Radix economy

The radix economy of a number in a particular base (or radix) is the number of digits needed to express it in that base, multiplied by the base (the number of possible values each digit could have). Various proposals have been made to quantify the relative costs of using different radices in representing numbers, especially in computer systems. Radix economy also has implications for organizational structure, networking, and other fields.

The radix economy E(b,N) for any particular number N in a given base b is defined as

where we use the floor function ${\displaystyle \lfloor \rfloor }$ and the base-b logarithm ${\displaystyle \log _{b}}$.

If both b and N are positive integers, then the radix economy ${\displaystyle E(b,N)}$ is equal to the number of digits needed to express the number N in base b, multiplied by base b. The radix economy thus measures the cost of storing or processing the number N in base b if the cost of each "digit" is proportional to b. A base with a lower average radix economy is therefore, in some senses, more efficient than a base with a higher average radix economy.

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