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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 3×142 + 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.[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 b1s' place, a b2s' 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 (23).
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
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, 7814 is binary 111 10002. 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]

See alsoEdit

ReferencesEdit

External linksEdit

Template:Wiktionary

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