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For other uses, see Radix (disambiguation).

In mathematical numeral systems, the radix or base is the number of unique digits, including zero, used to represent numbers in a positional numeral system. 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 positional numeral system (except unary, where the radix is 1), the number x and its base y are conventionally written as (x)_y, although for base ten the subscript is usually assumed and not written, as it is the most common way to express value. For example, (100)_{10} (in the decimal system) represents the number one hundred, whilst (100)_2 (in the binary system with base 2) represents the number four.[1]


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

In numeral systems[edit]

In the system with radix 13, for example, a string of digits such as 398 denotes the decimal number 3 \times 13^2 + 9 \times 13^1 + 8 \times 13^0.

More generally, in a system with radix b (b > 1), a string of digits d_1 \ldots d_n denotes the decimal number d_1 b^{n-1} + d_2 b^{n-2} + \cdots +  d_n b^0, where  0\leq d_i < b .[1]

Commonly used numeral systems include:

Base/Radix Name Description
10 decimal system the most used system of numbers in the world, is used in arithmetic. Its ten digits are "0–9". Used in most mechanical counters.
12 duodecimal (dozenal) system is often used due to divisibility by 2, 3, 4 and 6. It was traditionally used as part of quantities expressed in dozens and grosses.
2 binary numeral system used internally by nearly all computers, is base two. The two digits are "0" and "1", expressed from switches displaying OFF and ON respectively. Used in most electric counters.
16 hexadecimal system is often used in computing. The sixteen digits are "0–9" followed by "A–F".
8 octal system is occasionally used in computing. The eight digits are "0–7".
60 sexagesimal system originated in ancient Sumeria and passed to the Babylonians.[2] Used nowadays as the basis of our modern circular coordinate system (degrees, minutes, and seconds) and time measuring (hours, minutes, and seconds).
64 MIME Base64 is also used in computing, using as digits "A–Z", "a–z", "0–9", plus two more characters, often "+" and "/".[3][4]
85 PostScript Ascii85 used in computing to encode sequences of bits as base 85 numbers
256 byte is used internally by computers, actually grouping eight binary digits together. For reading by humans, a byte is usually shown as a pair of hexadecimal digits.[5]

For an advanced list, see List of numeral systems.

The octal, hexadecimal and base-64 systems are often used in computing because of their ease as shorthand for binary. For example, every hexadecimal digit has an equivalent 4 digit binary number.

Radices are usually natural numbers. However, other positional systems are possible, e.g. golden ratio base (whose radix is a non-integer algebraic number),[6] and negative base (whose radix is negative).[7]

See also[edit]

External links[edit]


  1. ^ a b M. Morris Mano and Charles Kime (2014). Logic and computer design fundamentals. (4th ed. ed.). Harlow: Pearson. pp. 13–14. ISBN 978-1-292-02468-4. 
  2. ^ Bertman, Stephen (2005). Handbook to life in ancient Mesopotamia (Paperback edition ed.). Oxford [u.a.]: Oxford Univ. Press. p. 257. ISBN 978-019-518364-1. 
  3. ^ The Base16,Base32,and Base64 Data Encodings. IETF. October 2006. RFC 4648. https://tools.ietf.org/html/rfc4648. Retrieved 2014-10-09. 
  4. ^ Multipurpose Internet Mail Extensions: (MIME) Part One: Format of Internet Message Bodies. IETF. November 1996. RFC 2045. https://tools.ietf.org/html/rfc2045. Retrieved 2014-10-09. 
  5. ^ Kingsley-Hughes, Adrian; Kingsley-Hughes, Kathie (2005), Beginning Programming, John Wiley & Sons, p. 56, ISBN 9780764597480 .
  6. ^ Bergman, George (1957). "A Number System with an Irrational Base". Mathematics Magazine 31 (2): 98–110. doi:10.2307/3029218. JSTOR 3029218. 
  7. ^ William J. Gilbert (September 1979). "Negative Based Number Systems" (PDF). Mathematics Magazine 52 (4): 240–244. Retrieved 7 February 2015.