Base (mathematics)

In arithmetic, the base refers to the number "b" in an expression of the form "b"^"n". The number "n" is called the exponent and the expression is known formally as exponentiation of "b" by "n" or the exponential of "n" with base "b". It is more commonly expressed as "the "n"th power of "b", "b" to the "n"th 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 numeral systems" for denoting numbers. For a given integer "b" the positional numeral system is called base "b" and "b" is also known as the radix.

In general, "b" and "n" can be arbitrary real or complex numbers.

The inverse function to exponentiation with base "b" (when it is well-defined) is called the logarithm with base "b", denoted log_"b". Thus log_"b"("b"^"n") = "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 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, widely used in computing, is base two. The two digits are "0" and "1".
* The octal system, which is base 8, is also often used in computing. The eight digits are "0-7".
* The decimal system, 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 system. It is base 16, and the 16 digits are "0-9" followed by "A-F".
* In the most common implementations of the Base64 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 for details.