Arithmetic

The identity elements for arithmetic operations make use of the only two integers that are neither composite nor prime, that is 0 and 1.

A number is a mathematical object used to count, measure, label.

Number theory studies the properties of numbers.

Properties of numbers are distinct from the properties of numerals.

During the 19th century, mathematicians began to develop many different abstractions which share certain properties of numbers and may be seen as extending the concept. Among the first were the hypercomplex numbers, which consist of various extensions or modifications of the complex number system. Today, number systems are considered important special examples of much more general categories such as rings and fields, and the application of the term "number" is a matter of convention, without fundamental significance.[1

In math, a number is a mathematical primitive i.e. an undefined concept without a physical manifestation. We rely on our intuition to understand the concept of a number alone, apart from its representation.

• Classes

  • Cardinal number

    • cardinality

    • transfinite cardinal numbers

  • Ordinal number, ordinality

• Numeral system

  • unary, 1

  • binary, 2

  • quinquagesimal, 50

• Factorization

  • non-prime and non-compound, 1

  • prime numbers, 2,3,5,7,11,13,...

  • compound numbers, 42=(2x3x7)

• Gaussian integer factorization: $(1+i)(1-i) (1+i)(1-i)$

• Primality, Prime: 1st

• Divisors: 1:{1}; 2:{1,2},

Notation

• Arabic numeral: 1; 2

• Greek numeral: 1:α'; 2:Β´

• Roman numeral: 1:I; 2:II

  • unicode: 1:Ⅰ; ⅰ 2:Ⅱ,ⅱ

notation	glyph
Arabic	1 2 3 4 5 6 7 8 9 0
Greek	α´ β´
Roman	Ⅰ ⅰ
Persian	١
Arab/Kurdish	−١ ٠ ١ ٢
Urdu	۰ ۱ ۲
Sindhi	١
Assamese/Bengali	−১ ০ ১ ২
Devanāgarī	१ २
Ge'ez	፩ ፪
Hebrew	ב א
Hindu	०
Kannada	೧
Khmer	០ ១ ២
Malayalam	൧
Telugu	౨
Thai	๐ ๑ ๒
Tamil	௧ ௨
Georgian	Ⴁ ⴁ ბ
Korean	일 이
Japanese	〇 一 二
Chinese	〇 一 二
Chinese/Japanese	负一 零 壹 貳

Prefixes

| nr |        Greek |          Latin |
|----|-------------:|---------------:|
| 1  |  mono, haplo |            uni |
| 2  |           di |        duo, bi |
| 3  |          tri |       tre, ter |
| 4  |        tetra |         quadri |
| 5  |        penta | quinque, quint |
| 6  |         hexa |           sexa |
| 7  |        hepta |         septua |
| 8  |         octa |           octo |
| 9  |        ennea |           nona |
| 10 |           deca |           deci |
| 11 |       hen-deca |         undeca |
| 12 |        do-deca |        duodeca |
| 13 |  tris-kai-deca |        tredeca |
| 14 | tetra-kai-deca |   quattuordeca |
| 15 | penta-kai-deca |                |
| 16 |  hexa-kai-deca |                |
| 17 | hepta-kai-deca |                |
| 18 |  octa-kai-deca |                |
| 19 | ennea-kai-deca |                |
| 20 | icos-a         |                |
| 21 | icosi-kai-hena |                |
| 22 | icosi-di       |                |
| 23 | icosi          |                |
| 24 | icosi          |                |
| 25 | icosi          |                |
| 26 | icosi          |                |
| 27 | icosi          |                |
| 28 | icosi          |                |
| 29 | icosi-ennea    |                |
| 30 | triaconta
| 40 | tetraconta
| 90 | enneaconta
|100 | hecto, hecato
|1000 | chilia
|10000 | myria


icosi-kai-hena = 21
icosi-di ... icosi-ennea = 22...29
triaconta = 30
tetraconta...enneaconta = 40..90
hecto | hecato = 100
chilia = 1000
myria = 10000

The Greek numerical prefixes are used for naming things like polygons, polyhedra and polynomials:

mono = 1 di = 2 tri = 3 tetra = 4 penta = 5 hexa = 6 hepta = 7 octa = 8 ennea = 9 deca = 10 hendeca = 11 dodeca = 12 triskaideca = 13 tetrakaideca...enneakaideca = 14...19 icosa = 20 icosikaihena = 21 icosidi...icosiennea = 22...29 triaconta = 30 tetraconta...enneaconta = 40..90 hecto | hecato = 100 chilia = 1000 myria = 10000

• The word kai, meaning and, is often omitted.

• Chemistry uses a slightly different set of prefixes, two of which come from latin:

nona = 9 undeca = 11

And the rest of which are pretty much made up:

henicosa = 21 docosa...nonacosa = 22...29 hentriaconta...nonatriaconta = 31...39 kilia = 1000 dilia...nonalia = 2000...9000

• For large numbers these prefixes are used:

bi = 2 tri = 3 quadra = 4 quinta = 5 sexta = 6 septa = 7 octo = 8 nona = 9 deca = 10 undeca = 11 dodeca = 12 tredeca = 13 quattuordeca = 14 quinquedeca = 15 sexdeca = 16 septendeca = 17 octodeca = 18 novemdeca = 19 viginti = 20 unviginti...novemviginti = 21...29 triginti = 30

Some of these suffices need checking, since these are not quite the latin names of the numbers; but in the actual compounds (e.g. octillion, vigintillion) they disappear.