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A ''numeral'' is a symbol or group of symbols, or a word in a natural language that represents a number. Numerals differ from numbers just as words differ from the things they refer to. The symbols "11", "eleven" and "XI" are different numerals, all representing the same number. This article attempts to explain the various systems of numerals. See also number names.
A 'numeral system' (or 'system of numeration') is a framework where a set of numbers are represented by numerals in a consistent manner. It can be seen as the ''context'' that allows the numeral "11" to be interpreted as the binary numeral for ''three'', the decimal numeral for ''eleven'', or other numbers in different bases.
Ideally, a numeral system will:

★ Represent a useful set of numbers (e.g. all whole numbers, integers, or real numbers)

★ Give every number represented a unique representation (or at least a standard representation)

★ Reflect the algebraic and arithmetic structure of the numbers.
For example, the usual decimal representation of whole numbers gives every whole number a unique representation as a finite sequence of digits, with the operations of arithmetic (addition, subtraction, multiplication and division) being present as the standard algorithms of arithmetic. However, when decimal representation is used for the rational or real numbers, the representation is no longer unique: many rational numbers have two numerals, a standard one that terminates, such as 2.31, and another that recurs, such as 2.309999999... . Numerals which terminate have no non-zero digits after a given position. For example, numerals like 2.31 and 2.310 are taken to be the same, except in the experimental sciences, where greater precision is denoted by the trailing zero.
Numeral systems are sometimes called ''number systems'', but that name is misleading, as it could refer to different systems of numbers, such as the system of real numbers, the system of complex numbers, the system of ''p''-adic numbers, etc. Such systems are not the topic of this article.

Contents

Types of numeral systems

History

Bases used

In computing

Five

Eight

Ten

Twelve

Twenty

Sixty

Dual base (five and twenty)

Base names

Positional systems in detail

Change of radix

Generalized variable-length integers

Properties of numerical systems with integer bases

See also

References

External links

Types of numeral systems

Nowadays, the most commonly used system of numerals is known as Hindu-Arabic numerals, and two great Indian mathematicians could be given credit for developing them. Aryabhatta of Kusumapura who lived during the 5th century developed the place value notation and Brahmagupta a century later introduced the symbol zero.
The simplest numeral system is the unary numeral system, in which every natural number is represented by a corresponding number of symbols. If the symbol ′ is chosen, for example, then the number seven would be represented by ′′′′′′′. Tally marks represent one such system still in common use. In practice, the unary system is normally only useful for small numbers, although it plays an important role in theoretical computer science. Also, 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.
The unary notation can be abbreviated by introducing different symbols for certain new values. Very commonly, these values are powers of 10; so for instance, if ′ stands for one, - for ten and + for 100, then the number 304 can be compactly represented as +++ ′′′′ and number 123 as + - - ′′′ without any need for zero. This is called sign-value notation. The ancient Egyptian system is of this type, and the Roman system is a modification of this idea.
More useful still are systems which employ special abbreviations for repetitions of symbols; for example, using the first nine letters of our alphabet for these abbreviations, with A standing for "one occurrence", B "two occurrences", and so on, we could then write C+ D′ for the number 304. The numeral system of English is of this type ("three hundred [and] four"), as are those of virtually all other spoken languages, regardless of what written systems they have adopted.
More elegant is a ''positional system'', also known as place-value notation. Again working in base 10, we use ten different digits 0, ..., 9 and use the position of a digit to signify the power of ten that the digit is to be multiplied with, as in 304 = 3×100 + 0×10 + 4×1. Note that zero, which is not needed in the other systems, is of crucial importance here, in order to be able to "skip" a power. The Hindu-Arabic numeral system, borrowed from India, is a positional base 10 system; it is used today throughout the world.
Arithmetic is much easier in positional systems than in the earlier additive ones; furthermore, additive systems have a need for a potentially infinite number of different symbols for the different powers of 10; positional systems need only 10 different symbols (assuming that it uses base 10).
The numerals used when writing numbers with digits or symbols can be divided into two types that might be called the arithmetic numerals 0,1,2,3,4,5,6,7,8,9 and the geometric numerals 1,10,100,1000,10000... respectively. The sign-value systems use only the geometric numerals and the positional system use only the arithmetic numerals. The sign-value system does not need arithmetic numerals because they are made by repetition (except for the Ionic system), and the positional system does not need geometric numerals because they are made by position. However, the spoken language uses ''both'' arithmetic and geometric numerals.
In certain 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 the empty string. This establishes a bijection between the set of all such digit-strings and the set of non-negative integers, avoiding the non-uniqueness caused by leading zeros. Bijective base-''k'' numeration is also called ''k''-adic notation, not to be confused with p-adic numbers. Bijective base-1 is the same as unary.

History

: ''See also: History of natural numbers and the status of zero''
Counting aids, especially the use of body parts (counting on fingers), were certainly used in prehistoric times as today. There are many variations. Besides counting 10 fingers, some cultures have counted knuckles, the space between fingers, and toes as well as fingers. The Oksapmin culture of New Guinea uses a system of 27 upper body locations to represent numbers.
To preserve numerical information, Tallies carved in wood, bone, and stone have been used since prehistoric times. Stone age cultures, including ancient American Indian groups, used tallies for gambling, personal services, and trade-goods.
A method of preserving numeric information in clay was invented by the Sumerians between 8000 and 3500 BC. This was done with small clay tokens of various shapes that were strung like beads on a string. Beginning about 3500 BC clay tokens were gradually replaced by number signs impressed with a round stylus at different angles in clay tablets (originally containers for tokens) which were then baked. About 3100 BC written numbers were dissociated from the things being counted and became abstract numerals.
Between 2700 BC and 2000 BC in Sumer, the round stylus was gradually replaced by a reed stylus that was used to press wedge-shaped cuneiform signs in clay. These cuneiform number signs resembled the round number signs they replaced and retained the additive sign-value notation of the round number signs. These systems gradually converged on a common sexagesimal number system; this was a place-value system consisting of only two impressed marks, the vertical wedge and the chevron, which could also represent fractions. This sexagesimal number system was fully developed at the beginning of the Old Babylonia period (about 1950 BC) and became standard in Babylonia.
Sexagesimal numerals were a mixed radix system that retained the alternating base 10 and base 6 in a sequence of cuneiform vertical wedges and chevrons. By 1950 BC this was a positional notation system. Sexagesimal numerals came to be widely used in commerce, but were also used in astronomical and other calculations. This system was exported from Babylonia and used throughout Mesopotamia, and by every Mediterranean nation that used standard Babylonian units of measure and counting, including the Greeks, Romans and Egyptians. Babylonian-style sexagesimal numeration is still used in modern societies to measure time (minutes per hour) and angles (degrees).
In China, armies and provisions were counted using modular tallies of prime numbers. Unique numbers of troops and measures of rice appear as unique combinations of these tallies. A great convenience of modular arithmetic is that it is easy to multiply, though quite difficult to add. This makes use of modular arithmetic for provisions especially attractive. Conventional tallies are quite difficult to multiply and divide. In modern times modular arithmetic is sometimes used in Digital signal processing.
The oldest Greek system was the that of the Attic numerals, but in the 4th century BC they began to use a quasidecimal alphabetic system (see Greek numerals). Jews began using a similar system (Hebrew numerals), with the oldest examples known being coins from around 100 BC.
The Roman empire used tallies written on wax, papyrus and stone, and roughly followed the Greek custom of assigning letters to various numbers. The Roman numerals system remained in common use in Europe until positional notation came into common use in the 1500s.
The Maya of Central America used a mixed base 18 and base 20 system, possibly inherited from the Olmec, including advanced features such as positional notation and a zero. They used this system to make advanced astronomical calculations, including highly accurate calculations of the length of the solar year and the orbit of Venus.
The Incan Empire ran a large command economy using quipu, tallies made by knotting colored fibers. Knowledge of the encodings of the knots and colors was suppressed by the Spanish conquistadors in the 16th century, and has not survived although simple quipu-like recording devices are still used in the Andean region.
Some authorities believe that positional arithmetic began with the wide use of the abacus in China. The earliest written positional records seem to be tallies of abacus results in China around 400. In particular, zero was correctly described by Chinese mathematicians around 932, and seems to have originated as a circle around a spot containing no beads.
The modern positional Hindu-Arabic numeral system was developed by mathematicians in India, and passed on to Muslim mathematicians, along with astronomical tables brought to Baghdad by an Indian ambassador around 773.
From India, the thriving trade between Islamic sultans and Africa carried the concept to Cairo. Arabic mathematicians extended the system to include decimal fractions, and wrote an important work about it in the 9th century. The modern Arabic numerals were introduced to Europe with the translation of this work in the 12th century in Spain and Leonardo of Pisa's ''Liber Abaci'' of 1201. In Europe, the complete Indian system with the zero was derived from the Arabs in the 12th century.
The binary system (base 2), was propagated in the 17th century by Gottfried Leibniz. Leibniz had developed the concept early in his career, and had revisited it when he reviewed a copy of the I ching from China. Binary numbers came into common use in the 20th century because of computer applications.

Bases used

In computing

Switches, mimicked by their electronic successors built of vacuum tubes, have only two possible states: "open" and "closed". Substituting open=1 and closed=0 (or the other way around) yields the entire set of binary digits. This base-2 system (binary) is the basis for digital computers. It is used to perform integer arithmetic in almost all digital computers; some exotic base-3 (ternary) and base-10 computers have also been built, but those designs were discarded early in the history of computing hardware.
Modern computers use transistors that represent two states with either 'high' or 'low' voltages. The smallest unit of memory for this binary state is called a bit. Bits are arranged in groups to aid in processing, and to make the binary numbers shorter and more manageable for humans. More recently these groups of bits, such as bytes and words, are sized in multiples of four. Thus base 16 (hexadecimal) is commonly used as shorthand. Base 8 (octal) has also been used for this purpose.
A computer does not treat all of its data as numerical. For instance, some of it may be treated as program instructions or data such as text. However, arithmetic and Boolean logic constitute most internal operations. Whole numbers are represented exactly, as integers. Real numbers, allowing fractional values, are usually approximated as floating point numbers. The computer uses different methods to do arithmetic with these two kinds of numbers.

Five

A base-5 system (quinary) has been used in many cultures for counting. Plainly it is based on the
number of fingers on a human hand. It may also be regarded as a sub-base of other bases, such as base 10 and
base 60.

Eight

A base-8 system (octal) was devised by the Yuki of Northern California, who used the spaces between the fingers to count. There is also linguistic evidence which suggests that the Bronze Age Proto-Indo Europeans (from whom most European and Indic languages descend) might have replaced a base 8 system (or a system which could only count up to 8) with a base 10 system. The evidence is that the word for 9, ''newm'', is suggested by some to derive from the word for 'new', ''newo-'', suggesting that the number 9 had been recently invented and called the 'new number' (Mallory & Adams 1997).

Ten

The base-10 system (decimal) is the one most commonly used today. It is assumed to have originated because humans have ten fingers. These systems often use a larger superimposed base. See Decimal superbase.

Twelve

Base-12 systems (duodecimal or dozenal) have been popular because multiplication and division are easier than in base-10, with addition just as easy. 12 is a useful base because it has many factors. It is the smallest multiple of one through four and of six. We still have a special word for "dozen" and just like there is a word for 10², ''hundred'', there is also a word for 12², ''gross''. Base-12 could have originated from the number of knuckles in the four fingers of a hand excluding the thumb, which is used as a pointer in counting.
There are 24 hours per day, usually counted till 12 until noon (a.m.) and once again until midnight (p.m.), often further divided per 6 hours in counting (for instance in Thailand) or as switches between using terms like 'night', 'morning', 'afternoon', and 'evening', whereas other languages use such terms with durations of 3 to 9 hours often according to switches at some of the 3 hour interval marks.
Multiples of 12 have been in common use as English units of resolution in the analog and digital printing world, where 1 point equals 1/72 of an inch and 12 points equal 1 pica, and printer resolutions like 360, 600, 720, 1200 or 1440 dpi (dots per inch) are common. These are combinations of base-12 and base-10 factors: (3×12)×10, 12×(5×10), (6×12)×10, 12×(10×10) and (12×12)×10.

Twenty

The Maya civilization and other civilizations of Pre-Columbian Mesoamerica used base-20 (vigesimal), possibly originating from the number of a person's fingers and toes. Evidence of base-20 counting systems is also found in the languages of central and western Africa.
Possible remnants of a base-20 system also exist in French, as seen in the names of the numbers from 60 through 99. For example, sixty-five is ''soixante-cinq'' (literally, "sixty [and] five"), while seventy-five is ''soixante-quinze'' (literally, "sixty [and] fifteen"). Furthermore, for any number between 80 and 99, the "tens-column" number is expressed as a multiple of twenty (somewhat similar to the archaic English manner of speaking of "scores"). For example, eighty-two is ''quatre-vingt-deux'' (literally, four twenty[s] [and] two), while ninety-two is ''quatre-vingt-douze'' (literally, four twenty[s] [and] twelve).
Danish numerals display a similar base-20 structure.

Sixty

Base 60 (sexagesimal) was used by the Sumerians and their successors in Mesopotamia and survives today in our system of time (hence the division of an hour into 60 minutes and a minute into 60 seconds) and in our system of angular measure (a degree is divided into 60 minutes and a minute is divided into 60 seconds). 60 also has a large number of factors, including the first six counting numbers. Base-60 systems are believed to have originated through the merging of base-10 and base-12 systems. The Chinese Calendar, for example, uses a base-60 Jia-Zi甲子 system to denote years, with each year within the 60-year cycle being named with two symbols, the first being base-10 (called Tian-Gan天干 or heavenly stems) and the second symbol being base 12 (called Di-Zhi地支 or earthly branches). Both symbols are incremented in successive years until the first pattern recurs 60 years later. The second symbol of this system is also related to the 12-animal Chinese zodiac system.

Dual base (five and twenty)

Many ancient counting systems use 5 as a primary base, almost surely coming from the number of fingers on a person's hand. Often these systems are supplemented with a secondary base, sometimes ten, sometimes twenty. In some African languages the word for 5 is the same as "hand" or "fist" (Dyola language of Guinea-Bissau, Banda language of Central Africa). Counting continues by adding 1, 2, 3, or 4 to combinations of 5, until the secondary base is reached. In the case of twenty, this word often means "man complete". This system is referred to as ''quinquavigesimal''. It is found in many languages of the Sudan region.

Base names

1 - unary
2 - binary
3 - ternary / trinary
4 - quaternary
5 - quinary / quinternary
6 - senary / heximal / hexary
7 - septenary / septuary
8 - octal / octonary / octonal / octimal
9 - nonary / novary / noval
10 - decimal / denary
11 - undecimal / undenary / unodecimal
12 - dozenal / duodecimal / duodenary
13 - tridecimal / tredecimal / triodecimal
14 - tetradecimal / quadrodecimal / quattuordecimal
15 - pentadecimal / quindecimal
16 - hexadecimal / sexadecimal / sedecimal
17 - septendecimal / heptadecimal
18 - octodecimal / decennoctal
19 - nonadecimal / novodecimal / decennoval
20 - vigesimal / bigesimal / bidecimal
21 - unovigesimal / unobigesimal
22 - duovigesimal
23 - triovigesimal
24 - quadrovigesimal / quadriovigesimal
26 - hexavigesimal / sexavigesimal
27 - heptovigesimal
28 - octovigesimal
29 - novovigesimal
30 - trigesimal / triogesimal
31 - unotrigesimal
(...repeat naming pattern...)
36 - hexatridecimal / sexatrigesimal
(...repeat naming pattern...)
40 - quadragesimal / quadrigesimal
41 - unoquadragesimal
(...repeat naming pattern...)
50 - quinquagesimal / pentagesimal
51 - unoquinquagesimal
(...repeat naming pattern...)
60 - sexagesimal
(...repeat naming pattern...)
64 - quadrosexagesimal
(...repeat naming pattern...)
70 - septagesimal / heptagesimal
80 - octagesimal / octogesimal
90 - nonagesimal / novagesimal
100 - centimal / centesimal
(...repeat naming pattern...)
110 - decacentimal
111 - unodecacentimal
(...repeat naming pattern...)
200 - bicentimal / bicentesimal
(...repeat naming pattern...)
210 - decabicentimal
211 - unodecabicentimal
(...repeat naming pattern...)
300 - tercentimal / tricentesimal
400 - quattrocentimal / quadricentesimal
500 - quincentimal / pentacentesimal
600 - hexacentimal / hexacentesimal
700 - heptacentimal / heptacentesimal
800 - octacentimal / octocentimal / octacentesimal / octocentesimal
900 - novacentimal / novacentesimal
1000 - millesimal
2000 - bimillesimal
(...repeat naming pattern...)
10000 - decamillesimal

Positional systems in detail

''Also see Positional notation.''
In a positional base-''b'' numeral system (with ''b'' a positive natural number known as the radix), ''b'' basic symbols (or digits) corresponding to the first ''b'' natural numbers including zero are used. To generate the rest of the numerals, the position of the symbol in the figure is used. The symbol in the last position has its own value, and as it moves to the left its value is multiplied by ''b''.
For example, in the decimal system (base 10), the numeral 4327 means ('4'×10³) + ('3'×10²) + ('2'×10¹) + ('7'×10⁰), noting that 10⁰ = 1.
In general, if ''b'' is the base, we write a number in the numeral system of base ''b'' by expressing it in the form ''a''_''n''''b''^''n'' + ''a''_{''n'' − 1}''b''^{''n'' − 1} + ''a''_{''n'' − 2}''b''^{''n'' − 2} + ... + ''a''₀''b''⁰ and writing the enumerated digits ''a''_n''a''_{''n'' − 1}''a''_{''n'' − 2} ... ''a''₀ in descending order. The digits are natural numbers between 0 and ''b'' − 1, inclusive.
If a text (such as this one) discusses multiple bases, and if ambiguity exists, the base (itself represented in base 10) is added in subscript to the right of the number, like this: number_base. Unless specified by context, numbers without subscript are considered to be decimal.
By using a dot to divide the digits into two groups, one can also write fractions in the positional system. For example, the base-2 numeral 10.11 denotes 1×2¹ + 0×2⁰ + 1×2⁻¹ + 1×2⁻² = 2.75.
In general, numbers in the base ''b'' system are of the form:
:$$
(a_na_{n-1}cdots a_1a_0.c_1 c_2 c_3cdots)_b =
sum_{k=0}^n a_kb^k + sum_{k=1}^infty c_kb^{-k}

The numbers ''b''^''k'' and ''b''^−''k'' are the weights of the corresponding digits. The ''position k'' is the logarithm of the corresponding ''weight w'', that is $k\; =\; log\_\{b\}\; w\; =\; log\_\{b\}\; b^k$. The highest used position is close to the order of magnitude of the number.
The number of tally marks required in the unary numeral system for ''describing the weight'' would have been 'w'. In the positional system the number of digits required to describe it is only $k\; +\; 1\; =$'$log\_\{b\}\; w$'$+\; 1$, for $k\; ge\; 0$. E.g. to describe the weight 1000 then 4 digits are needed since $log\_\{10\}\; 1000\; +\; 1\; =\; 3\; +\; 1$. The number of digits required to ''describe the position'' is $log\_\{b\}\; k\; +\; 1\; =\; log\_\{b\}\; log\_\{b\}\; w\; +\; 1$ (in positions 1, 10, 100... only for simplicity in the decimal example).
{| class="wikitable"
|-
! width="170"| Position
|width="40"|3
|width="40"|2
|width="40"|1
|width="40"|0
|width="40"|-1
|width="40"|-2
|width="40"|...
|-
! Weight
| $b^3$ || $b^2$ || $b^1$ || $b^0$ || $b^\{-1\}$ || $b^\{-2\}$ || ...
|-
! Digit
| $a\_3$ || $a\_2$ || $a\_1$ || $a\_0$ || $c\_1$ || $c\_2$ || ...
|-
! Decimal example weight
| 1000 || 100 || 10 || 1 || 0.1 || 0.01 || ...
|-
! Decimal example digit
| 4 || 3 || 2 || 7 || 0 || 0 || ...
|}
Note that a number has a terminating or repeating expansion if and only if it is rational; this does not depend on the base. A number that terminates in one base may repeat in another (thus 0.3₁₀ = 0.0100110011001...₂). An irrational number stays unperiodic (infinite amount of unrepeating digits) in all integral bases. Thus, for example in base 2, π = 3.1415926...₁₀ can be written down as the unperiodic 11.001001000011111...₂.
If ''b'' = ''p'' is a prime number, one can define base-''p'' numerals whose expansion to the left never stops; these are called the p-adic numbers.

Change of radix

A simple algorithm for converting integers between positive-integer radices is repeated division by the target radix; the remainders give the "digits" starting at the least significant. E.g., 1020304 base 10 into base 7:

1020304 / 7 = 145757 r 5

 145757 / 7 =  20822 r 3

  20822 / 7 =   2974 r 4

   2974 / 7 =    424 r 6

    424 / 7 =     60 r 4

     60 / 7 =      8 r 4

      8 / 7 =      1 r 1

      1 / 7 =      0 r 1   => 11446435

E.g., 10110111 base 2 into base 5:

10110111 / 101 = 100100 r 11  (3)

  100100 / 101 =    111 r  1  (1)

     111 / 101 =      1 r 10  (2)

       1 / 101 =      0 r  1  (1)  => 1213

To convert a "decimal" fraction, do repeated multiplication, taking the protruding integer parts as the "digits". Unfortunately a terminating fraction in one base may not terminate in another. E.g., 0.1A4C base 16 into base 9:

0.1A4C × 9 = 0.ECAC

0.ECAC × 9 = 8.520C

0.520C × 9 = 2.E26C

0.E26C × 9 = 7.F5CC

0.F5CC × 9 = 8.A42C 

0.A42C × 9 = 5.C58C  => 0.082785...

Generalized variable-length integers

More general is using a notation (here written little-endian) like $a\_0\; a\_1\; a\_2$ for $a\_0\; +\; a\_1\; b\_1\; +\; a\_2\; b\_1\; b\_2$, etc.
This is used in punycode, one aspect of which is the representation of a sequence of non-negative integers of arbitrary size in the form of a sequence without delimiters, of "digits" from a collection of 36: a-z and 0-9, representing 0-25 and 26-35 respectively. A digit lower than a threshold value marks that it is the most-significant digit, hence the end of the number. The threshold value depends on the position in the number. For example, if the threshold value for the first digit is b (i.e. 1) then a (i.e. 0) marks the end of the number (it has just one digit), so in numbers of more than one digit the range is only b-9 (1-35), therefore the weight ''b''₁ is 35 instead of 36. Suppose the threshold values for the second and third digit are c (2), then the third digit has a weight 34 × 35 = 1190 and we have the following sequence:
a (0), ba (1), ca (2), .., 9a (35), bb (36), cb (37), .., 9b (70), bca (71), .., 99a (1260), bcb (1261), etc.
Note that unlike a regular base-35 numeral system, we have numbers like 9b where 9 and b each represent 35; yet the representation is unique because ac and aca are not allowed.
The flexibility in choosing threshold values allows optimization depending on the frequency of occurrence of numbers of various sizes.
The case with all threshold values equal to 1 corresponds to bijective numeration, where the zeros correspond to separators of numbers with digits which are nonzero.

Properties of numerical systems with integer bases

Numeral systems with base ''A'', where ''A'' is a positive integer, possess the following properties:
:If ''A'' is even and ''A/2'' is odd, all integral powers greater than zero of the number ''(A/2)+1'' will contain ''(A/2)+1'' as their last digit
:If both ''A'' and ''A/2'' are even, then all integral powers greater than or equal to zero of the number ''(A/2)+1'' will alternate between having ''(A/2)+1'' and 1 as their last digit. (For odd powers it will be ''(A/2)+1'', for even powers it will be 1)
Proof of the first property:
Define $\{A\; over\; 2\}\; +\; 1\; =\; x$ Then x is even, and all $x^p$ for p greater than 0 must be even. The property is equivalent to
:$!\; x^p\; equiv\; x\; (mbox\{mod\}\; a)$
We first check the case for p=1
:$!\; x\; equiv\; x\; (mbox\{mod\}\; A)$
x is less than A, so the result is trivial. We then check for p=2:
:$!\; x^2\; =\; xx$
:$!\; x^2\; =\; x(x-1)\; +\; x$
Since $x-1\; =\; (\{A\; over\; 2\}\; +\; 1)\; -\; 1\; =\; \{A\; over\; 2\}$, then for all even N:
:$!\; \{NA\; over\; 2\}\; =\; N(x-1)\; equiv\; 0\; (mbox\{mod\}\; A)\; (1)$
Because x is even, then $x(x-1)$ is congruent to zero modulo a. Therefore:
:$!\; x^2\; equiv\; x\; (mbox\{mod\}\; A)$
Using induction, assuming that the property holds for p-1:
:$!\; x^p\; =\; \{x^\{p-1\}\}x\; =\; \{x^\{p-1\}\}(x-1)\; +\; x^\{p-1\}$
Since the case holds for p-1, then $\{x^\{p-1\}\}\; equiv\; x\; (mbox\{mod\}\; A)$. Since
:$!\; \{x^\{p-1\}\}(x-1)$
is a case of Equation 1, then $\{x^\{p-1\}\}(x-1)\; equiv\; 0\; (mbox\{mod\}\; A)$. This leaves, for all p greater than 0,
:$!\; x^p\; equiv\; x\; (mbox\{mod\}\; a)$
Q.E.D.
Proof of the second property:
Define $\{A\; over\; 2\}\; +\; 1\; =\; x$ Then x is odd, and all $x^p$ for p greater than or equal to 0 must be odd. The property is equivalent to
:$!\; x^p\; equiv\; 1\; (mbox\{mod\}\; A);\; mbox\{if\}\; p\; equiv\; 0\; (mbox\{mod\}\; 2)$
:$!\; x^p\; equiv\; x\; (mbox\{mod\}\; A);\; mbox\{if\}\; p\; equiv\; 1\; (mbox\{mod\}\; 2)$
Since $x-1\; =\; (\{A\; over\; 2\}\; +\; 1)\; -\; 1\; =\; \{A\; over\; 2\}$, then for all odd E:
:$!\; \{EA\; over\; 2\}\; =\; E(x-1)\; equiv\; \{A\; over\; 2\}\; (mbox\{mod\}\; A)\; (2)$
The case is first checked for p=0:
:$!\; x^0\; =\; 1$
:$!\; 1\; equiv\; 1\; (mbox\{mod\}\; A)$
This result is trivial
Next, for p=1:
:$!\; x^1\; =\; x$
:$!\; x\; equiv\; x\; (mbox\{mod\}\; A)$
This result is also trivial
Next, for p=2:
:$!\; x^2\; =\; xx\; =\; x(x-1)\; +\; x$
Because x is odd, then x(x-1) is a case of Equation 2,
:$x(x-1)\; +\; x\; equiv\; \{\{A\; over\; 2\}\; +\; x\}\; (mbox\{mod\}\; A)$
:$!\; \{A\; over\; 2\}\; +\; x\; =\; \{A\; over\; 2\}\; +\; \{A\; over\; 2\}\; +\; 1\; =\; A+1$
:$!\; A+1\; equiv\; 1\; (mbox\{mod\}\; A),\; (mbox\{so\}\; x(x-1)\; +\; x\; =\; x^2\; equiv\; 1\; (mbox\{mod\}\; A)$
Next, for p=3:
:$!\; x^3\; =\; \{x^2\}x\; =\; \{x^2\}(x-1)\; +\; x^2$
Because $x^2$ is odd, $\{x^2\}(x-1)\; +\; x^2$ is a case of Equation 2,
:$!\; \{x^2\}(x-1)\; +\; x^2\; equiv\; \{\{A\; over\; 2\}\; +\; x^2\}\; (mbox\{mod\}\; A)$
Since $x^2\; equiv\; 1\; (mbox\{mod\}\; A)$,
:$!\; \{x^2\}(x-1)\; +\; x^2\; equiv\; \{\{A\; over\; 2\}\; +\; 1\}\; (mbox\{mod\}\; A)$
$\{\{A\; over\; 2\}\; +\; 1\}\; =\; x$, so $x^3\; equiv\; x\; (mbox\{mod\}\; A)$.
Using induction, assuming that the property holds for p-1:
:$!\; x^p\; equiv\; \{x^\{p-1\}\}(x-1)\; +\; x^\{p-1\}$
If p is odd:
:$!\; x^\{p-1\}\; equiv\; 1\; (mbox\{mod\}\; A)$
Since $\{x^\{p-1\}\}(x-1)$ is a case of Equation (2), $\{x^\{p-1\}\}(x-1)\; +\; x^\{p-1\}\; equiv\; \{\{A\; over\; 2\}\; +\; 1\}\; (mbox\{mod\}\; A)$, so
:$x^p\; equiv\; x\; (mbox\{mod\}\; A)$
If p is even:
:$!\; x^\{p-1\}\; equiv\; x\; (mbox\{mod\}\; A)$
Since $\{x^\{p-1\}\}(x-1)$ is a case of Equation (2), $\{x^\{p-1\}\}(x-1)\; +\; x^\{p-1\}\; equiv\; \{\{A\; over\; 2\}\; +\; x\}\; (mbox\{mod\}\; A)$.
$\{A\; over\; 2\}\; +\; x\; =\; \{A\; over\; 2\}\; +\; \{A\; over\; 2\}\; +\; 1\; =\; A+1$
$A+1\; equiv\; 1\; (mbox\{mod\}\; A)$, so
:$x^p\; equiv\; 1\; (mbox\{mod\}\; A)$
Q.E.D.

See also

★ Computer numbering formats

★ Subtractive notation

★ Recurring decimal

★ Quipu

★ Babylonian numerals – a sexagesimal (base-60) system

★ Golden ratio base

★ Grammatical number

★ Zenzizenzizenzic

References

★ Georges Ifrah. ''The Universal History of Numbers : From Prehistory to the Invention of the Computer'', Wiley, 1999. ISBN 0-471-37568-3.

★ D. Knuth. ''The Art of Computer Programming''. Volume 2, 3rd Ed. Addison-Wesley. pp.194–213, "Positional Number Systems".

★ J.P. Mallory and D.Q. Adams, ''Encyclopedia of Indo-European Culture'', Fitzroy Dearborn Publishers, London and Chicago, 1997.

★ Hans J. Nissen, P. Damerow, R. Englund, ''Archaic Bookkeeping'', University of Chicago Press, 1993, ISBN 0-226-58659-6.

★ Denise Schmandt-Besserat, ''How Writing Came About'', University of Texas Press, 1992, ISBN 0-292-77704-3.

★ Claudia Zaslavsky, ''Africa Counts: Number and Pattern in African Cultures'', Lawrence Hill Books, 1999, ISBN 1-55652-350-5.

External links

★ History of Counting and Numeral Systems-PlainMath.Net

★ Online Converter for Different Numeral Systems (Base 2-36, JavaScript, GPL)

★ Online Converter for Decimal/Roman Numerals (JavaScript, GPL)

★ Number Sense & Numeration Lessons

★ Counting Systems of Papua New Guinea

★ Numerical Mechanisms and Children's Concept of Numbers