The word "base" in mathematics is used to refer to a particular mathematical object that is used as a building block. The most common uses are the related concepts of the number system whose digits are used to represent numbers and the number system in which logarithms are defined. It can also be used to refer to the bottom edge or surface of a geometric figure.

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, ..., 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, ...).

Let the base b representation of a number x be written

(e.g., ). Then, for example, the number 10 is written in various bases as

since, for example,
and so on.

Common bases are given special names based on the value of b, as summarized in the following table. The most common bases are binary and hexadecimal (used by computers) and decimal (used by people).

The index of the leading digit needed to represent the number is

where is the floor function. Now, recursively compute the successive digits

where and

for i = n, , ..., 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 .

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).

Bergman (1957/58) considered an irrational base, and Knuth (1998) considered transcendental bases. This leads to some rather unfamiliar results, such as equating to 1 in "base ," .

The base of a logarithm is a number b used to define the number system in which the logarithm is computed. In general, the logarithm of a number x in base b is written . The symbol is an abbreviation regrettably used both for the common logarithm (by engineers and physicists and indicated on pocket calculators) and for the natural logarithm (by mathematicians). denotes the natural logarithm (as used by engineers and physicists and indicated on pocket calculators), and denotes . In this work, the notations and are used.

To convert between logarithms in different bases, the formula

can be used.

 

Binary, Common Logarithm, Decimal, Decimal Expansion, Digit, Duodecimal, e, Exponentiation, Hereditary Representation, Hexadecimal, Lg, Ln, Logarithm, Napierian Logarithm, Natural Logarithm, Number System, Octal, Quaternary, Radix, Sexagesimal, Ternary, Vigesimal