Comments:
<0> Cerveza : imho the 2 sqrt are not interpreted in the same way
<1> There is no mathematical way to distinguish between i and -i; you have to include both of them, or else you inevitably find yourself in difficulties.
<2> matekuten: first find the answer for 1-digit, 2-digit, etc
<3> 1 digit = 3
<3> 1 digit = 2
<0> yrlnry : but surely you can distinguish them (in fact you just did)
<1> No, I did not.
<2> ok, now do 2 digits
<1> Calling them "i" and "-i" just means that one is the negative of the other.
<3> that's some wicked arithmetic dude
<2> yrlnry: Im(i)=1, Im(-i)=-1
<1> That is a completely symmetric statement.
<0> i and -i look quite different to me
<3> i'm guessing 4
<2> dont guess
<3> why not?
<3> you gotta guess a root sometimes
<1> llll: So? That is begging the question, because the definition of Im() is arbitrary.
<2> becasue guessing wont help you find your answer
<1> Any property of Im() is also shared by -Im().
<2> yrlnry: so you can distinguish them. i know what you are trying to say, but you didnt say it quite right :)
<0> Cerveza : i think the confusion arises from the difference between a complex (square) root and the the real valued root function
<0> Cerveza : unfortunately they use both the same symbol
<4> I see what you mean
<2> eg if we define the complex numbers as pairs of reals, then i=(0,1) and -i=(0,-1), they are quite different
<0> sqrt(i^2)=+/- i is correct as here sqrt stand for the multivalued complex square root
<0> which is different from sqrt function
<1> Well, you could also define the complex numbers as kinds of carrots, and i would be an orange carrot and -i would be a purple carrot, and then they would be quite different too, as one would be orange and the other purple.
<4> right, which is essentially the positive root
<0> _llll_ : exactly they are 2 different, distinguishable numbers
<0> Cerveza : yes
<2> what is true is that you could write i for either of the roots of x^2+1, and define the complex numbers
<2> once you choose a name for one root, you can of course distinguish the other, since it's -1*i
<3> I think it's 2^8
<2> but if someone else chose "the otehr root" it wouldnt matter
<5> chopin, i'm done the proof, and it's a mean one
<1> yes, I understand that what is really meant is that there is a field automorphism on C mapping i to -i and vice versa.
<0> and sqrt(-1) stems from the original root function imho and was then defined to be i
<2> i preer to define C as R[x]/(x^2+1)
<4> matekuten, it is, and there is an abovious reason wy if you work in base 2
<1> That I did not say it that way initially does not mean that what i said was untrue.
<2> then "i" is the image of x under the obvious quotient map, and -i is the image of -x
<5> worth doing for a comp sci student though
<2> you said "there is no way mathematically to distinguish them", which didnt make much sense to me, since the Im function does distinguish them
<1> llll: What you've just said is equivalent to saying that they are unequal.
<2> no!
<1> Yes.
<3> how about the number of 8-digit binary strings that contains exactly 3 zeros?
<2> 1 and 2 are not equal, but Im does not distinguish them
<1> Two numbers are equal if and only if some map distinguishes them.
<0> yrlnry : they are unequal as numbers
<1> So you pull out some map, say look, this map distinguishes them, you are saying no more and no less than that they are unequal.
<2> yes, so what?
<2> that was my point, that they are distinguishable..
<1> So your thing about Im() is totally irrelevant. If what you want to say is that they are numerically unequal, you should say that. The thing about the map is just an obscuring pedantry.
<0> matekuten : argh i did that and forgot it
<3> shame on you
<2> what i want to say is that saying "these things are mathematically indistinguishable" is a bit rude to mathematics
<1> If you want to interpret it as "rude to mathematics", I will not dispute you, since I have no idea what that could possibly mean.
<1> I hope the mighty Goddess of Mathematics doesn't take offense at my bad manners.
<2> it means that mathematics can distinguish them, so implying it cannot is, well
<3> these guy means buisness, he won't even type can't
<5> what makes you so sure (s)he's a Goddess, and not a God?
<5> aside from its being so fickle
<3> this
<1> I picture you in your group theory cl*** arguing that Z2 x Z2 and D2 are different groups, because one contains an element named (0,0) and the other does not.
<3> why would a diety have a gender?
<2> but i and -i are in *the same group*
<5> good question
<1> And then inventing the (0,0)-counting function, which returns 1 for the former and 0 for the latter, and declaring that that proves that they are thereby distinguishable.
<1> llll: Yes, but as numbers, they have *identical properties*.
<2> no they do not ahve identical properties, since one has imaginary part greater than zero, and the otehr does not
<1> Z2xZ2 has an element (0,0), and D2 does not!
<2> so whhat? the groups are isomorphic
<1> and i and -i are isomorphic elements under the unique field automorphism of C.
<0> matekuten : actually i think the answer is just C(8,3), i had different but related problem in mind (that was about consecutive zeros, which makes it more complicated)
<2> what is an isomoprhic element?
<3> C is the complement?
<0> yrlnry : why does that matter
<2> i and =i are not isomoprhic if C is regarded as a poset
<1> C is the complex numbers.
<3> ok
<3> that doesnt make sense
<2> i dont know of any way to regard C as a category in which isomorphism would differ from equality
<0> yrlnry : you can distinguish just as members of a set (completely independent from which other views they may appear equal)
<2> note that a set is a category in which all maps are identities, and hence isomorphism is just equality
<0> yrlnry : all you need is _1_ context that allows you to distinguish them
<1> kmh: Yes, so? I did not say they were *identical*.
<3> C stands for combinations
<2> D2 and Z2xZ2 are objects in the categories of groups, in which isomoprhism means something more useful than equality
<0> yrlnry : not identical => they _can_ be distinguished
<0> yrlnry : that's the "so"
<1> I did not say they could not be distinguished. As you yourself pointed out, one is written i, and the other as -i.
<2> you did say that, hence this whole pointless discussion tho
<1> The only puzzle is, which one is written with which notation?
<0> <1> There is no mathematical way to distinguish between i and -i; <--- this is only part that caused the objections (that kind of formulation)
<6> <-'s karma lowered to -2.
<7> hehe
<2> yes, that is what i was objecting to
<1> The qualifier "mathematical" is important in that sentence. I did not say that they were indistringuishable.
<2> ok now you are being silly
<4> you guys are still arguing about this/
<5> <-++
<2> Im is part of my mathematics
<6> <-'s karma raised to -1.
<5> let's not be harsh
<1> Cerveza: I am somewhat amazed myself.
<4> I left to go have lunch and I come back and nothing changed
<2> i didnt think anyone was being harsh
<0> let's just move on to new topivs and greener pastures
<5> taking away karma points from <- for no reason
<2> <-++
<6> <-'s karma raised to 0.
<4> who is <-?
<7> <--
<6> <'s karma lowered to -1.
<0> lol
<7> <++
<6> <'s karma raised to 0.
<4> ok, stop that
<4> its annoying
<7> (sorry, just wanted to check)
<4> and I have a real question
<0> the unknown karma conjecture :-P
<8> well it is necessary to make an arbitrary choice in distinguishing i and -i
<4> I have a distribution I got from an pseudo rng, and I think its a gamma distribution, but how do I prove it?
<8> you could say that this choice is 'unmathematical'
<2> you could say it is un-natural
<2> but choosing an arbitrary element of a set happens all the time
<0> Cerveza : you cannot "prove" it
<8> example _llll_
<2> this is kind of how we got category theory worked out, because E and M wanted to understand how some isomorphisms were "natural" and some were "unnatural"
<0> Cerveza : but you can reason and narrow it downs in terms of probabilities/likelihoods
<4> ok, how do I do that?
<0> Cerveza : the simplest reasoning would just to compare plots
<4> you mean eyeball it
<4> there isn't a more mathamatical way?
<8> i think maybe the point is that this choice is unmathematical in the sense that it has to be made 'outside of mathematics'
<0> of you actual random data and the gamma distribution, if they visually match you've got a good reason to believe it is one
<2> eg a finite abelian group G is isomorphic to its group of characters
<2> but there is not "natural" way to choose an isomorphism
<0> Cerveza : you could try looking for some statistucal tests
<4> how do I test a distribution?
<0> Cerveza : that allow to test a gamma function hypothesis (against being some other distribution)
<0> Cerveza : but i don't know specific test for this particular case
<9> @. elite keal
<6> i 1aCx IN V3R8Al AnD $O(iAl ExpREzsIoN
<0> all the test does to confirm your guess from the plot in terms of error probabilities
<10> ouch
<4> you mean like alpha and beta values?
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