Base (mathematics)

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Base (mathematics)

In arithmetic

Arithmetic

Arithmetic or arithmetics is the oldest and most elementary branch of mathematics, used by almost everyone, for tasks ranging from simple day-to-day counting to advanced science and business calculations....
, the base refers to the number b in an expression of the form bⁿ. The number n is called the exponent and the expression is known formally as exponentiation

Exponentiation

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In arithmetic

Arithmetic

Exponentiation

Exponentiation is a mathematics operation , written 'an', involving two numbers, the base a and the exponent n....
of b by n or the exponential of n with base b. It is more commonly expressed as "the nth power of b", "b to the nth power" or "b to the power n". The term power strictly refers to the entire expression, but is sometimes used to refer to the exponent.

When b is an integer bigger than 1, this process is particularly important in positional
Positional notation

A positional notation or place-value notation system is a numeral system in which each position is related to the next by a constant multiplier, Geometric progression, called the radix or radix of that numeral system....
numeral systems for denoting numbers. For a given integer b the positional numeral system is called base b
Radix

In numeral system, the base or radix is usually the number of unique Numerical digit, including zero, that a Positional notation numeral system uses to represent numbers....
and b is also known as the radix.

In general, b and n can be arbitrary real

Real number

In mathematics, the real numbers may be described informally in several different ways. The real numbers include both rational numbers, such as 42 and −23/129, and irrational numbers, such as pi and the square root of two; or, a real number can be given by an infinite decimal representation, such as 2.4871773339...., where the digits co...
or complex

Complex number

In mathematics, the complex numbers are an extension of the real numbers obtained by adjoining an imaginary unit, denoted i, which satisfies:...
numbers.

The inverse function

Inverse function

In mathematics, if ƒ is a function from A to B then an inverse function for ƒ is a function in the opposite direction, from B to A, with the property that a round trip from A to B to A returns each element of the initial set to itself....
to exponentiation with base b (when it is well-defined

Well-defined

In mathematics, the term well-defined is used to specify that a certain concept or object is defined in a mathematical or logical way using a set of base axioms in an entirely unambiguous way and satisfies the properties it is required to satisfy....
) is called the logarithm

Logarithm

In mathematics, the logarithm of a number to a given base is the Power or exponent to which the base must be raised in order to produce the number....
with base b, denoted log_b. Thus log_b(bⁿ) = n.

Bases and positional numeral systems

Also see adjacent table.

In order to discuss bases other than the decimal system (base ten), a distinction needs to be made between a number and the digit representing that number. In the decimal positional numeral system, there are ten possible digits in each position. These are "0", "1", "2", "3", "4", "5", "6", "7", "8" , and "9" (henceforth "0-9"). In other bases, the digits used may be unfamiliar to us or may be used to indicate numbers other than those they represent in the decimal system. For example, in the base 32

Base 32

Base32 is a base-32 transfer encoding using the twenty-six letters A-Z and six digits 2-7....
numeral system, there are 32 possible digits for each position. These combinations are the numbers 0-31, but they could be signified (in ascending order) first by the symbols A-Z and then by the symbols 2-7. So A represents 0, Z represents the number 25, 2 represents the number 26, 3 represents 27, etc. Because of the ubiquitousness of the decimal system, it is common that numbers are written in base ten, and unless otherwise indicated, most numbers encountered are normally assumed to be decimal numbers. However, any real number can be represented with any base. Some commonly used positional numeral systems with bases other than 10 are:

The binary numeral system
Binary numeral system

The binary numeral system, or notation with a radix of 2. Owing to its straightforward implementation in digital electronic circuitry using logic gates, the binary system is used internally by all modern computers....
, widely used in computing
Computing

Computing is usually defined as the activity of using and developing computer technology, computer hardware and computer software. It is the computer-specific part of information technology....
, is base two. The two digits are "0" and "1".
The octal system
Octal

The octal numeral system, or oct for short, is the radix-8 number system, and uses the digits 0 to 7. Numerals can be made from Binary numeral system numerals by grouping consecutive digits into groups of three ....
, which is base 8, is also often used in computing. The eight digits are "0-7".
The decimal system
Decimal system

Decimal system may refer to:* The decimal number system, used in mathematics for writing numbers and performing arithmetic.* The Dewey Decimal Classification, a subject classification system used in libraries....
, the most used system of numbers in the world, is used in arithmetic. Its ten digits are "0-9".
Also in widespread use in the computing world is the hexadecimal
Hexadecimal

In mathematics and computer science, hexadecimal is a numeral system with a radix, or base, of 16. It uses sixteen distinct symbols, most often the symbols 0–9 to represent values zero to nine, and A, B, C, D, E, F to represent values ten to fifteen....
system. It is base 16, and the 16 digits are "0-9" followed by "A-F".
In the most common implementations of the Base64
Base64

The term Base64 refers to a specific MIME#Content-Transfer-Encoding. It is also used as a generic term for any similar encoding scheme that encodes binary data by treating it numerically and translating it into a base 64 representation....
system, the 64 digits are "A-Z", followed by "a-z", followed by "0-9", followed by "+" and "/". A is zero, Z is 25, a is 26, z is 51, 0 is 52, 9 is 61, + is 62 and / is 63; for a total of 64 combinations, including 0. In the case of Base64, things are even more complicated, because Base64 isn't just a base 64 numeral system, but a specific encoding, whereby the base 64 numerical string is translated to an 8 bit character code (and vice versa); see Base64
Base64

The term Base64 refers to a specific MIME#Content-Transfer-Encoding. It is also used as a generic term for any similar encoding scheme that encodes binary data by treating it numerically and translating it into a base 64 representation....
for details.