## Quick Reference

Although early number systems were not positional, all of the number systems most commonly used today are positional systems: the value of a number in such a system is determined not just by the digits in the number but also by the position in the number of each of the digits. If a positional system has a **fixed radix** (or **fixed base**) *R* then each digit *a** _{i}* in any number

*a*

_{n}*a*

*…*

_{n−1}*a*

_{0}is an integer in the range 0 to (

*R*– 1) and the number is interpreted as

*a*

_{n}*R*

*+*

^{n}*a*

_{n−1}*R*

*+ … +*

^{n−1}*a*

_{1}

*R*

^{1}+

*a*

_{0}

*R*

^{0}Since this is a polynomial in

*R*, such numbers are sometimes called

**polynomial numbers**. The decimal and binary systems are both fixed-radix systems, with a radix of 10 and 2, respectively.

*a*_{n}*a** _{n−1}* …

*a*

_{0}

*a*_{n}*R** ^{n}* +

*a*

_{n−1}*R*

*+ … +*

^{n−1}*a*

_{1}

*R*

^{1}+

*a*

_{0}

*R*

^{0}

Fractional values can also be represented in a fixed-radix system. Thus, 0·*a*_{1}*a*_{2}…*a** _{n}* is interpreted as

*a*

_{1}

*R*

^{-1}+

*a*

_{2}

*R*

^{-2}+ … +

*a*

_{n}*R*

^{-n}

0·*a*_{1}*a*_{2}…*a*_{n}

*a*_{1}*R*^{-1} + *a*_{2}*R*^{-2} + … + *a*_{n}*R*^{-n}

In a **mixed-radix** (or **mixed-base**) **system**, the digit *a** _{i}* in any number

*a*

_{n}*a*

*…*

_{n-1}*a*

_{0}lies in the range 0 to

*R*

*, where*

_{i}*R*

*is not the same for every*

_{i}*i*. The number is then interpreted as (…(

*a*

_{n}*R*

*) +*

_{n- 1}*a*

*)*

_{n-1}*R*

*+…+…+*

_{n-2}*a*

_{1})

*R*

_{0}+

*a*

_{0}for example, 122 days 17 hours 35 minutes 22 seconds is equal to ((((1×10) + 2)10 + 2)24 + 17)60 + 35)60 + 22 seconds

*a*_{n}*a** _{n-1}*…

*a*

_{0}

(…(*a*_{n}*R** _{n- 1}*) +

*a*

*)*

_{n-1}*R*

*+…+…+*

_{n-2}*a*

_{1})

*R*

_{0}+

*a*

_{0}

((((1×10) + 2)10 + 2)24 + 17)60 + 35)60 + 22 seconds

**From:**
number system
in
A Dictionary of Computing »

*Subjects:*
Computing.

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