Base

The word "base" in mathematics has several meanings, generally referring to an object chosen as a reference or building block.

In geometry, a base is a designated side of a plane figure or planar boundary region of a solid, often used as the reference from which an altitude is measured. In two dimensions, any side of a triangle may be chosen as its base, while the parallel sides of a trapezoid are its two bases. In three dimensions, a prism or cylinder has two congruent parallel bases, whereas a pyramid or cone has one base. A base need not be the lowest side or boundary region in a drawing.

In arithmetic, "base" has two closely related uses: the base of a numeral system and the base of a logarithm.

A real number can be represented using an integer base with (sometimes also called a radix or scale). The choice of a base yields a representation of numbers known as a numeral 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, ...).

The decimal system uses decimal digits, while binary uses binary digits. Their place values are powers of 10 and 2, respectively.

The digits of a number in base (for integer ) can be obtained in the Wolfram Language using IntegerDigits[x, b].

Let the base representation of a number be written

(1)

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

			(2)
			(3)
			(4)
			(5)
			(6)
			(7)
			(8)
			(9)
			(10)
			(11)

since, for example,

			(12)
			(13)
			(14)

and so on.

Common bases are given special names based on the value of , 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

(15)

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

(16)

where and

(17)

for , , ..., 1, 0, .... This gives the base representation of . Note that if is an integer, then need only run through 0, and that if 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 numeral 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 ," . Even more unexpectedly, the representation of a given integer in an irrational base may be nonunique, for example

			(18)
			(19)

where is the golden ratio.

It is also possible to consider negative bases such as negabinary and negadecimal (e.g., Allouche and Shallit 2003). The digits in a negative base may be obtained with the Wolfram Language code

  NegativeIntegerDigits[0, n_Integer?Negative] := {0}
  NegativeIntegerDigits[i_, n_Integer?Negative] :=
    Rest @ Reverse @ Mod[
      NestWhileList[(# - Mod[#, -n])/n& ,
        i, # != 0& ],
    -n]

The base of a logarithm is a number used to define the numeral system in which the logarithm is computed. In general, the logarithm of a number in base 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

(20)

can be used.