Base (Number)

A Real Number $x$ can be represented using any Integer number $b$ 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 $b$, the Digits 0, 1, ..., $b-1$ 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 $b$ representation of a number $x$ be written

$$(a_n\,a_{n-1}\,\ldots\,a_0.\,a_{-1}\,\ldots)_b,$$ (1)

(e.g., ${123.456}_{10}$), then the index of the leading Digit needed to represent the number is

$$n\equiv \left\lfloor{\log_b x}\right\rfloor ,$$ (2)

where $\left\lfloor{x}\right\rfloor$ is the Floor Function. Now, recursively compute the successive Digits

$$a_i=\left\lfloor{r_i\over b^i}\right\rfloor ,$$ (3)

where $r_n\equiv x$ and

$$r_{i-1}=r_i-a_ib^i$$ (4)

for $i=n$, $n-1$, ..., 1, 0, .... This gives the base $b$ representation of $x$. Note that if $x$ is an Integer, then $i$ need only run through 0, and that if $x$ has a fractional part, then the expansion may or may not terminate. For example, the Hexadecimal representation of 0.1 (which terminates in Decimal notation) is the infinite expression $0.19999\ldots_h$.

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 $\pi$ to 1 in ``base $\pi$,'' $\pi=1_\pi$.

See also Binary, Decimal, Hereditary Representation, Hexadecimal, Octal, Quaternary, Sexagesimal, Ternary, Vigesimal

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.