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 ai in any number anan−1 …a0 is an integer in the range 0 to (R– 1) and the number is interpreted as anRn + an−1Rn−1 + … + a1R1 + a0R0 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.
anRn + an−1Rn−1 + … + a1R1 + a0R0
Fractional values can also be represented in a fixed-radix system. Thus, 0·a1a2…an is interpreted as a1R-1 + a2R-2 + … + anR-n
a1R-1 + a2R-2 + … + anR-n
In a mixed-radix (or mixed-base) system, the digit ai in any number anan-1…a0 lies in the range 0 to Ri, where Ri is not the same for every i. The number is then interpreted as (…(anRn- 1) + an-1)Rn-2+…+…+a1)R0 + a0 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
(…(anRn- 1) + an-1)Rn-2+…+…+a1)R0 + a0
((((1×10) + 2)10 + 2)24 + 17)60 + 35)60 + 22 seconds