# Radix

Numeral systems |
---|

Hindu–Arabic numeral system |

East Asian |

Alphabetic |

Former |

Positional systems by base |

Non-standard positional numeral systems |

List of numeral systems |

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 standard positional numeral system, 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, while (100)_{2} (in the binary system with base 2) represents the number four.^{[1]}

## 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 number 3 × 13^{2} + 9 × 13^{1} + 8 × 13^{0}.

More generally, in a system with radix *b* (*b* > 1), a string of digits *d*_{1} … *d _{n}* denotes the number

*d*

_{1}

*b*

^{n−1}+

*d*

_{2}

*b*

^{n−2}+ … +

*d*

_{n}b^{0}, where 0 ≤

*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 sometimes advocated 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" or "a–f". |

8 | octal system | is occasionally used in computing. The eight digits are "0–7". |

20 | vigesimal | traditional numeral system in several cultures, still used by some for counting. |

60 | sexagesimal system | originated in ancient Sumeria and passed to the Babylonians.^{[2]} Used today as the basis of our modern circular coordinate system (degrees, minutes, and seconds) and time measuring (hours, minutes, and seconds). |

*For a larger list, see List of numeral systems.*

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 78_{16} is binary 1111000_{2}. A similar relationship holds between every octal digit and every possible sequence of three binary digits, since eight is the cube of two.

Radices are usually natural numbers. However, other positional systems are possible, e.g. golden ratio base (whose radix is a non-integer algebraic number),^{[3]} and negative base (whose radix is negative).^{[4]}

## See also

## References

- 1 2 Mano, M. Morris; Kime, Charles (2014).
*Logic and Computer Design Fundamentals*(4th ed.). Harlow: Pearson. pp. 13–14. ISBN 978-1-292-02468-4. - ↑ Bertman, Stephen (2005).
*Handbook to Life in Ancient Mesopotamia*(Paperback ed.). Oxford [u.a.]: Oxford Univ. Press. p. 257. ISBN 978-019-518364-1. - ↑ Bergman, George (1957). "A Number System with an Irrational Base".
*Mathematics Magazine*.**31**(2): 98–110. doi:10.2307/3029218. JSTOR 3029218. - ↑ William J. Gilbert (September 1979). "Negative Based Number Systems" (PDF).
*Mathematics Magazine*.**52**(4): 240–244. Retrieved 7 February 2015.

## External links

Look up in Wiktionary, the free dictionary.radix |