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I don't remember where but I have read that a number system can have a matrix as base. Is it true? What is the intuition behind having such a system?

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I just did some googling: compalg.elte.hu/projects/binsys –  ziyuang Jun 3 '11 at 17:34
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What would you even mean by "number system" in this context? –  Qiaochu Yuan Jun 3 '11 at 17:35
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As @Qiaochu I'm not quite sure what you mean. Do matrix representations of an algebra, e.g. the complex numbers via skew-symmetric real $2\times 2$-matrices or the quaternions via the Pauli matrices qualify as examples? –  t.b. Jun 3 '11 at 17:44

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up vote 13 down vote accepted

I don't know where you read that a number system can have matrices as a base. I did come across what follows, but I don't think the claim is that "any" ("every") number system can have matrices as its base, nor that if a number system exists with all matrices serving as its base; rather, if such a base exists, the criteria for determining the matrices that comprise it is quite exclusive.

You might want to have a look at the "BinSys Project: Generalized binary number systems"; see BinSys Project for more detailed description, and additional links.

Here's an excerpt (below), confirming that, indeed, work is underway to develop a "number system with a matrix as a base, in which [certain] vectors are the digits...". I haven't read through all the details regarding the BinSystem Project, but the intro below may help answer your question.


Introduction

"Let n be an integer greater than one. When we speak of number systems in the original sense, we use the fact that each natural number z can be written uniquely in the finite form

$$ z = \sum_j d_jn^j, \text{ where}\;\; d_j = 0, 1, \dots, n-1$$

"We say that n is the base of the number system, the $d_j$ are called the digits. If $n = 2$ then we speak of a binary number system. These systems are too poor to represent negative numbers so we need a sign. If we allow the base to be a negative integer, a representation of all integers may become possible. Such, for example if we use the base -2, each integer has a form $$z = \sum_j d-j(-2)^j, \text {where}\;\; d_j = 0 \text{ or } 1$$

"This can be generalized for the algebraic integers of a finite extension of the rational number field. A simple example: all the Gaussian integers (complex numbers of the form $x+yi$, where $x,y$ are integers) can be written uniquely in the base $(-1+i)$ as follows: $$z = \sum_j d_j(-1 + i)^j, \text{ where}\;\; d_j = 0 \text{ or } 1$$

"Using linear algebra we can define number systems in an even more general way. The base is now a matrix and the digits are vectors. We can reformulate the previous example. Each two-dimensional integer vector has a representation as a finite sum: $$v = \sum_jM^jd_j,$$ where $$M = \left(\matrix{-1, -1 \\1, -1}\right) \text{ and } d_j =\left( \matrix{0 \\ 0}\right)\text{ or } \left( \matrix{1 \\ 0}\right)$$ "...We speak of a binary system if the determinant of "M" is ±2. In this case there are only two digits, one of them being the origin. This means that if we have a number system then every integer vector can be represented as a finite series of 0s and 1s."

"Not every matrix M can be a number system base. Until now no characterisation of ”good” matrices have been given. There are sufficient conditions and there are necessary ones but the gap between them is too large. There is no known efficient method of dealing with matrices that fulfill necessary conditions but fail sufficient conditions. One thing to note is that if we fix the determinant and the dimension then roughly speaking there are only a finite number of possible matrices.

Expected results

"The program aims at finding many generalized binary number systems. An extensive search is performed in the finite set of matrices of given size fulfilling some necessary conditions. The difficulty is that the size of this finite set is an exponential function of the dimension. It now seems possible to attack the case of 11 X 11 matrices. To check further necessary conditions the program performs a lot of floating-point calculation. Thus, a lot of CPU time is needed. Luckily, parallelization is possible and we can benefit of running on several machines.

"The program outputs a list of matrices (being more precise characteristic polynomials) that are already likely to be number system bases. This list is processed by another program (which does not need so much CPU). The final result is then a (complete) list of binary number systems in a fixed dimension.

"Thereafter we perform information theoretical analysis. The number systems provide a binary representation of integer vectors. Using coordinates we have another (more standard) representation. The two representations usually differ in length. Besides, vectors close to each other in the space can have binary representations that look very different. These observations suggest that one could apply number systems in data compression, coding or cryptography.

"Number systems are interesting from a geometrical point of view, too. If we allow negative powers of M to appear in the binary representation we get a possibly infinite representation of real vectors (we could say that we use a radix point). The boundary of the set of vectors with binary representation containing only negative powers of M (the set H of numbers with zero integer part) has mostly fractional dimension (it is a „fractal”). The output of the program can be used to analyze these sets. This means topological analysis, e.g. calculation of the dimension, connectedness etc. If we use the matrix M above, we get the following set:

image

"Finally, knowing all matrices up to a given dimension could help us to a deeper understanding of the mathematics of generalized number systems."


Project Name: Generalized binary number systems

Owner: Eötvös Loránd University, Faculty of Informatics, Department of Computer Algebra

Leader: Attila Kovács, PhD.

Partner: SZTAKI Desktop GRID
Laboratory of Parallel and Distributed systems

Members: Péter Burcsi, Norbert Podhorszki PhD, Gábor Vida, Ádám Kornafeld

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Wish I could give $100(\sin^2 x + \cos^2 x)$ thumbs up! –  Amzoti May 17 '13 at 1:36
    
Go for more than 10 –  Babak S. Jul 10 '13 at 5:38

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