In this chapter, we discuss how numbers can be represented by strings over a finite alphabet. Our emphasis is on the representation of integers, although we briefly discuss representations for real numbers in Section 3.4.

We start with the classical base-k representation, and then discuss less familiar representations such as representation in base —k, Fibonacci representation, and representation in complex bases.

Numeration Systems

A numeration system is a way of expressing an integer n (or, more generally, an element of a given semiring S) as a finite linear combination of base elements ui. The ai are called the generalized digits, or just digits. The finite string of digits arar-1 … a1a0 is then said to be a representation of the number n. Note that our convention is to write representations starting with the most significant digit, although admittedly this choice is somewhat arbitrary.

For example, in ordinary decimal notation the base elements are the powers of 10. As is certainly familiar to most readers, every non-negative integer can be expressed as a non-negative integer linear combination with 0 ≤ ai < 10.

The leading-zeros problem is a minor annoyance we must deal with. For example, each of the strings 101, 0101, 00101, … is a different way to express the number 5 in base 2.