Numeral system
Ideally, a numeral system will: For example, the usual decimal representation gives every nonzero natural number a unique representation as a finite sequence of digits, beginning with a non-zero digit.
This system was established by the 7th century in India,[1] but was not yet in its modern form because the use of the digit zero had not yet been widely accepted.
Instead of a zero sometimes the digits were marked with dots to indicate their significance, or a space was used as a placeholder.
[2] The original numerals were very similar to the modern ones, even down to the glyphs used to represent digits.
[1] By the 13th century, Western Arabic numerals were accepted in European mathematical circles (Fibonacci used them in his Liber Abaci).
The exact age of the Maya numerals is unclear, but it is possible that it is older than the Hindu–Arabic system.
The Mayas had no equivalent of the modern decimal separator, so their system could not represent fractions.
[6] Later sources introduced conventions for the expression of zero and negative numbers.
The use of a round symbol 〇 for zero is first attested in the Mathematical Treatise in Nine Sections of 1247 AD.
Indian mathematicians are credited with developing the integer version, the Hindu–Arabic numeral system.
The system slowly spread to other surrounding regions like Arabia due to their commercial and military activities with India.
Middle-Eastern mathematicians extended the system to include negative powers of 10 (fractions), as recorded in a treatise by Syrian mathematician Abu'l-Hasan al-Uqlidisi in 952–953, and the decimal point notation was introduced[when?]
by Sind ibn Ali, who also wrote the earliest treatise on Arabic numerals.
The unary system is only useful for small numbers, although it plays an important role in theoretical computer science.
Elias gamma coding, which is commonly used in data compression, expresses arbitrary-sized numbers by using unary to indicate the length of a binary numeral.
Zero, which is not needed in the other systems, is of crucial importance here, in order to be able to "skip" a power.
[10] The positional decimal system is presently universally used in human writing.
For very large integers, bases 232 or 264 (grouping binary digits by 32 or 64, the length of the machine word) are used, as, for example, in GMP.
Unary numerals used in the neural circuits responsible for birdsong production.
[11] The nucleus in the brain of the songbirds that plays a part in both the learning and the production of bird song is the HVC (high vocal center).
The command signals for different notes in the birdsong emanate from different points in the HVC.
In some areas of computer science, a modified base k positional system is used, called bijective numeration, with digits 1, 2, ..., k (k ≥ 1), and zero being represented by an empty string.
Bijective base-k numeration is also called k-adic notation, not to be confused with p-adic numbers.
In a positional base b numeral system (with b a natural number greater than 1 known as the radix or base of the system), b basic symbols (or digits) corresponding to the first b natural numbers including zero are used.
By using a dot to divide the digits into two groups, one can also write fractions in the positional system.
A number has a terminating or repeating expansion if and only if it is rational; this does not depend on the base.
Putting overscores, n, or dots, ṅ, above the common digits is a convention used to represent repeating rational expansions.
It is also possible to define a variation of base b in which digits may be positive or negative; this is called a signed-digit representation.
More generally, if tn is the threshold for the n-th digit, it is easy to show that
So we have the following sequence of the numbers with at most 3 digits: a (0), ba (1), ca (2), ..., 9a (35), bb (36), cb (37), ..., 9b (70), bca (71), ..., 99a (1260), bcb (1261), ..., 99b (2450).
