Radix
In a positional numeral system, the radix or base is the number of unique digits, including the digit zero, used to represent numbers. For example, for the decimal/denary system the radix is ten, because it uses the ten digits from 0 through 9.
In any standard positional numeral system, a number is conventionally written as with x as the string of digits and y as its base, although for base ten the subscript is usually assumed, as it is the most common way to express value. For example, 10 is equivalent to 100 and represents the number one hundred, while 2 represents the number four.
Etymology
is a Latin word for "root". Root can be considered a synonym for base, in the arithmetical sense.In numeral systems
In the system with radix 13, for example, a string of digits such as 398 denotes the number = 632.More generally, in a system with radix b, a string of digits denotes the number, where. In contrast to decimal, or radix 10, which has a ones' place, tens' place, hundreds' place, and so on, radix b would have a ones' place, then a b1s' place, a b2s' place, etc.
Commonly used numeral systems include:
| Base/radix | Name | Description |
| 2 | Binary numeral system | Used internally by nearly all computers, is base 2. The two digits are "0" and "1", expressed from switches displaying OFF and ON respectively. Used in most electric counters. |
| 8 | Octal system | Used occasionally in computing. The eight digits are "0"–"7" and represent 3 bits. |
| 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 system | Sometimes advocated due to divisibility by 2, 3, 4, and 6. It was traditionally used as part of quantities expressed in dozens and grosses. |
| 16 | Hexadecimal system | Often used in computing as a more compact representation of binary. The sixteen digits are "0"–"9" followed by "A"–"F" or "a"–"f". |
| 20 | Vigesimal system | Traditional numeral system in several cultures, still used by some for counting. |
| 60 | Sexagesimal system | Originated in ancient Sumer and passed to the Babylonians. Used today as the basis of modern circular coordinate system and time measuring by analogy to the rotation of the Earth. |
The octal and hexadecimal systems are often used in computing because of their ease as shorthand for binary. Every hexadecimal digit corresponds to a sequence of four binary digits, since sixteen is the fourth power of two; for example, hexadecimal 7816 is binary 2. Similarly, every octal digit corresponds to a unique sequence of three binary digits, since eight is the cube of two.
This representation is unique. Let b be a positive integer greater than 1. Then every positive integer a can be expressed uniquely in the form
where m is a nonnegative integer and the r's are integers such that
Radices are usually natural numbers. However, other positional systems are possible, for example, golden ratio base, and negative base.
A negative base allows the representation of negative numbers without the use of a minus sign. For example, let b = −10. Then a string of digits such as 19 denotes the number = −1.