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 
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, ...).
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).
| base | number system |
| 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 |
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 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 
," 
.
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 number 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.
