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In a positional numeral system, the radix or base is the number of unique digits, including the digit zero, used to represent numbers. For example, for the decimal/denary system (the most common system in use today) the radix (base number) is ten, because it uses the ten digits from 0 through 9. In any standard positional numeral system, a number is conventionally written as with ''x'' as the
string String or strings may refer to: * String (structure), a long flexible structure made from threads twisted together, which is used to tie, bind, or hang other objects Arts, entertainment, and media Films * ''Strings'' (1991 film), a Canadian ani ...
of digits and ''y'' as its base, although for base ten the subscript is usually assumed (and omitted, together with the pair of
parentheses A bracket is either of two tall fore- or back-facing punctuation marks commonly used to isolate a segment of text or data from its surroundings. Typically deployed in symmetric pairs, an individual bracket may be identified as a 'left' or 'r ...
), as it is the most common way to express
value Value or values may refer to: Ethics and social * Value (ethics) wherein said concept may be construed as treating actions themselves as abstract objects, associating value to them ** Values (Western philosophy) expands the notion of value bey ...
. 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 A binary system is a system of two astronomical bodies which are close enough that their gravitational attraction causes them to orbit each other around a barycenter ''(also see animated examples)''. More restrictive definitions require that t ...
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

In the system with radix 13, for example, a string of digits such as 398 denotes the (decimal) number = 632. More generally, in a system with radix ''b'' (), a string of digits denotes the number , where . 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 ''b''1s' place, a ''b''2s' place, etc. 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 2. 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 :$a = r_m b^m + r_ b^ + \dotsb + r_1 b + r_0,$ where ''m'' is a nonnegative integer and the ''rs are integers such that :0 < ''r''''m'' < ''b'' and 0 ≤ ''r''''i'' < ''b'' for ''i'' = 0, 1, ... , ''m'' − 1. Radices are usually natural numbers. However, other positional systems are possible, for example,
golden ratio base Golden ratio base is a non-integer positional numeral system that uses the golden ratio (the irrational number  ≈ 1.61803399 symbolized by the Greek letter φ) as its base. It is sometimes referred to as base-φ, golden mean base, ...
algebraic number An algebraic number is a number that is a root of a non-zero polynomial in one variable with integer (or, equivalently, rational) coefficients. For example, the golden ratio, (1 + \sqrt)/2, is an algebraic number, because it is a root of the po ...
), and
negative base A negative base (or negative radix) may be used to construct a non-standard positional numeral system. Like other place-value systems, each position holds multiples of the appropriate power of the system's base; but that base is negative—that i ...
(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.

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Base (exponentiation) In exponentiation, the base is the number b in an expression of the form bn. Related terms The number n is called the exponent and the expression is known formally as exponentiation of b by n or the exponential of n with base b. It is more com ...
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Mixed radix Mixed radix numeral systems are non-standard positional numeral systems in which the numerical base varies from position to position. Such numerical representation applies when a quantity is expressed using a sequence of units that are each a m ...
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Polynomial In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An examp ...
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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). This is one of various proposals that have been ...
* Radix sort * Non-standard positional numeral systems

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