# number system

**number system** 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.

Fractional values can also be represented in a fixed-radix system. Thus, ·

*a*

_{1}

*a*

_{2}…

*a*

_{n}is interpreted as

*a*

_{1}

*R*

^{–1}+

*a*

_{2}

*R*

^{–2}+ … +

*a*

_{n}*R*

^{–}

^{n}In a

*mixed-radix*(or

*mixed-base*)

*system*, the digit

*a*in any number

_{i}*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

# number systems

**number systems.** J.-J. Rousseau introduced a system of numerical notation in which the first 8 numerals are substituted for the 8 notes in the scale. Nos. are popular among 20th-cent. composers, because of the concept of ‘parameters’, in which mus. sounds are regarded as the sum of several components (pitch, duration, intensity, timbre, and position in space). What is called the Fibonacci series (each no. the sum of the previous 2) has been used to control these components by such composers as Krenek, Stockhausen, Maxwell Davies, and Nono.

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**number system**