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Numeral systems by culture 


Positional systems by base 
Decimal (10) 
Nonstandard positional numeral systems 
List of numeral systems 
In mathematical numeral systems, the radix or base is the number of unique digits, including zero, that a positional numeral system uses to represent numbers. For example, for the decimal system (the most common system in use today) the radix is ten, because it uses the ten digits from 0 through 9.
In any numeral system (except unary, where radix is 1), the base will always be written as . For example, (in the decimal system) represents the number ten(in English and most natural languages, decimal is assumed); whilst (in the binary system) represents the number two.^{[1]}
Radix is a Latin word for "root". Root can be considered a synonym for base in the arithmetical sense.
In the system with radix 13, for example, a string of digits such as 398 denotes the decimal number .
More generally, in a system with radix b (b > 1), a string of digits denotes the decimal number , where .
Commonly used numeral systems include:
Base  Name  Description 

10  decimal system  the most used system of numbers in the world, is used in arithmetic. Its ten digits are "0–9". Used in most mechanical counters. 
12  duodecimal (dozenal) system  is often used due to divisibility by 2, 3, 4 and 6. It was traditionally used as part of quantities expressed in dozens and grosses. 
2  binary numeral system  used internally by nearly all computers, is base two. The two digits are "0" and "1", expressed from switches displaying OFF and ON respectively. Used in most electric counters. 
16  hexadecimal system  is often used in computing. The sixteen digits are "0–9" followed by "A–F". 
8  octal system  is occasionally used in computing. The eight digits are "0–7". 
60  sexagesimal system  originated in ancient Sumeria and passed to the Babylonians. It is still used as the basis of our modern circular coordinate system (degrees, minutes, and seconds) and time measuring (hours, minutes, and seconds). 
64  Base 64  is also used in computing, using as digits "A–Z", "a–z", "0–9", plus two more characters, often "+" and "/".^{[citation needed]} 
256  byte  is used internally by computers, actually grouping eight binary digits together. For reading by humans, a byte is usually shown as a pair of hexadecimal digits.^{[citation needed]} 
The octal, hexadecimal and base64 systems are often used in computing because of their ease as shorthand for binary. For example, every hexadecimal digit has an equivalent 4 digit binary number.
Radices are usually natural numbers. However, other positional systems are possible, e.g. golden ratio base (whose radix is a noninteger algebraic number), and negative base (whose radix is negative).
Many devices are built to accept numbers in decimal representation and display results in decimal. Often such devices convert from decimal to some internal radix on input, do all internal operations in that radix, and then convert the results from the internal radix to decimal on output. Such devices could in principle use any radix internally. The people who design such computing devices sometimes wonder what would be the "best" radix to use internally  radix economy.
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