12- 4 What Is the Most Efficient Base?

Suppose you are building a computer and you are trying to decide what base to use to represent integers. For the registers you have available circuits that are 2-state (binary), 3-state, 4-state, and so on. Which should you use?

Let us assume that the cost of a b-state circuit is proportional to b. Thus, a 3-state circuit costs 50% more than a binary circuit, a 4-state circuit costs twice as much as a binary circuit, and so on.

Suppose you want the registers to be able to hold integers from 0 to some maximum M. Encoding integers from 0 to M in base b requires digits (e.g., to represent all integers from 0 to 999,999 in decimal requires log₁₀(1,000,000) = 6 digits).

One would expect the cost of a register to be equal to the product of the number of digits required times the cost to represent each digit:

where c is the cost of a register and k is a constant of proportionality. For a given M, we wish to find b that minimizes the cost.

The minimum of this function occurs for that value of b that makes dc/db = 0. Thus, we have

This is zero when lnb = 1, or b = e.

This is not a very satisfactory result. Because e 2.718, 2 and 3 must be the most efficient integral bases. Which is more efficient? The ratio of the cost of a base 2 register to the cost of a base 3 register is

Thus, base 2 is more costly than base 3, but only by a small amount.

By the same analysis, base 2 is more costly than base e by a factor of about 1.062.