Radix

In a positional numeral system, the radix (: radices) or base is the number of unique digits, including the digit zero, used to represent numbers. For example, for the decimal system (the most common system in use today) the radix is ten, because it uses the ten digits from 0 through 9.

In any standard positional numeral system, a number is conventionally written as (x)y with x as the string of digits and y as its base, although for base ten the subscript is usually assumed (and omitted, together with the pair of parentheses), as it is the most common way to express value. For example, (100)10 is equivalent to 100 (the decimal system is implied in the latter) and represents the number one hundred, while (100)2 (in the binary system with base 2) represents the number four.

Etymology
Radix is a Latin word for "root". Root can be considered a synonym for base, in the arithmetical sense.

In numeral systems
Generally, in a system with radix b (b > 1), a string of digits d1 ... dn denotes the number d1bn−1 + d2bn−2 + … + dnb0, where 0 ≤ di < b. In contrast to decimal, or radix 10, which has a ones' place, tens' place, hundreds' place, and so on, radix b would have a ones' place, then a b1s' place, a b2s' place, etc.

For example, if b = 12, a string of digits such as 59A (where the letter "A" represents the value of ten) would represent the value 5 × 122 + 9 × 121 + 10 × 120 = 838 in base 10.

Commonly used numeral systems include:

The octal and hexadecimal systems are often used in computing because of their ease as shorthand for binary. Every hexadecimal digit corresponds to a sequence of four binary digits, since sixteen is the fourth power of two; for example, hexadecimal 7816 is binary 111 10002. Similarly, every octal digit corresponds to a unique sequence of three binary digits, since eight is the cube of two.

This representation is unique. Let b be a positive integer greater than 1. Then every positive integer a can be expressed uniquely in the form


 * $$a = r_m b^m + r_{m-1} b^{m-1} + \dotsb + r_1 b + r_0,$$

where m is a nonnegative integer and the r's are integers such that


 * 0 < rm < b and 0 ≤ ri < b for i = 0, 1, ..., m − 1.

Radices are usually natural numbers. However, other positional systems are possible, for example, golden ratio base (whose radix is a non-integer algebraic number), and negative base (whose radix is negative). A negative base allows the representation of negative numbers without the use of a minus sign. For example, let b = −10. Then a string of digits such as 19 denotes the (decimal) number 1 × (−10)1 + 9 × (−10)0 = −1.