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 numerical digit, digits, including the digit zero, used to represent numbers. For example, for the decimal, 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 (computer science), string of digits and ''y'' as its base. For base ten, the subscript is usually assumed and omitted (together with the enclosing parentheses), as it is the most common way to express value (mathematics), 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 (numeral), binary system with base 2) represents the number four.

Etymology

:wikt:radix, ''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 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.

Table of bases

Different bases are especially used in connection with computers. The commonly used bases are 10 (decimal), 2 (Binary number, binary), 8 (octal), and 16 (hexadecimal). A byte with 8 bits can represent values from 0 to 255, often expressed with leading zeros 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

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