## Division (mathematics) |

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**Division** is one of the four basic operations of ^{[1]}^{:7} For example, in the picture on the right, the 20 apples are divided into four groups of five apples, meaning that *twenty divided by five gives four*, or *four is the result of division of twenty by five*. This is denoted as 20 / 5 = 4, 20 ÷ 5 = 4, or 20/5 = 4.^{[2]}

Division can be viewed

Division is the inverse of multiplication; if *a* × *b* = *c*, then *a* = *c* ÷ *b*, as long as *b* is not zero. ^{[3]}^{:246} because if *b* = 0, then *a* cannot be deduced from *b* and *c*, as then *c* will always equal zero regardless of *a*. In some contexts, division by zero can be defined although to a limited extent, and ^{[a]}^{[2]}^{[4]}

In division, the *dividend* is divided by the *divisor* to get a *quotient*. In the above example, 20 is the dividend, five is the divisor, and four is the quotient. In some cases, the divisor may not be contained fully by the dividend; for example, 10 ÷ 3 leaves a *remainder* of one, as 10 is not a multiple of three. Sometimes this remainder is added to the quotient as a ^{[5]}

Besides dividing apples, division can be applied to other physical and abstract objects. Division has been defined in several contexts, such as for the

Division is the most mentally difficult of the four basic operations of arithmetic^{[citation needed]}, but the discipline and mastery of it provides an educational bridge from arithmetic to

- notation
- computing
- properties
- euclidean division
- of integers
- of rational numbers
- of real numbers
- by zero
- of complex numbers
- of polynomials
- of matrices
- abstract algebra
- calculus
- see also
- notes
- references
- external links

Calculation results | |
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Division (÷) | |

nth root | |

Division is often shown in algebra and science by placing the *dividend* over the *divisor* with a horizontal line, also called a *a* divided by *b* is written

This can be read out loud as "*a* divided by *b*", "*a* by *b*" or "*a* over *b*". A way to express division all on one line is to write the *dividend* (or numerator), then a *divisor* (or denominator), like this:

This is the usual way to specify division in most computer

A typographical variation halfway between these two forms uses a

Any of these forms can be used to display a *numerator* and *denominator*), and there is no implication that the division must be evaluated further. A second way to show division is to use the

This form is infrequent except in elementary arithmetic. *Teutsche Algebra*.^{[6]}^{:211}

In some non-*a* : *b*.^{[7]} This notation was introduced by *Acta eruditorum*.^{[6]}^{:295} Leibniz disliked having separate symbols for ratio and division. However, in English usage the *a* is to *b*").

Since the 19th century US textbooks have used or to denote *a* divided by *b*, especially when discussing ^{[8]}

Other Languages

Alemannisch: Division (Mathematik)

العربية: قسمة (رياضيات)

aragonés: División

অসমীয়া: হৰণ

Aymar aru: Jaljayaña

azərbaycanca: Bölmə (riyaziyyat)

تۆرکجه: بؤلمه

বাংলা: ভাগ

Bân-lâm-gú: Tû-hoat

башҡортса: Бүлеү

беларуская: Дзяленне

беларуская (тарашкевіца): Дзяленьне

български: Деление

བོད་ཡིག: བགོ་རྩིས།

bosanski: Dijeljenje (matematika)

brezhoneg: Rannadur

català: Divisió

čeština: Dělení

chiShona: Kugovanisa

Cymraeg: Rhannu (mathemateg)

dansk: Division (matematik)

Deutsch: Division (Mathematik)

eesti: Jagamine

Ελληνικά: Διαίρεση

español: División (matemática)

Esperanto: Divido

euskara: Zatiketa (matematika)

فارسی: تقسیم

føroyskt: Brot

français: Division

Gàidhlig: Roinn (matamataig)

galego: División (matemáticas)

贛語: 除法

客家語/Hak-kâ-ngî: Chhù-fap

хальмг: Хувалһан

한국어: 나눗셈

हिन्दी: विभाजन (गणित)

hrvatski: Dijeljenje

Ido: Divido (matematiko)

Bahasa Indonesia: Pembagian

interlingua: Division (mathematica)

isiXhosa: Ukwaba/ukwahlula-hlula

íslenska: Deiling

italiano: Divisione (matematica)

עברית: חילוק

Basa Jawa: Paran

ಕನ್ನಡ: ಭಾಗಾಕಾರ

қазақша: Бөлу

Кыргызча: Бөлүү

Latina: Divisio (mathematica)

latviešu: Dalīšana

lietuvių: Dalyba

magyar: Osztás

македонски: Делење

മലയാളം: ഹരണം

مصرى: قسمه

Mìng-dĕ̤ng-ngṳ̄: Dṳ̀-huák

Mirandés: Debison (matemática)

Nederlands: Delen

日本語: 除法

norsk: Divisjon (matematikk)

norsk nynorsk: Divisjon i matematikk

Novial: Divisione

occitan: Division

ଓଡ଼ିଆ: ହରଣ (ଗଣିତ)

ਪੰਜਾਬੀ: ਤਕਸੀਮ

پښتو: تقسيم

Patois: Divijan (matimatix)

polski: Dzielenie

português: Divisão

română: Împărțire (matematică)

Runa Simi: Rakiy

русский: Деление (математика)

Scots: Diveesion (mathematics)

sicilianu: Spartuta

Simple English: Division (mathematics)

slovenčina: Delenie (matematika)

slovenščina: Deljenje

Soomaaliga: U qeybin

کوردی: دابەشکردن

српски / srpski: Дељење

srpskohrvatski / српскохрватски: Dijeljenje

suomi: Jakolasku

svenska: Division (matematik)

Tagalog: Paghahati

தமிழ்: வகுத்தல் (கணிதம்)

తెలుగు: భాగహారం

ไทย: การหาร

Türkçe: Bölme

українська: Ділення

اردو: تقسیم (ریاضی)

Vahcuengh: Cawzfap

vèneto: Divixion

Tiếng Việt: Phép chia

Winaray: Pagtunga-tunga

吴语: 除法

ייִדיש: צעטיילונג

粵語: 除

中文: 除法

Lingua Franca Nova: Divide (matematica)