# Division (mathematics)

20 ÷ 4 = 5 with apples. This is said verbally, "twenty divided by four equals five.

Division is one of the four basic operations of arithmetic, the others being addition, subtraction, and multiplication.

At an elementary level the division of two natural numbers is –among other possible interpretations– the process of calculating the number of times one number is contained within another one.[1]:7 This number of times is not always an integer, and this led to two different concepts.

The division with remainder or Euclidean division of two natural numbers provides a quotient, which is the number of times the second one is contained in the first one, and a remainder, which is the part of the first number that remains, when in the course of computing the quotient, no further full chunk of the size of the second number can be allocated.

For a modification of this division to yield only one single result, the natural numbers must be extended to rational numbers or real numbers. In these enlarged number systems, division is the inverse operation to multiplication, that is a = c ÷ b means a × b = c, as long as b is not zero—if b = 0, then this is a division by zero, which is not defined.[a][4]:246

Both forms of divisions appear in various algebraic structures. Those in which a Euclidean division (with remainder) is defined are called Euclidean domains and include polynomial rings in one indeterminate. Those in which a division (with a single result) by all nonzero elements is defined are called fields and division rings. In a ring the elements by which division is always possible are called the units; e.g., within the ring of integers the units are 1 and –1.

## Introduction

In its most simple form, division can be viewed either as a quotition or a partition. In terms of quotition, 20 ÷ 5 means the number of 5s that must be added to get 20. In terms of partition, 20 ÷ 5 means the size of each of 5 parts into which a set of size 20 is divided. For example, 20 apples divide into four groups of five apples, meaning that twenty divided by five is equal to four. This is denoted as 20 / 5 = 4, 20 ÷ 5 = 4, or 20/5 = 4.[2] Notationally, the dividend is divided by the divisor to get a quotient. In the example, 20 is the dividend, five is the divisor, and four is the quotient.

Unlike the other basic operations, when dividing natural numbers there is sometimes a remainder that will not go evenly into 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 fractional part, so 10 ÷ 3 is equal to 31/3 or 3.33..., but in the context of integer division, where numbers have no fractional part, the remainder is kept separately or discarded.[5] When the remainder is kept as a fraction, it leads to a rational number. The set of all rational numbers is created by every possible division using integers. In modern mathematical terms, this is known as extending the system.

Unlike multiplication and addition, Division is not commutative, meaning that a ÷ b is not always equal to b ÷ a.[6] Division is also not associative, meaning that when dividing multiple times, the order of the division changes the answer to the problem.[7] For example, (20 ÷ 5) ÷ 2 = 2, but 20 ÷ (5 ÷ 2) = 8, where the parentheses mean that the operation inside the parentheses is performed before the operations outside.

Division is, however, distributive. This means that (a+b) ÷ c = (a ÷ c) + (b ÷ c) for every number. Specifically, division has the right-distributive property over addition and subtraction. That means:

${\displaystyle {\frac {a+b}{c}}=(a+b)\div c={\frac {a}{c}}+{\frac {b}{c}}}$

This is the same as multiplication: ${\displaystyle (a+b)\times c=a\times c+b\times c}$. However, division is not left-distributive:

${\displaystyle {\frac {a}{b+c}}=a\div (b+c)=\left({\frac {b}{a}}+{\frac {c}{a}}\right)^{-1}\neq {\frac {a}{b}}+{\frac {a}{c}}}$

This is unlike multiplication.

If there are multiple divisions in a row the order of calculation traditionally goes from left to right[8][9], which is called left-associative:

${\displaystyle a\div b\div c=(a\div b)\div c=a\div (b\times c)=a\times b^{-1}\times c^{-1}}$.

Other Languages
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།
català: Divisió
čeština: Dělení
chiShona: Kugovanisa
eesti: Jagamine
Ελληνικά: Διαίρεση
Esperanto: Divido
فارسی: تقسیم
føroyskt: Brot
français: Division

хальмг: Хувалһан
한국어: 나눗셈
hrvatski: Dijeljenje
Bahasa Indonesia: Pembagian
íslenska: Deiling
עברית: חילוק
Basa Jawa: Paran
ಕನ್ನಡ: ಭಾಗಾಕಾರ
қазақша: Бөлу
Кыргызча: Бөлүү
latviešu: Dalīšana
lietuvių: Dalyba
Lingua Franca Nova: Divide (matematica)
magyar: Osztás
македонски: Делење
മലയാളം: ഹരണം
مصرى: قسمه
Mìng-dĕ̤ng-ngṳ̄: Dṳ̀-huák
Nederlands: Delen

norsk nynorsk: Divisjon i matematikk
Novial: Divisione
occitan: Division
ଓଡ଼ିଆ: ହରଣ (ଗଣିତ)
ਪੰਜਾਬੀ: ਤਕਸੀਮ
پښتو: تقسيم
polski: Dzielenie
português: Divisão
Runa Simi: Rakiy
sicilianu: Spartuta
Simple English: Division (mathematics)
slovenščina: Deljenje
Soomaaliga: U qeybin
کوردی: دابەشکردن
српски / srpski: Дељење
srpskohrvatski / српскохрватски: Dijeljenje