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Thread: Complex base numeral systems?

  1. #1
    Sep 2008

    Complex base numeral systems?

    I am only in precalc but I am doing work with complex forces and I think that using a complex base numeral system would simplify much of my work, and i'm just interested in the topic.

    I am having trouble following what is going on in the paper. My problems start at Definition 1. I dont understand what surjective/pseudoinjective mean, I dont know what the backwards E, the sideways union, the upsidedown A, the epsilon, the backwards epsilon, or for that matter the mu represent. I realize that I am in over my head, but what better way to learn right?

    Also, why are certain boxes shaded in the integer part picture, and what do the Z^x's and stuff represent, im having trouble following.

    If anybody has the time I would appreciate the help, but I understand if nobody wants to take on the difficult task of explaining something this complex to someone with such little foundation to work with.

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  2. #2
    MHF Contributor
    Opalg's Avatar
    Aug 2007
    Leeds, UK
    First comment is that this is an advanced research paper by a professional mathematician (from Poland, where they take their mathematics pretty seriously). You'll be doing very well indeed if you can make sense of all 20 pages of it.

    To answer your specific questions:

    1. surjective (in the context of the paper, which is talking about representations of numbers in some base) means that every number has a representation. (So for example, in the real numbers to the usual base 10, every number has a decimal representation, such as $\displaystyle \pi = 3.14159\ldots$.)

    2. pseudoinjective means that most, but not quite all, numbers have a unique representation. (For example, 0.999... (recurring) is the same as 1.000..., so for the number 1 the representation is not unique.)

    3. $\displaystyle \exists$ means "there exists", $\displaystyle \subset$ means "contained in", $\displaystyle \forall$ means "for all", $\displaystyle x\in S$ means "x is an element of the set S" and $\displaystyle S\ni x$ is another way of saying the same thing. In this paper, $\displaystyle \mu_2(S)$ means the 2-dimensional measure of the set S (measure being a generalisation of the concept of area). So the sentence "$\displaystyle \exists_{S\subset\mathbb{C}}\ \mu_2(S)=0:\forall_{x\in\mathbb{C}\setminus S}\ x$ has at most one representation" means "There is a very small* set S of complex numbers with the property that for all complex numbers x except those in S, x has at most one representation."

    * More precisely "very small" means "having measure zero".

    In the first part of the paper, the shaded boxes represent the complex numbers (with integer real and imaginary parts) that can be represented in the form $\displaystyle a_0+a_1z+a_2z^2+\ldots+a_kz^k$, where z (the base) is a given complex number, and the coefficients $\displaystyle a_0,\ldots, a_k$ are integers between 0 and n-1, for some given n, and for some number k (n=2 and k=5 in the first two figures). I haven't tried to read beyond the first couple of pages, so I don't know what the rest of the paper contains (but it has some nice fractal pictures).
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  3. #3
    Super Member
    Aug 2008
    Hey guys, that concept is pretty interesting although that paper isn't easy for me to read. Apparently he's investiagting "completeness" of number representations in base-z where $\displaystyle z=-1+i$ or $\displaystyle 1+i$ or no? I found another reference that helps explain it:

    Couldn't help check this example: $\displaystyle 2007=1110000000001110011011101_{-1+i}$

    That means $\displaystyle 2007=1(-1+i)^0+0(-1+i)+1(-1+i)^2+1(-1+i)^3+\cdots+1(-1+i)^{24}$

    That's easy to check in Mathematica:

    val = 1110000000001110011011101; 
    alist = IntegerDigits[val]; 
    Simplify[Sum[alist[[25 - n]]*(-1 + I)^n, 
       {n, 0, 24}]]
    How about $\displaystyle 2007i=10000000011001101000111_{-1+i}$

    Which I can check again in Mathematica:

    val2 = 10000000011001101000111; 
    alist2 = IntegerDigits[val2]; 
    len = Length[alist2]; 
    Simplify[Sum[alist2[[len - n]]*(-1 + I)^n, 
       {n, 0, len - 1}]]
    . . . never did any of this in any of the computer science classes I've taken. Pretty cool.
    Last edited by shawsend; Oct 23rd 2008 at 03:57 AM.
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  4. #4
    Sep 2008
    Thanks for the replies, I wasn’t expected such an enthusiastic response .
    Anyways I’m starting to understand what is happening, taking the summation to represent converting each digit to its complex base, but why is it broken down into an integral part and a fractional part?

    I see that set R represents real numbers and set C represents complex numbers, but what does set N represent? Is it the number of digits in the base z number?

    What does it mean when he says we can use the 6 youngest digits for n=2 and z = j+1?

    I also see how on the graph, Z^x, x represents the number of spaces away from zero on the plane. What is the significance of the j and >1 being next to the zero, and what does >1 mean in that case?

    What does the upside down union sign mean?

    That should be enough questions for one day lol. I really appreciate the help btw, I don’t think I could learn this on my own.

    [EDIT] also, shawsend, the article you are talking about is showing how you can apply this math to simplify computer operations using complex numbers, its a really cool application of the complex number shift. I'm reading that article now and it seems pretty interesting. The only difference is that the article you have given is for converting base-2 numbers to base j-1 numbers, whereas the other article is how to convert base...only base 10 i beleive?? into any complex base.
    Last edited by gtozoom; Oct 23rd 2008 at 05:26 AM.
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  5. #5
    Oct 2008
    I know this doesn't address all of your questions, but it might help regarding your questions about symbols:

    Table of mathematical symbols
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  6. #6
    Grand Panjandrum
    Nov 2005
    You might do well to look at reference 1 in the paper in question and see what Don Knuth has to say.

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