Algebraic Bases

The algebraic base is a mathematical extension of real bases, particularly the factorisations of b^n-a^n. Such a base is described as base b/a. When a=1, such follow the real bases, the equations expanding to a repetions of the omega digit. Mathematicians study the behaviour of primes relative to particular bases. In practice, they do not construct a number system representing the base, but rather consider the factors of b^n-a^n. A prime p is said to have a period of q, means that p divides b^n-a^n  whenever q divides n. For this reason, very little discussion is on how to implement an arithmetic system for large bases, and the readers are supposed to extend the decimal digits sufficiently to cover all spots from 0 to B-1 inclusive. Given this, we see that the term digit is meant to mean such symbols, this greatly impedes the search where the digital calculations happen at levels different to the base arithmetic.

Fractional Bases
The general form for a fractional base is where b>a>1.

Because b/a = c mod p is an integer, the period of each prime is identical to that of c. The values of the algebraic roots can still be found from the formulae, if one supposes that the sum of powers in each term is constant. Thus if the largest term is b^8, the sum of powers on b and a must add to 8, so eg b^4 becomes ''b^4*a^4. ''

Class-2 Bases
In a class-2 base, the values of a^2+b^2 and ab are integral. The more common form is for ab=1, which is the integral case, and when ab>1, the fractional case.

Class-2 bases have an interesting connection with the polygons, in that the modular arithmetic against p can represent the product of chords of a polygon.