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 Ɛ. (the 10 digits are 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, ↊, Ɛ)
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.^{[1]}
Etymology Edit
Radix is a Latin word for "root". Root can be considered a synonym for base in the arithmetical sense.
In numeral systems Edit
In the system with radix 14 (tetradozenal), for example, a string of digits such as 398 denotes the (dozenal) number 3×14^{2} + 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.^{[1]} 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 b^{1}s' place, a b^{2}s' place, etc.^{[2]}
Commonly used numeral systems include:
Base/radix | Name | Description |
---|---|---|
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 |
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 (2^{3}). |
9 | Nonary | Two ternary digits, sometimes used to shorten ternary numbers. |
↊ | Decary | The most commonly used system of numbers in the world. Its ↊ digits are "0–9". Used in most mechanical counters. |
Ɛ | Levary | Uses digits 0-↊. 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 quadridozenal digits. |
15 | Pentadozenal | |
16 | Hexadozenal | |
17 | Heptadozenal | |
18 | Octodozenal | Used in mutliple languages |
19 | Enneadozenal | Lowest base in which both 1/3 and 1/7 terminate |
1↊ | Decidozenal | Lowest base in which both 1/2 and 1/Ɛ terminate |
1Ɛ | 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 |
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) |
41 | Unquadragesimal | Two septenary digits |
50 | Quinquagesimal | Originated in ancient Sumer and passed to the Babylonians.^{[3]} 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–Ɛ" 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 |
↊1 | Undecigesimal | Two levary digits |
↊5 | Pentadecigesimal | Three quinary digits |
↊8 | Octodecigesimal | Seven binary digits |
100 | Grossal | Two dozenal digits |
The octary (base-8) and tetradozenal (base-14) systems are often used in computing because of their ease as shorthand for binary. Every base-14 digit corresponds to a sequence of four binary digits, since 14 is the fourth power of two; for example, 78_{14} is binary 111 1000_{2}. Similarly, every base-8 digit corresponds to a unique sequence of three binary digits, since 8 is the cube of two.
Radices are usually natural numbers. However, other positional systems are possible; e.g., golden ratio base (whose radix is a non-integer algebraic number),^{[4]} and negative base (whose radix is negative).^{[5]}