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!=0 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, ...).

The digits of a number x in base b (for integer b>1) can be obtained in the Wolfram Language using IntegerDigits[x, b].

Let the base b representation of a number x be written


(e.g., 123.456_(10)). 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).

basenumber system

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


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


where r_n=x and


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..._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).

Bergman (1957/58) considered an irrational base, and Knuth (1998) considered transcendental bases. This leads to some rather unfamiliar results, such as equating pi to 1 in "base pi," pi=10_pi. Even more unexpectedly, the representation of a given integer in an irrational base may be nonunique, for example


where phi 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& ],

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 log_bx. The symbol logx is an abbreviation regrettably used both for the common logarithm log_(10)x (by engineers and physicists and indicated on pocket calculators) and for the natural logarithm log_ex (by mathematicians). lnx denotes the natural logarithm log_ex (as used by engineers and physicists and indicated on pocket calculators), and lgx denotes log_2x. In this work, the notations logx=log_(10)x and lnx=log_ex are used.

To convert between logarithms in different bases, the formula


can be used.

See also

Binary, Common Logarithm, Decimal, Decimal Expansion, Digit, Duodecimal, e, Exponentiation, Hereditary Representation, Hexadecimal, Lg, Ln, Logarithm, Napierian Logarithm, Natural Logarithm, Negabinary, Negadecimal, Number System, Octal, Phi Number System, Quaternary, Radix, Sexagesimal, Ternary, Vigesimal Explore this topic in the MathWorld classroom

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Allouche, J.-P. and Shallit, J. "Representations in Negative Bases." §3.7 in Automatic Sequences: Theory, Applications, Generalizations. Cambridge, England: Cambridge University Press, pp. 103-105, 2003.Abramowitz, M. and Stegun, I. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 28, 1972.Bergman, G. "A Number System with an Irrational Base." Math. Mag. 31, 98-110, 1957/58.Bogomolny, A. "Base Converter.", D. E. "Positional Number Systems." §4.1 in The Art of Computer Programming, Vol. 2: Seminumerical Algorithms, 3rd ed. Reading, MA: Addison-Wesley, pp. 195-213, 1998.Lauwerier, H. Fractals: Endlessly Repeated Geometric Figures. Princeton, NJ: Princeton University Press, pp. 6-11, 1991.

Referenced on Wolfram|Alpha


Cite this as:

Weisstein, Eric W. "Base." From MathWorld--A Wolfram Web Resource.

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