Radix Notation

Decimal notation for both integers and fractions is an everyday familiarity. It is, however a specific case of a more general notation in which digits are used in a way where their positions carry information as well as their typography. 

In the more general, radix-r notation, a number is denoted by a sequence of digits7 di... d-k, each digit in the range 0... r — 1 to have the meaning 

There are certain useful checks for divisibility of integers, well known in radix 10. For example, divisibility by 9 is checked by recursively checking the divisibility of the sum of the digits, and divisibility by 11 by taking the sum of the digits in even places and subtracting from it the sum of those in odd places, and checking the result for divisibility by 11. 

These are just special cases of relationships which work equally well in this extension of the system where r y 10. 

If r — 1 is composite, then we can test for divisibility by one of its factors by initially multiplying the candidate by the cofactor and then testing for divisibility by r — 1. The remainder when the recursion finishes will either be 

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Because of certain properties of divisibility of r2 — 1, numbers expressed in radix notation can be tested for divisibility rather easily. The relevance here is that the radix might be algebraic, rather than a fixed number, and all of these results still apply. 

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