In a

positional numeral system Positional notation, also known as place-value notation, positional numeral system, or simply place value, usually denotes the extension to any base of the Hindu–Arabic numeral system (or decimal system). More generally, a positional 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 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 anim ...

of digits and ''y'' as its base. For base ten, the subscript is usually assumed and omitted (together with the enclosing

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. They come in four main pairs of shapes, as given in the box to the right, which also gives their n ...

), 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'' (), 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. 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 = 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 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 number In mathematics, the natural numbers are the numbers 0, 1, 2, 3, and so on, possibly excluding 0. Some start counting with 0, defining the natural numbers as the non-negative integers , while others start with 1, defining them as the positive in ...

s. 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 \frac ≈ 1.61803399 symbolized by the Greek letter φ) as its base. It is sometimes referred to as base-φ, golden mean b ...

(whose radix is a non-integer

algebraic number In mathematics, an algebraic number is a number that is a root of a function, root of a non-zero polynomial in one variable with integer (or, equivalently, Rational number, rational) coefficients. For example, the golden ratio (1 + \sqrt)/2 is ...

), 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.

Table of bases

Different bases are especially used in connection with computers. The commonly used bases are 10 (

decimal The decimal numeral system (also called the base-ten positional numeral system and denary or decanary) is the standard system for denoting integer and non-integer numbers. It is the extension to non-integer numbers (''decimal fractions'') of th ...

), 2 ( binary), 8 (

octal Octal (base 8) is a numeral system with eight as the base. In the decimal system, each place is a power of ten. For example: : \mathbf_ = \mathbf \times 10^1 + \mathbf \times 10^0 In the octal system, each place is a power of eight. For ex ...

), and 16 (

hexadecimal Hexadecimal (also known as base-16 or simply hex) is a Numeral system#Positional systems in detail, positional numeral system that represents numbers using a radix (base) of sixteen. Unlike the decimal system representing numbers using ten symbo ...

). A

byte The byte is a unit of digital information that most commonly consists of eight bits. Historically, the byte was the number of bits used to encode a single character of text in a computer and for this reason it is the smallest addressable un ...

with 8 bits can represent values from 0 to 255, often expressed with

leading zero A leading zero is any 0 digit that comes before the first nonzero digit in a number string in positional notation.. For example, James Bond's famous identifier, 007, has two leading zeros. Any zeros appearing to the left of the first non-zero dig ...

s in base 2, 8 or 16 to give the same length. The first row in the tables is the base written in decimal.

Notes

References

External links