Section 2.3
The a-ary number system

We are used to representing integers in the decimal system. But there are also other systems.

Definition

Let a > 1 be an integer. An a-ary representation of an integer m 0 is a sequence of numbers b₀, ..., b_k with 0 b_i < a (i=0, ..., k), such that

m = b_ka^k + b_k-1a^k-1 + ··· + b₁a + b₀.

We write

m = (b_k ... b₀)_a.

We speak of the a-ary number system.

 >>> Exercise Gapplet

Every positive number can be written in the a-ary number system in precisely one way.

Theorem

Let a > 1 be an integer. Every integer m 0 has an a-ary representation. Furthermore, this representation is unique if m > 0 and if we require that b_k 0 for the `most significant' (i.e., most left) digit in

m = (b_k ... b₀)_a.

 >>> Proof Example Exercise Gapplet