Positional number system

Positional number systems are number systems where the value of each digit is dependent on its position in the number. Each digit has a value chosen from a range denoted by the radix or base of the numeral system.

Numbers
A number in a positional number system is dependent on the values of its positions relative to digits. For example, a decimal number can be written like this:

Standard Bases
These are the bases where the radix b is a natural number 2 ≤ b. Each digit can have a value v of 0 ≤ v ≤ b - 1. Names of the standard bases between 11 and 9,999 are formed through a mutated decimal name for the number that sits at the base. The prefixes are applied from the most significant digit to the least significant digit, with the exception that the unit digit and the tens digit are switched. Bases higher than base 9999 (Nonamillenonacentenonanonagesimal) are referred to using the term "Base n", where n is the number. Although base names for them can be made, the words will be so monstrously long that for the most part it's not worth it.

Non-standard Bases
Standard Bases are created when an integer equal to or higher than 2 is the base. However, other numbers can also be put as the radix of a positional number system, or the digits can be changed to different values (rather than 0 to b-1).

Negative bases
Main article: Negative base

A negative base is a positional numeral system where the radix b is a negative integer such that -2 ≥ b. Each digit can have a value v of 0 ≤ v ≤ |b| - 1.

As with natural bases, each digit is multiplied by a power of the base relative to that digit's position in the number. With negative bases, this means that powers where the exponent is an even number will yield positive values, while odd powers will yield negative numbers. As such, an advantage of negative bases is that, while unintuitive, they can express negative and positive numbers without a sign (- or +).

Signed bases
Main article: Signed digit representation

A signed base is a positional numeral systems where, while the radix remains a natural number, the digits range from negative numbers to positive numbers. For example, balanced ternary uses digits with values of -1, 0 and 1, while balanced quinary uses digits of -2, -1, 0, 1 and 2.

Bijective bases
Main article: Bijective base

A bijective base is a positional numeral system where, while the radix remains a natural number, the digits express values v of 1 ≤ v ≤ b. (Zero is skipped.) The value of zero is expressed with a special symbol (usually the greek letter lambda; λ). Unlike a standard base, 1 can be the radix of a bijective base (Unary).

Mixed bases
Main article: Mixed radix

A mixed base (or mixed radix) is a positional numeral system where the maximum digit (or value) of each position is different. For example, a mixed radix of 2 and 10 (repeating) will count: 1, 10, 11, 20, 21, 30... 90, 91, 100, 101, 110, 111, 120... 190, 191, 1000. Mixed bases can mix any number of different radixes. For example, Factorial base is a mixed base where the maximum digit on each position is equal to the distance from the LSD (the LSD has a maximum value of 0, the second LSD 1, the next one 2, etc.)

Non-integer bases
Main article: Non-integer base

Non-integer bases are bases where the radix is not an integer. Oftentimes, the radix is a mathematical constant, such as π. Examples include: Base Infinity, Golden ratio base, Base square root of 2 and Base e. Non-integer bases are very hard to understand for humans.

Skew bases
Main article: Skew base

Skew bases are bases that, rather than representing powers of the radix as a 1 followed by a string of 0s, represent in that way that number minus one. This leads to some interesting properties, such as the "lending" of an additional digit from a higher base, which can appear only once in the number.

Properties
All standard bases share a few similar properties which help in examining the usefulness of them.

Universal divisibility rules
There are some universal divisibility rules which apply to all standard bases. These are based on the alpha and omega relationships to the base - i.e. one above the base, and one below the base. These divisibility rules also apply to every factor of these numbers. The base is always written as 10, while the alpha is always written as 11. The omega is equal to the highest digit available. Other, less fundamental divisibility tests can be derived through relationships with a power of the base. Note: While a square omega test (summing blocks of 2 digits) can be derived, it is worth noting that a number below the square of another number n (n>2) cannot be prime regardless of the base, because (a+1)(a-1) = a2-1. This property can also be easily explained by the fact that if we assume the maximum digit of a given base is x, then base2-1 will be written as "xx", which is clearly x * 11. As such, the test is easily substitued by applying the test for the omega and alpha, and if both hold, then the number is divisible by b2-1.

Similar tests can also be derived from relationships to the cube or any higher power of the base, but, this time both the alpha and omega tests will be redundant (as the numbers cannot be prime (if base > 2)). However, the tests are still notable, as it is not as easy to factorize these numbers.

Fractions
In any given standard base, fractions will only terminate if the factorization of the denominator contains only divisors of the base. For example, in decimal, only fractions with denominators that factorize as 2x*5y will terminate, while in dozenal, fractions whose denominators factorize as 2x*3y will. Exact divisors will terminate with a single digit, while numbers that do not divide 10 evenly but do divide 10n, will terminate with n digits.

All other fractions will be recurring fractions. The length of the period depends on the lowest omega that divides the number evenly. This makes use of the fact that 0.9999...=110 (which holds for any base, just replace the 9s with the omega digit in the base). Each fraction 1/n can have a period of at most n-1 digits. Fractions with the maximum possible period yield cyclic numbers (for exaple, 14285710, which appears in the decimal expansion of 1/7, is a cyclic number). (denominators are given in decimal)

Arithmetic
Arithmethic experiences little change across bases. The main change is the carry point, which is set to the base.

There exists an infinite multiplication and exponentation table, which dictates the results for all calculations as long as none of the digits included in the result exceed the omega digit.