A Real Number can be represented using any Integer number as a base (sometimes also called a Radix or Scale). The choice of a base yields to a representation of numbers known as a Number System. In base , the Digits 0, 1, ..., are used (where, by convention, for bases larger than 10, the symbols A, B, C, ...are generally used as symbols representing the Decimal numbers 10, 11, 12, ...).

Base | Name |

2 | Binary |

3 | Ternary |

4 | Quaternary |

5 | Quinary |

6 | Senary |

7 | Septenary |

8 | Octal |

9 | Nonary |

10 | Decimal |

11 | Undenary |

12 | Duodecimal |

16 | Hexadecimal |

20 | Vigesimal |

60 | Sexagesimal |

Let the base representation of a number be written

(1) |

(2) |

(3) |

(4) |

Some number systems use a mixture of bases for counting. Examples include the Mayan calendar and the old British monetary system (in which ha'pennies, pennies, threepence, sixpence, shillings, half crowns, pounds, and guineas corresponded to units of 1/2, 1, 3, 6, 12, 30, 240, and 252, respectively).

Knuth has considered using Transcendental bases. This leads to some rather unfamiliar results, such as equating to 1 in ``base ,'' .

**References**

Abramowitz, M. and Stegun, C. A. (Eds.).
*Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing.*
New York: Dover, p. 28, 1972.

Bogomolny, A. ``Base Converter.'' http://www.cut-the-knot.com/binary.html.

Lauwerier, H. *Fractals: Endlessly Repeated Geometric Figures.* Princeton, NJ: Princeton University Press,
pp. 6-11, 1991.

Weisstein, E. W. ``Bases.'' Mathematica notebook Bases.m.

© 1996-9

1999-05-26