Overview

In mathematics a base, also called a radix, is the number of distinct digit symbols used in a positional numeral system. The base determines how place values are weighted: each position represents a power of the base. The most familiar example today is the decimal system, which uses ten symbols (0–9) and is widely used in everyday life. Bases are most commonly positive integers greater than one, though mathematical generalizations explore other values and structures.

Positional notation and how bases work

In a positional system the value of a written numeral is the sum of its digits multiplied by powers of the base. If a sequence of digits is d_n d_{n-1} … d_1 d_0 in base b, its value is d_n·b^n + d_{n-1}·b^{n-1} + … + d_1·b + d_0. Each digit must be an integer between 0 and b−1 in the ordinary case. Fractional parts use negative powers (for example, in base 10 the numeral 0.5 equals 5·10^−1). A base is often indicated by writing the base as a subscript, for example 23_8 denotes the octal number equal to 2·8 + 3 = 19 in base 10.

Common bases and examples

Several bases are especially important in different contexts:

  • Binary (base 2): digits 0 and 1; fundamental in digital electronics and computing.
  • Octal (base 8): digits 0–7; historically convenient for grouping binary digits three at a time.
  • Decimal (base 10): digits 0–9; the everyday arithmetic system for most cultures.
  • Hexadecimal (base 16): digits 0–9 and letters A–F for values ten to fifteen; widely used in computer engineering for compact binary representation.

Variations, extensions, and notation

Although an integer base greater than one is the norm, mathematics permits several extensions. Unary or tally systems (sometimes described as base 1) are non-positional and represent numbers by repeated marks. Negative bases (for example, negabinary) and complex bases have been studied, providing representations where sign need not be stored separately. Non-integer and irrational bases are also a subject of theoretical research: such systems can represent numbers but do not share all convenient properties of integer positional systems. When necessary, the base of a numeral is indicated explicitly by a subscript or a context label so that the same digit string can be interpreted in different bases.

History and cultural variation

Numeral systems have arisen independently in many societies, often influenced by counting practices or tools. Decimal likely became widespread because of ten fingers. Other cultures developed different bases: the Babylonians used a sexagesimal (base 60) system for astronomy and time, while the Maya used vigesimal (base 20) elements in their inscriptions. The choice of base affects arithmetic algorithms, notation, and the set of convenient fractions.

Uses, conversion, and important distinctions

Bases matter practically and theoretically. In computer science, base 2, 8, and 16 are routine; programmers convert between binary and hexadecimal when debugging or designing systems. Converting a number from base b to base 10 follows the positional expansion described above; conversion between arbitrary bases is typically done by converting first to an intermediate base (often base 10 or base 2) then to the target base. It is important to distinguish the abstract concept of a base from the particular symbols used: the base counts the available digit values, while the symbols are a human convention for writing them.

For formal discussion and further examples see resources on numeral systems and number representation, where notation like 1011_2 (binary) or FF_16 (hexadecimal) illustrates how the same quantities appear in different bases. A base is usually a whole number greater than one in elementary treatments, but more general definitions are available in advanced texts.