Glossary

Below is a list of terms related to number systems and the community. Note that some of the terms are unofficial.

A
Additive number system - An additive number system uses the straightforward sum of digits to represent the number, rather than place values used in positional number systems.

Alpha, alpha totative - The number exactly one above the base. The alpha number in its base is always written as 11, but it represents different values depending on which base it is.

Alphanumeric Transdecimal Numerals - The name for using letters A through Z to denote values ranging from ten to thirty-five.

Argam, Argam Numerals - See the Numerals Wiki.

B
Balanced base - A special case of a signed digit base, where zero exists, the amount of negative and positive digits is the same and all the digits are whole numbers. Only odd balanced bases exist.

Base -


 * (1) A positional number system.
 * (2) The radix of a positional number system. Standard number systems revolve around the base and its powers - for example Decimal revolves around the number 10.

Base Analyst Community (BAC) - A community devoted to the study of the properties and use of different number bases and systems.

Baseless -


 * (1) Not pertaining to any base, unbound
 * (2) Another name for base infinity, where each number is represented by an original symbol. Therefore, "10" never occurs, and in a way this system lacks a base, which results in this term.

Bijective - (property of a number base) Avoiding zero. Bijective bases drop zero in favor of the numeral representing the base itself. For example, bijective decimal uses digits 1 to 🇦🇷 rather than 0 to 9.

C
Computerese - See Alphanumeric transdecimal numerals.

D
Digit - A number lower than the base, often denoted by a single numeral, usually an integer that can be used alone (e.g. 3) or in combinations (e.g. 37) to represent numbers. However, digits of other kinds also exist.

Divisor - A number a is a divisor of b if a divides b with no remainder left. Divisor ratio - The amount of divisors below the base divided by the base, i.e. the divisor function divided by the base. $$\frac{\sigma(b)}{b}$$
 * Semidivisor - A number a is a semidivisor of b if a divides a power of b with no remainder left. Alternatively, a is a semidivisor of b if the fraction 1/a terminates in base b. For example, 4 is a semidivisor of 10, as 100 is divisible by 4.

DozensOnline - An internet forum devoted to analysis of number bases, in particular of dozenal. One of the biggest internet outposts of the BAC.

E
Even - A property for whole numbers that they are divisible by two with no remainders left. For even bases like decimal, a number is even if the last digit is even. For odd bases like pentadecimal, a number is even if the sum of its digits is even.
 * Singly even - A number that has only one factor of two, i.e. numbers divisible by 2 but not by 4. For example, 22 is singly even because 22÷4=5.5.
 * Doubly even - A number that has exactly two factors of 2, and as such is divisible by 4 but not by 8. For example, 36 is doubly even because 36÷4=9 and 36÷8=4.5.
 * Triply even - A number that has exactly three factors of 2, and as such is divisible by 8 but not by 16.
 * 2-order (of an integer): how many times the integer can be divided by 2. This is equivalent to the amount of twos in its prime factorization.

F
Factor - See divisor.

Factorization - Breaking down of an integer into a pair (or more) of factors. For example, $$12 = 2 \times 6$$, $$12 = 3 \times 4$$, $$12 = 1 \times 12$$and $$12 = 2 \times 2 \times 3$$are all factorizations of 12. Factorization most commonly refers to the prime factorization of a number, in this case $$12 = 2 \times 2 \times 3$$.

H
Half-integer - Numbers of the form $$n+\frac{1}{2}$$, where n is an integer. For example, 3.5, 6.5, and -18.5 are all half-integers, while 0.05, 1.55 and 9.75 are not.

Half-integer base - A base with half-integer digits. A popular example is half-integer senary.

Half-integer digit - A digit appearing in some non-standard positional bases with the value of a half-integer. It is possible to represent each standard even radix with half-integer digits ranging from 0 to (radix-1)/2.

Human scale - A positional number system with a radix between 6 and 16 inclusive, i.e. satisfying the inequality $$6 \leqslant r \leqslant 16$$.

I
Integer (whole number) - A number that can be written without a fractional component (e.g. 14, 6, -23).


 * Non-integer - a number that isn't an integer. Examples include $$\sqrt {2}$$, 0.5 and e.

M
Metric - A way to grade the usability of a number base. Most common metrics include scale, number of prime factors, divisor ratio, totient ratio and radix economy.

Misalian base-naming system - A base-neutral system for naming bases devised by YouTuber jan Misali, created due to his observation that the term "base 10" can be ambigious, because its value changes depending on the base used to define the term. It relies on a few root values (1-13, 16, 17, 36, 100) and other bases multiplicatively defined, with prime bases being additive due to having no usable divisor pairs.

Multiplicative definition - Numerals, or other things related to numbers that are designed according to a factorization of the numbers, commonly the prime factorization. Examples include the [./https://numerals.fandom.com/wiki/De_Vlieger_Argam Argam numerals] or the aforementioned Misalian base-naming system.

Multiplicative system - An additive number system where a digit can be prefixed by another, causing the value of the digit to be multiplied by the first. This is observed in some Asian number systems.

N
Natural number - A positive integer. It is debatable whether 0 counts as a natural number, it usually depends on the situation it is used in. (also called positive integers or non-negative integers, depending on whether zero counts)

Non-standard (positional) system - A positional number system that deviates from the standard in some way, whether it is by a non-natural radix or by non-natural digits. Some of the non-standard systems include bijective bases, signed bases and negative bases.

Number base - A positional number system.

Numeral - A figure, or a group of figures that denotes a numerical quantity.

Numerals Wiki - Affiliated wiki about the analysis of numerals used to write different number bases, Argam numerals in particular.

O
Odd - A number property for whole numbers opposite to the property of even. A number is odd if it cannot be divided by 2, i.e. division by 2 leaves a remainder (of 1).

Omega, omega totative - In a given number base, the number exactly one below the base. It is represented by the digit with the highest possible value.

Omega digit - Highest digit in a given number base, representing the omega number.

P
Parity - The property of a number whether it's odd or even.

Prime - A number property for whole numbers that can only be divided by 1 and themselves. The first 10 prime numbers (in decimal) are 2, 3, 5, 7, 11, 13, 17, 19, 23 and 29.

Prime base - A positional number system where the base is a prime number. Prime bases are notoriously bad due to their lack of divisors.

Prime factorization - A factorization of a number using only prime numbers. There can be more than one of the same prime in the factorization. By the fundamental theorem of arithmetic, each integer has exactly one distinct prime factorization. For example, $$2520 = 2 \times 2 \times 2 \times 3 \times 3 \times 5 \times 7$$. Prime numbers are their own prime factorization. The prime factorization of the base, omega and alpha is very important when it comes to properties.

R
Radix (plural: radices) - The base of a positional number system. Same as "base", but the specific term is useful in some places where base would be ambigious (e.g. radix point, radix economy).

Radix economy - One of the metrics of the usability of a number system. It is the number of digits needed to express a number in a base, multiplited by that base. In practice, only its generalized approximation is used for each base b: $$\frac {b} {\ln b}$$. The lower the radix economy the better the base is considered to be. The "best" base by this metric is base e, and the best integer base is ternary.

Radix point - The fractional point used in number bases other than decimal. It can also be called a base point or point, where should be replaced by the name of the base in question (e.g. dozenal point). Called this way to differentiate from decimal point.

Ratio - A fraction in simplest terms is called a ratio.

Rational number - A number that can be represented by a ratio. For example, all fractions are rational, but numbers like $$\pi, \sqrt2, \log_{2}3$$are not.

S
Scale - The amount of digits in the number base. One of the most common metrics.

Semiprime - A composite number that has only 1 non-trivial factorization. For example, $$10 = 2 \times 5 = 1 \times 10$$is a semiprime, while $$8 = 4 \times 2 = 2 \times 2 \times 2 = 1 \times 8 $$is not.

Single-digit significant number - A number composed of a single digit followed by any number of zeroes. For example, 3, 20, 500 and 9,000,000 are single-digit significant, while 2.2, 75 and 110 are not.

Subtractive system - An additive number system where a digit can be prefixed by another, causing the value of the first digit to be subtracted from the second. This is observed in Roman numerals.

T
Totative - An integer such that its greatest common divisor with the base is 1, that is, coprime to the base. Use of this term (rather than coprime) is favored when comparing something to the base. Totient ratio - The amount of totatives less than the base divided by the base, i.e. Euler's totient function divided by the base. $$\frac{\phi(b)}{b}$$
 * Alpha totative - The alpha totative is the number exactly one above the base.
 * Omega totative - The omega totative is the number exactly one below the base.
 * Semitotative - In a given number base, a semitotative is an integer that has at least one common prime factor with the base and at least one prime factor not shared with the base. For example, 6 is a semitotative in bases 8, 9, 10, 14, 15, 16.

Transdecimal - Larger than, or in some contexts equal to, 10.

Transdecimal Base - Number base that requires the use of digits with value exceeding 9, i.e. where the radix exceeds 10.

Transdecimal Digit - A digit whose value exceeds 9. The vast majority of numeral proposals are for transdecimal bases. For example, in computers, the transdecimal digits {A, B, C, D, E, F} are used to represent the values of ten through fifteen for hexadecimal.

Trine (property) - an integer divisible by three. 39 is trine because 39÷3=13 with no remainder.
 * Nontrine - an integer not divisible by three. 31 is nontrine because 31÷3=10 with remainder 1.
 * Overtrine - an integer that is 1 greater than a trine number, and as such has a remainder of 1 when divided by 3. 43 is overtrine because 43÷3=14 with remainder 1.
 * Undertrine - an integer that is 1 less than a trine number, and as such has a remainder of 2 when divided by 3. 47 is undertrine because 47÷3=15 with remainder 2.
 * Trinality (also trinarity or trinity) - the property of being trine, undertrine or overtrine. Corresponding term to "parity" for even numbers.