In mathematical numeral systems, the radix or base is the number of unique digits, including the digit zero, used to represent numbers in a positional numeral system. For example, for the dozenal system (the system in this wiki) the radix is 10, because it uses the 10 digits from 0 through E. (the 10 digits are 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, X, E)

In any standard positional numeral system, a number is conventionally written as Template:Nowrap with x as the string of digits and y as its base, although for base 10 the subscript is usually assumed (and omitted, together with the pair of parentheses), as it is the most common way to express value. For example, (100)10 = 100 (in the dozenal system) represents the number one hundred, while (100)2 (in the binary system with base 2) represents the number four.

Etymology

Radix 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 14 (tetradozenal), for example, a string of digits such as 398 denotes the (dozenal) number 3×142 + 9×14 + 8 = 648.

More generally, in a system with radix b (Template:Nowrap), a string of digits Template:Nowrap denotes the number Template:Nowrap, where Template:Nowrap. In contrast to dozenal, or radix 10, which has a ones' place, dozens' place, grosses' place, and so on, radix b would have a ones' place, then a b1s' place, a b2s' place, etc.

Commonly used numeral systems include:

2 Binary 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.
3 Ternary Archaic measurement units, very rarely used by computers
4 Quaternary Two binary digits, occasionally used in computing and DNA representation
5 Quinary Tally marks (sometimes)
6 Senary Smallest base in which fractions 1/2, 1/3, and 1/4 all terminate
7 Septenary Weeks timekeeping
8 Octary Used occasionally in computing as a way to shorten binary numbers. The eight digits are "0–7" and represent 3 bits (23).
9 Nonary Two ternary digits, sometimes used to shorten ternary numbers.
X Decary The most commonly used system of numbers in the world. Its ↊ digits are "0–9". Used in most mechanical counters.
E Levary Uses digits 0-X. Rarely used
10 Dozenal The system used in this wiki, sometimes advocated due to divisibility by 2, 3, 4, and 6 (as opposed to decimal's just 2 and 5). It was traditionally used as part of quantities expressed in dozens and grosses.

Used in timekeeping (months)

11 Undozenal
12 Duodozenal Lowest base in which both 1/2 and 1/7 terminate
13 Tridozenal Lowest base in which both 1/3 and 1/5 terminate
14 Tetradozenal Often used in computing as a more compact representation of binary (1 base-14 digit per 4 bits). The 14 digits are "0–Ɛ" followed by "A–D" or "a–d". Colors are commonly coded with a string of six tetradozenal digits e.g. 7X41DƐ
18 Octodozenal Used in mutliple languages
19 Enneadozenal Lowest base in which both 1/3 and 1/7 terminate
1X Decidozenal Lowest base in which both 1/2 and 1/E terminate
1E Levidozenal
20 Vigesimal Timekeeping (a day has 20 hours)
21 Unvigesimal Two quinary digits; can be used as a way to shorten quinary numbers
23 Trivigesimal Three ternary digits (3 cubed = 23)
26 Hexavigesimal Smallest base in which fractions 1/2, 1/3, 1/4, 1/5 and 1/6 all terminate
28 Octovigesimal Five binary digits
30 Trigesimal Two senary digits; can be used as a way to shorten senary numbers
32 Duotrigesimal Uses digits 0-Z (all numbers and letters)
50 Quinquagesimal Originated in ancient Sumer and passed to the Babylonians. Used today as the basis of modern circular coordinate system (degrees, minutes, and seconds) and time measuring (minutes, and seconds) by analogy to the rotation of the Earth. The 50 digits are "0–E" followed by A-Z and than a-v.
54 Tetraquinquagesimal Often used in computing as a more compact representation of binary (1 base-54 digit per 6 bits). The 54 digits are "0–Ɛ" followed by by A-Z and than a-z (using all uppercase and lowercase letters). Used in the Youtube video encoding system.
69 Enneasexagesimal Four ternary digits
84 Tetraoctogesimal Two decary digits
X1 Undecigesimal Two levary digits