## Number |

A **number** is a ^{[1]} A written ^{[2]} A *number* may refer to a symbol, a word or phrase, or the mathematical object.

In ^{[3]} ^{[4]} ^{[5]} such as ^{[6]} which extend the real numbers with a ^{[4]}

Besides their practical uses, numbers have cultural significance throughout the world.^{[7]}^{[8]} For example, in Western society, the number ^{[7]} Though it is now regarded as ^{[9]} Numerology heavily influenced the development of ^{[9]}

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 ^{[10]}

- history
- main classification
- subclasses of the integers
- subclasses of the complex numbers
- extensions of the concept
- see also
- notes
- references
- external links

Numbers should be distinguished from **numerals**, the symbols used to represent numbers. The Egyptians invented the first ciphered numeral system, and the Greeks followed by mapping their counting numbers onto Ionian and Doric alphabets.^{[11]} Roman numerals, a system that used combinations of letters from the Roman alphabet, remained dominant in Europe until the spread of the superior ^{[12]} The key to the effectiveness of the system was the symbol for zero, which was developed by ancient ^{[12]}

Bones and other artifacts have been discovered with marks cut into them that many believe are ^{[13]} These tally marks may have been used for counting elapsed time, such as numbers of days, lunar cycles or keeping records of quantities, such as of animals.

A tallying system has no concept of place value (as in modern

The first known system with place value was the ^{[14]}

The first known documented use of zero dates to AD 628, and appeared in the * Brāhmasphuṭasiddhānta*, the main work of the

Brahmagupta's Brahmasphuṭasiddhanta is the first book that mentions zero as a number, hence Brahmagupta is usually considered the first to formulate the concept of zero. He gave rules of using zero with negative and positive numbers, such as 'Zero plus a positive number is a positive number, and a negative number plus zero is the negative number'. The Brahmasphutasiddhanta is the earliest known text to treat zero as a number in its own right, rather than as simply a placeholder digit in representing another number as was done by the Babylonians or as a symbol for a lack of quantity as was done by Ptolemy and the Romans.

The use of 0 as a number should be distinguished from its use as a placeholder numeral in *nfr* to denote zero balance in *Shunye* or *shunya* to refer to the concept of *void*. In mathematics texts this word often refers to the number zero.^{[15]} In a similar vein,

There are other uses of zero before Brahmagupta, though the documentation is not as complete as it is in the Brahmasphutasiddhanta.

Records show that the

The late

By 130 AD, *documented* use of a true zero in the Old World. In later *Syntaxis Mathematica* (*Almagest*), the Hellenistic zero had morphed into the

Another true zero was used in tables alongside *nulla* meaning *nothing*, not as a symbol. When division produced 0 as a remainder, *nihil*, also meaning *nothing*, was used. These medieval zeros were used by all future medieval

The abstract concept of negative numbers was recognized as early as 100–50 BC in China. * The Nine Chapters on the Mathematical Art* contains methods for finding the areas of figures; red rods were used to denote positive

During the 600s, negative numbers were in use in * Brāhmasphuṭasiddhānta* 628, who used negative numbers to produce the general form

* Liber Abaci*, 1202) and later as losses (in

As recently as the 18th century, it was common practice to ignore any negative results returned by equations on the assumption that they were meaningless, just as

It is likely that the concept of fractional numbers dates to *Elements*

The concept of

The earliest known use of irrational numbers was in the ^{[18]} The first existence proofs of irrational numbers is usually attributed to ^{[19]}

The 16th century brought final European acceptance of * Crelle,* 74),

The search for roots of

The existence of transcendental numbers^{[20]} was first established by *e* is transcendental and

The earliest known conception of mathematical

* Two New Sciences* discussed the idea of

In the 1960s,

A modern geometrical version of infinity is given by

The earliest fleeting reference to square roots of negative numbers occurred in the work of the mathematician and inventor

This was doubly unsettling since they did not even consider negative numbers to be on firm ground at the time. When

seemed capriciously inconsistent with the algebraic identity

which is valid for positive real numbers *a* and *b*, and was also used in complex number calculations with one of *a*, *b* positive and the other negative. The incorrect use of this identity, and the related identity

in the case when both *a* and *b* are negative even bedeviled *i* in place of to guard against this mistake.

The 18th century saw the work of

while

The existence of complex numbers was not completely accepted until *De Algebra tractatus*.

Also in 1799, Gauss provided the first generally accepted proof of the

*a* + *bi*, where *a* and *b* are integral, or rational (and *i* is one of the two roots of *x*^{2} + 1 = 0). His student, *a* + *bω*, where *ω* is a complex root of *x*^{3} − 1 = 0. Other such classes (called *x*^{k} − 1 = 0 for higher values of *k*. This generalization is largely due to

In 1850

*Elements* to the theory of primes; in it he proved the infinitude of the primes and the

In 240 BC,

In 1796,

Other Languages

Afrikaans: Getal

Alemannisch: Zahl

አማርኛ: ቁጥር

Ænglisc: Rīm

Аҧсшәа: Анатуралтә ахыԥхьаӡара

العربية: عدد

aragonés: Numero

ܐܪܡܝܐ: ܡܢܝܢܐ

armãneashti: Numiru

অসমীয়া: সংখ্যা

asturianu: Númberu

Atikamekw: Akitasowin

Avañe'ẽ: Papaha

azərbaycanca: Ədəd

বাংলা: সংখ্যা

Bahasa Banjar: Wilangan

Bân-lâm-gú: Sò͘-ba̍k

башҡортса: Һан

беларуская: Лік

беларуская (тарашкевіца): Лік

български: Число

Boarisch: Zoih

བོད་ཡིག: གྲངས་ཀ།

bosanski: Broj

brezhoneg: Niver

буряад: Тоо

català: Nombre

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čeština: Číslo

chiShona: Nhamba

Choctaw: Hohltina

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dansk: Tal

Deutsch: Zahl

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eesti: Arv

Ελληνικά: Αριθμός

emiliàn e rumagnòl: Nómmer

español: Número

Esperanto: Nombro

euskara: Zenbaki

فارسی: عدد

føroyskt: Tal

français: Nombre

Frysk: Getal

Fulfulde: Limle

Gaeilge: Uimhir

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galego: Número

贛語: 數

хальмг: Тойг

한국어: 수 (수학)

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हिन्दी: संख्या

hrvatski: Broj

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Ilokano: Numero

Bahasa Indonesia: Bilangan

interlingua: Numero

Ирон: Нымæц

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íslenska: Tala (stærðfræði)

italiano: Numero

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ಕನ್ನಡ: ಸಂಖ್ಯೆ

ქართული: რიცხვი

қазақша: Сан

Kiswahili: Namba

Kreyòl ayisyen: Nonm

kurdî: Hejmar

лакку: Аьдад

ລາວ: ຈຳນວນ

Latina: Numerus

latviešu: Skaitlis

Lëtzebuergesch: Zuel

lietuvių: Skaičius

Limburgs: Getal

Lingua Franca Nova: Numero

la .lojban.: namcu

Luganda: Ennamba

magyar: Szám

मैथिली: अंक

македонски: Број

Malagasy: Isa

മലയാളം: സംഖ്യ

मराठी: संख्या

Bahasa Melayu: Nombor

Mirandés: Númaro

монгол: Тоо

မြန်မာဘာသာ: ကိန်း

Nāhuatl: Tlapōhualli

Nederlands: Getal (wiskunde)

नेपाली: अंक

नेपाल भाषा: ल्याः

日本語: 数

Nordfriisk: Taal

norsk: Tall

norsk nynorsk: Tal

Nouormand: Neunmétho

Novial: Nombre

occitan: Nombre

олык марий: Шотпал

oʻzbekcha/ўзбекча: Son

ਪੰਜਾਬੀ: ਅੰਕ

Pangasinan: Numero

پنجابی: نمبر

پښتو: عدد

Patois: Nomba

Plattdüütsch: Tahl

polski: Liczba

português: Número

română: Număr

Runa Simi: Yupay

русиньскый: Чісло

русский: Число

саха тыла: Ахсаан

संस्कृतम्: संख्याः

Sängö: Nömörö

Scots: Nummer

Sesotho sa Leboa: Nomoro

shqip: Numri

sicilianu: Nùmmuru

Simple English: Number

slovenčina: Číslo (matematika)

slovenščina: Število

ślůnski: Nůmera

Soomaaliga: Tiro

کوردی: ژمارە

српски / srpski: Број

srpskohrvatski / српскохрватски: Broj

Basa Sunda: Wilangan

suomi: Luku

svenska: Tal

Tagalog: Bilang

தமிழ்: எண்

Taqbaylit: Amḍan

tarandíne: Numere

татарча/tatarça: Сан

తెలుగు: సంఖ్య

ไทย: จำนวน

ትግርኛ: ቁጽሪ

тоҷикӣ: Адад

ತುಳು: ಸಂಖ್ಯೆ

Türkçe: Sayı

Türkmençe: San

українська: Число

اردو: عدد

vèneto: Nùmaro

Tiếng Việt: Số

Võro: Arv

文言: 數

Winaray: Ihap

吴语: 数

Xitsonga: Nomboro

ייִדיש: צאל

Yorùbá: Nọ́mbà

粵語: 數

žemaitėška: Skaitlios

中文: 数