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Base (mathematics)

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	Topics Base (mathematics) Topic Home In arithmeticArithmetic Overview Arithmetic or arithmetics is the oldest and simplest branch of mathematics, used by almost everyone, for tasks rangin... , 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 exponentiationExponentiation Overview Exponentiation is a mathematical operation, written a'n, involving two numbers, the base a and the ... 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 positionalPositional notation Positional notation or place-value notation is a numeral system in which each position is related to the next by a con... numeral systems for denoting numbers. For a given integer b the positional numeral system is called base bRadix In mathematical numeral systems, the base or radix is usually the number of various unique digits, including zero, th... and b is also known as the radix. In general, b and n can be arbitrary realReal number In mathematics, the set of real numbers, denoted R, is the set of all rational numbers and irrational numbers.... or complexComplex number In mathematics, a complex number is a number of the form ... numbers. The inverse functionInverse function In mathematics, an inverse function is in simple terms a function which "does the reverse" of a given function.... to exponentiation with base b (when it is well-definedWell-defined In mathematics, the term well-defined is used to specify that a certain concept is defined in a mathematical or logical way ... ) is called the logarithmLogarithm The logarithm is the mathematical operation that is the inverse of exponentiation .... 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 32Base 32 Base 32 is a positional notation using a base of 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 systemBinary numeral system The binary numeral system represents numeric values using two symbols, typically 0 and 1.... , widely used in computingComputing Originally, the word *computing* was synonymous with counting and calculating, and a science and technology that deals wit... , is base two. The two digits are "0" and "1". The octal systemOctal The octal numeral system is the base-8 number system, and uses the digits 0 to 7.... , which is base8, is also often used in computing. The eight digits are "0-7". Also in widespread use in the computing world is the hexadecimalHexadecimal In mathematics and computer science, base-, hexadecimal, or simply hex, is a numeral system with a radix or base... system. It is base16, and the 16 digits are "0-9" followed by "A-F". In the most common implementations of the Base64Base64 Base 64 is a Positional notation using a base of 64.... 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 Base64Base64 Base 64 is a Positional notation using a base of 64.... for details.