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
We say that n is the base of the number system, the
dj 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
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 uiquely in the base (-1+i) as follows
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
where
and
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 fulfil 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.
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Last edited by Youth on 2006-4-3 at 19:44 ]
