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Division is one of the four basic operations of
arithmetic Arithmetic () is an elementary part of mathematics that consists of the study of the properties of the traditional operations on numbers— addition, subtraction, multiplication, division, exponentiation, and extraction of roots. In the 19th ...
, the ways that numbers are combined to make new numbers. The other operations are
addition Addition (usually signified by the plus symbol ) is one of the four basic operations of arithmetic, the other three being subtraction, multiplication and division. The addition of two whole numbers results in the total amount or '' sum'' ...
,
subtraction Subtraction is an arithmetic operation that represents the operation of removing objects from a collection. Subtraction is signified by the minus sign, . For example, in the adjacent picture, there are peaches—meaning 5 peaches with 2 taken ...
, and
multiplication Multiplication (often denoted by the cross symbol , by the mid-line dot operator , by juxtaposition, or, on computers, by an asterisk ) is one of the four elementary mathematical operations of arithmetic, with the other ones being addi ...
. At an elementary level the division of two
natural number In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called '' cardinal ...
s is, among other possible interpretations, the process of calculating the number of times one number is contained within another. This number of times need not be an integer. For example, if 20 apples are divided evenly between 4 people, everyone receives 5 apples (see picture). The division with remainder or
Euclidean division In arithmetic, Euclidean division – or division with remainder – is the process of dividing one integer (the dividend) by another (the divisor), in a way that produces an integer quotient and a natural number remainder strictly smaller than ...
of two
natural numbers In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called '' cardinal ...
provides an integer ''quotient'', which is the number of times the second number is completely contained in the first number, 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 example, if 21 apples are divided between 4 people, everyone receives 5 apples again, and 1 apple remains. For division to always yield one number rather than a quotient plus a remainder, the natural numbers must be extended to
rational number In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all rati ...
s or
real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one- dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Ever ...
s. In these enlarged
number system A number is a mathematical object used to count, measure, and label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with number words. More universally, individual numbers can ...
s, division is the inverse operation to multiplication, that is means , as long as is not zero. If , then this is a
division by zero In mathematics, division by zero is division where the divisor (denominator) is zero. Such a division can be formally expressed as \tfrac, where is the dividend (numerator). In ordinary arithmetic, the expression has no meaning, as there is ...
, which is not defined. In the 21-apples example, everyone would receive 5 apple and a quarter of an apple, thus avoiding any leftover. Both forms of division appear in various
algebraic structure In mathematics, an algebraic structure consists of a nonempty set ''A'' (called the underlying set, carrier set or domain), a collection of operations on ''A'' (typically binary operations such as addition and multiplication), and a finite set ...
s, different ways of defining mathematical structure. Those in which a Euclidean division (with remainder) is defined are called
Euclidean domain In mathematics, more specifically in ring theory, a Euclidean domain (also called a Euclidean ring) is an integral domain that can be endowed with a Euclidean function which allows a suitable generalization of the Euclidean division of integers. ...
s and include
polynomial ring In mathematics, especially in the field of algebra, a polynomial ring or polynomial algebra is a ring (which is also a commutative algebra) formed from the set of polynomials in one or more indeterminates (traditionally also called variable ...
s in one indeterminate (which define multiplication and addition over single-variabled formulas). Those in which a division (with a single result) by all nonzero elements is defined are called fields and
division ring In algebra, a division ring, also called a skew field, is a nontrivial ring in which division by nonzero elements is defined. Specifically, it is a nontrivial ring in which every nonzero element has a multiplicative inverse, that is, an element ...
s. In a ring the elements by which division is always possible are called the units (for example, 1 and −1 in the ring of integers). Another generalization of division to algebraic structures is the
quotient group A quotient group or factor group is a mathematical group obtained by aggregating similar elements of a larger group using an equivalence relation that preserves some of the group structure (the rest of the structure is "factored" out). For examp ...
, in which the result of "division" is a group rather than a number.

# Introduction

The simplest way of viewing division is in terms of
quotition and partition In arithmetic, quotition and partition are two ways of viewing fractions and division. In quotition division one asks, "how many parts are there?"; While in partition division one asks, "what is the size of each part?". For example, the expressio ...
: from the quotition perspective, means the number of 5s that must be added to get 20. In terms of partition, means the size of each of 5 parts into which a set of size 20 is divided. For example, 20 apples divide into five groups of four apples, meaning that ''twenty divided by five is equal to four''. This is denoted as , or . What is being divided is called the ''dividend'', which is divided by the ''divisor'', and the result is called the ''quotient''. In the example, 20 is the dividend, 5 is the divisor, and 4 is the quotient. Unlike the other basic operations, when dividing natural numbers there is sometimes a
remainder In mathematics, the remainder is the amount "left over" after performing some computation. In arithmetic, the remainder is the integer "left over" after dividing one integer by another to produce an integer quotient (integer division). In algeb ...
that will not go evenly into the dividend; for example, leaves a remainder of 1, as 10 is not a multiple of 3. Sometimes this remainder is added to the quotient as a
fractional part The fractional part or decimal part of a non‐negative real number x is the excess beyond that number's integer part. If the latter is defined as the largest integer not greater than , called floor of or \lfloor x\rfloor, its fractional part can ...
, so is equal to or , but in the context of integer division, where numbers have no fractional part, the remainder is kept separately (or exceptionally, discarded or rounded). When the remainder is kept as a fraction, it leads to a
rational number In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all rati ...
. The set of all rational numbers is created by extending the integers with all possible results of divisions of integers. Unlike multiplication and addition, division is not
commutative In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name of ...
, meaning that is not always equal to . Division is also not, in general,
associative In mathematics, the associative property is a property of some binary operations, which means that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a valid rule of replaceme ...
, meaning that when dividing multiple times, the order of division can change the result. For example, , but (where the use of parentheses indicates that the operations inside parentheses are performed before the operations outside parentheses). Division is traditionally considered as left-associative. That is, if there are multiple divisions in a row, the order of calculation goes from left to right:George Mark Bergman
Order of arithmetic operations
Education Place

: $a / b / c = \left(a / b\right) / c = a / \left(b \times c\right) \;\ne\; a/\left(b/c\right)= \left(a\times c\right)/b.$ Division is right-distributive over addition and subtraction, in the sense that : $\frac = \left(a \pm b\right) / c = \left(a/c\right)\pm \left(b/c\right) =\frac \pm \frac.$ This is the same for
multiplication Multiplication (often denoted by the cross symbol , by the mid-line dot operator , by juxtaposition, or, on computers, by an asterisk ) is one of the four elementary mathematical operations of arithmetic, with the other ones being addi ...
, as $\left(a + b\right) \times c = a \times c + b \times c$. However, division is ''not'' left-distributive, as : $\frac = a / \left(b + c\right) \;\ne\; \left(a/b\right) + \left(a/c\right) = \frac.$   For example $\frac = \frac = 2 ,$ but $\frac + \frac = 6+3 = 9 .$ This is unlike the case in multiplication, which is both left-distributive and right-distributive, and thus distributive.

# Notation

Division is often shown in algebra and science by placing the ''dividend'' over the ''divisor'' with a horizontal line, also called a fraction bar, between them. For example, "''a'' divided by ''b''" can written as: :$\frac ab$ which can also be read out loud as "divide ''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 slash, then the ''divisor'' (or denominator), as follows: :$a/b$ This is the usual way of specifying division in most computer
programming language A programming language is a system of notation for writing computer programs. Most programming languages are text-based formal languages, but they may also be graphical. They are a kind of computer language. The description of a programmin ...
s, since it can easily be typed as a simple sequence of
ASCII ASCII ( ), abbreviated from American Standard Code for Information Interchange, is a character encoding standard for electronic communication. ASCII codes represent text in computers, telecommunications equipment, and other devices. Because of ...
characters. (It is also the only notation used for
quotient object In category theory, a branch of mathematics, a subobject is, roughly speaking, an object that sits inside another object in the same category. The notion is a generalization of concepts such as subsets from set theory, subgroups from group theory, ...
s in
abstract algebra In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field. The ter ...
.) Some
mathematical software Mathematical software is software used to model, analyze or calculate numeric, symbolic or geometric data. Evolution of mathematical software Numerical analysis and symbolic computation had been in most important place of the subject, but other ...
, such as
MATLAB MATLAB (an abbreviation of "MATrix LABoratory") is a proprietary multi-paradigm programming language and numeric computing environment developed by MathWorks. MATLAB allows matrix manipulations, plotting of functions and data, implementa ...
and
GNU Octave GNU Octave is a high-level programming language primarily intended for scientific computing and numerical computation. Octave helps in solving linear and nonlinear problems numerically, and for performing other numerical experiments using a lang ...
, allows the operands to be written in the reverse order by using the
backslash The backslash is a typographical mark used mainly in computing and mathematics. It is the mirror image of the common slash . It is a relatively recent mark, first documented in the 1930s. History , efforts to identify either the origin o ...
as the division operator: :$b\backslash a$ A typographical variation halfway between these two forms uses a
solidus Solidus (Latin for "solid") may refer to: * Solidus (coin) The ''solidus'' (Latin 'solid';  ''solidi'') or nomisma ( grc-gre, νόμισμα, ''nómisma'',  'coin') was a highly pure gold coin issued in the Late Roman Empire and Byz ...
(fraction slash), but elevates the dividend and lowers the divisor: :$^\!/_$ Any of these forms can be used to display a
fraction A fraction (from la, fractus, "broken") represents a part of a whole or, more generally, any number of equal parts. When spoken in everyday English, a fraction describes how many parts of a certain size there are, for example, one-half, eight ...
. A fraction is a division expression where both dividend and divisor are integers (typically called the ''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
division sign The division sign () is a symbol consisting of a short horizontal line with a dot above and another dot below, used in Anglophone countries to indicate mathematical division. However, this usage, though widespread in some countries, is not u ...
(÷, also known as
obelus An obelus (plural: obeluses or obeli) is a term in typography that refers to a historical mark which has resolved to three modern meanings: * Division sign * Dagger * Commercial minus sign (limited geographical area of use) The word "obelu ...
though the term has additional meanings), common in arithmetic, in this manner: :$a \div b$ This form is infrequent except in elementary arithmetic. ISO 80000-2-9.6 states it should not be used. This division sign is also used alone to represent the division operation itself, as for instance as a label on a key of a
calculator An electronic calculator is typically a portable electronic device used to perform calculations, ranging from basic arithmetic to complex mathematics. The first solid-state electronic calculator was created in the early 1960s. Pocket-sized ...
. The obelus was introduced by Swiss mathematician
Johann Rahn Johann Rahn (Latinised form Rhonius) (10 March 1622 – 25 May 1676) was a Swiss mathematician who is credited with the first use of the division sign, ÷ (a repurposed obelus variant) and the therefore sign, ∴. The symbols were used in '' ...
in 1659 in ''Teutsche Algebra''. The ÷ symbol is used to indicate subtraction in some European countries, so its use may be misunderstood. In some non-
English English usually refers to: * English language * English people English may also refer to: Peoples, culture, and language * ''English'', an adjective for something of, from, or related to England ** English national ...
-speaking countries, a colon is used to denote division: :$a : b$ This notation was introduced by
Gottfried Wilhelm Leibniz Gottfried Wilhelm (von) Leibniz . ( – 14 November 1716) was a German polymath active as a mathematician, philosopher, scientist and diplomat. He is one of the most prominent figures in both the history of philosophy and the history of ma ...
in his 1684 ''Acta eruditorum''. Leibniz disliked having separate symbols for ratio and division. However, in English usage the colon is restricted to expressing the related concept of
ratio In mathematics, a ratio shows how many times one number contains another. For example, if there are eight oranges and six lemons in a bowl of fruit, then the ratio of oranges to lemons is eight to six (that is, 8:6, which is equivalent to the ...
s. Since the 19th century, US textbooks have used $b\right)a$ or $b \overline$ to denote ''a'' divided by ''b'', especially when discussing long division. The history of this notation is not entirely clear because it evolved over time.

# Computing

## Manual methods

Division is often introduced through the notion of "sharing out" a set of objects, for example a pile of lollies, into a number of equal portions. Distributing the objects several at a time in each round of sharing to each portion leads to the idea of ' chunking' a form of division where one repeatedly subtracts multiples of the divisor from the dividend itself. By allowing one to subtract more multiples than what the partial remainder allows at a given stage, more flexible methods, such as the bidirectional variant of chunking, can be developed as well. More systematically and more efficiently, two integers can be divided with pencil and paper with the method of short division, if the divisor is small, or long division, if the divisor is larger. If the dividend has a fractional part (expressed as a
decimal fraction The decimal numeral system (also called the base-ten positional numeral system and denary or decanary) is the standard system for denoting integer and non-integer numbers. It is the extension to non-integer numbers of the Hindu–Arabic numer ...
), one can continue the procedure past the ones place as far as desired. If the divisor has a fractional part, one can restate the problem by moving the decimal to the right in both numbers until the divisor has no fraction, which can make the problem easier to solve (e.g., 10/2.5 = 100/25 = 4). Division can be calculated with an
abacus The abacus (''plural'' abaci or abacuses), also called a counting frame, is a calculating tool which has been used since ancient times. It was used in the ancient Near East, Europe, China, and Russia, centuries before the adoption of the Hin ...
. Logarithm tables can be used to divide two numbers, by subtracting the two numbers' logarithms, then looking up the antilogarithm of the result. Division can be calculated with a
slide rule The slide rule is a mechanical analog computer which is used primarily for multiplication and division, and for functions such as exponents, roots, logarithms, and trigonometry. It is not typically designed for addition or subtraction, which i ...
by aligning the divisor on the C scale with the dividend on the D scale. The quotient can be found on the D scale where it is aligned with the left index on the C scale. The user is responsible, however, for mentally keeping track of the decimal point.

## By computer

Modern
calculator An electronic calculator is typically a portable electronic device used to perform calculations, ranging from basic arithmetic to complex mathematics. The first solid-state electronic calculator was created in the early 1960s. Pocket-sized ...
s and
computer A computer is a machine that can be programmed to carry out sequences of arithmetic or logical operations (computation) automatically. Modern digital electronic computers can perform generic sets of operations known as programs. These pr ...
s compute division either by methods similar to long division, or by faster methods; see Division algorithm. In
modular arithmetic In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" when reaching a certain value, called the modulus. The modern approach to modular arithmetic was developed by Carl Friedrich Gauss in his bo ...
(modulo a prime number) and for
real numbers In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
, nonzero numbers have a
multiplicative inverse In mathematics, a multiplicative inverse or reciprocal for a number ''x'', denoted by 1/''x'' or ''x''−1, is a number which when multiplied by ''x'' yields the multiplicative identity, 1. The multiplicative inverse of a fraction ''a''/'' ...
. In these cases, a division by may be computed as the product by the multiplicative inverse of . This approach is often associated with the faster methods in computer arithmetic.

# Division in different contexts

## Euclidean division

Euclidean division is the mathematical formulation of the outcome of the usual process of division of integers. It asserts that, given two integers, ''a'', the ''dividend'', and ''b'', the ''divisor'', such that ''b'' ≠ 0, there are unique integers ''q'', the ''quotient'', and ''r'', the remainder, such that ''a'' = ''bq'' + ''r'' and 0 ≤ ''r'' < , where denotes the
absolute value In mathematics, the absolute value or modulus of a real number x, is the non-negative value without regard to its sign. Namely, , x, =x if is a positive number, and , x, =-x if x is negative (in which case negating x makes -x positive), a ...
of ''b''.

## Of integers

Integers are not closed under division. Apart from division by zero being undefined, the quotient is not an integer unless the dividend is an integer multiple of the divisor. For example, 26 cannot be divided by 11 to give an integer. Such a case uses one of five approaches: # Say that 26 cannot be divided by 11; division becomes a
partial function In mathematics, a partial function from a set to a set is a function from a subset of (possibly itself) to . The subset , that is, the domain of viewed as a function, is called the domain of definition of . If equals , that is, if is ...
. # Give an approximate answer as a floating-point number. This is the approach usually taken in numerical computation. # Give the answer as a
fraction A fraction (from la, fractus, "broken") represents a part of a whole or, more generally, any number of equal parts. When spoken in everyday English, a fraction describes how many parts of a certain size there are, for example, one-half, eight ...
representing a
rational number In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all rati ...
, so the result of the division of 26 by 11 is $\tfrac$ (or as a
mixed number A fraction (from la, fractus, "broken") represents a part of a whole or, more generally, any number of equal parts. When spoken in everyday English, a fraction describes how many parts of a certain size there are, for example, one-half, eight ...
, so $\tfrac = 2 \tfrac 4.$) Usually the resulting fraction should be simplified: the result of the division of 52 by 22 is also $\tfrac$. This simplification may be done by factoring out the
greatest common divisor In mathematics, the greatest common divisor (GCD) of two or more integers, which are not all zero, is the largest positive integer that divides each of the integers. For two integers ''x'', ''y'', the greatest common divisor of ''x'' and ''y'' is ...
. # Give the answer as an integer ''
quotient In arithmetic, a quotient (from lat, quotiens 'how many times', pronounced ) is a quantity produced by the division of two numbers. The quotient has widespread use throughout mathematics, and is commonly referred to as the integer part of a ...
'' and a ''
remainder In mathematics, the remainder is the amount "left over" after performing some computation. In arithmetic, the remainder is the integer "left over" after dividing one integer by another to produce an integer quotient (integer division). In algeb ...
'', so $\tfrac = 2 \mbox 4.$ To make the distinction with the previous case, this division, with two integers as result, is sometimes called ''
Euclidean division In arithmetic, Euclidean division – or division with remainder – is the process of dividing one integer (the dividend) by another (the divisor), in a way that produces an integer quotient and a natural number remainder strictly smaller than ...
'', because it is the basis of the
Euclidean algorithm In mathematics, the Euclidean algorithm,Some widely used textbooks, such as I. N. Herstein's ''Topics in Algebra'' and Serge Lang's ''Algebra'', use the term "Euclidean algorithm" to refer to Euclidean division or Euclid's algorithm, is an ef ...
. # Give the integer quotient as the answer, so $\tfrac = 2.$ This is the ''
floor function In mathematics and computer science, the floor function is the function that takes as input a real number , and gives as output the greatest integer less than or equal to , denoted or . Similarly, the ceiling function maps to the least int ...
'' applied to case 2 or 3. It is sometimes called integer division, and denoted by "//". Dividing integers in a
computer program A computer program is a sequence or set of instructions in a programming language for a computer to execute. Computer programs are one component of software, which also includes documentation and other intangible components. A computer program ...
requires special care. Some
programming language A programming language is a system of notation for writing computer programs. Most programming languages are text-based formal languages, but they may also be graphical. They are a kind of computer language. The description of a programmin ...
s treat integer division as in case 5 above, so the answer is an integer. Other languages, such as
MATLAB MATLAB (an abbreviation of "MATrix LABoratory") is a proprietary multi-paradigm programming language and numeric computing environment developed by MathWorks. MATLAB allows matrix manipulations, plotting of functions and data, implementa ...
and every
computer algebra system A computer algebra system (CAS) or symbolic algebra system (SAS) is any mathematical software with the ability to manipulate mathematical expressions in a way similar to the traditional manual computations of mathematicians and scientists. The d ...
return a rational number as the answer, as in case 3 above. These languages also provide functions to get the results of the other cases, either directly or from the result of case 3. Names and symbols used for integer division include div, /, \, and %. Definitions vary regarding integer division when the dividend or the divisor is negative:
rounding Rounding means replacing a number with an approximate value that has a shorter, simpler, or more explicit representation. For example, replacing $with$, the fraction 312/937 with 1/3, or the expression with . Rounding is often done to ob ...
may be toward zero (so called T-division) or toward −∞ (F-division); rarer styles can occur – see
modulo operation In computing, the modulo operation returns the remainder or signed remainder of a division, after one number is divided by another (called the '' modulus'' of the operation). Given two positive numbers and , modulo (often abbreviated as ) is th ...
for the details.
Divisibility rule A divisibility rule is a shorthand and useful way of determining whether a given integer is divisible by a fixed divisor without performing the division, usually by examining its digits. Although there are divisibility tests for numbers in any ra ...
s can sometimes be used to quickly determine whether one integer divides exactly into another.

## Of rational numbers

The result of dividing two
rational number In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all rati ...
s is another rational number when the divisor is not 0. The division of two rational numbers ''p''/''q'' and ''r''/''s'' can be computed as :$= \times = .$ All four quantities are integers, and only ''p'' may be 0. This definition ensures that division is the inverse operation of
multiplication Multiplication (often denoted by the cross symbol , by the mid-line dot operator , by juxtaposition, or, on computers, by an asterisk ) is one of the four elementary mathematical operations of arithmetic, with the other ones being addi ...
.

## Of real numbers

Division of two
real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one- dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Ever ...
s results in another real number (when the divisor is nonzero). It is defined such that ''a''/''b'' = ''c'' if and only if ''a'' = ''cb'' and ''b'' ≠ 0.

## Of complex numbers

Dividing two
complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...
s (when the divisor is nonzero) results in another complex number, which is found using the conjugate of the denominator: :$= = = + i.$ This process of multiplying and dividing by $r-is$ is called 'realisation' or (by analogy) rationalisation. All four quantities ''p'', ''q'', ''r'', ''s'' are real numbers, and ''r'' and ''s'' may not both be 0. Division for complex numbers expressed in polar form is simpler than the definition above: :$= = e^.$ Again all four quantities ''p'', ''q'', ''r'', ''s'' are real numbers, and ''r'' may not be 0.

## Of polynomials

One can define the division operation for
polynomial In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An exampl ...
s in one variable over a field. Then, as in the case of integers, one has a remainder. See Euclidean division of polynomials, and, for hand-written computation,
polynomial long division In algebra, polynomial long division is an algorithm for dividing a polynomial by another polynomial of the same or lower degree, a generalized version of the familiar arithmetic technique called long division. It can be done easily by hand, beca ...
or
synthetic division In algebra, synthetic division is a method for manually performing Euclidean division of polynomials, with less writing and fewer calculations than long division. It is mostly taught for division by linear monic polynomials (known as the Ruffin ...
.

## Of matrices

One can define a division operation for matrices. The usual way to do this is to define , where denotes the inverse of ''B'', but it is far more common to write out explicitly to avoid confusion. An elementwise division can also be defined in terms of the Hadamard product.

### Left and right division

Because
matrix multiplication In mathematics, particularly in linear algebra, matrix multiplication is a binary operation that produces a matrix from two matrices. For matrix multiplication, the number of columns in the first matrix must be equal to the number of rows in the ...
is not
commutative In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name of ...
, one can also define a left division or so-called ''backslash-division'' as . For this to be well defined, need not exist, however does need to exist. To avoid confusion, division as defined by is sometimes called ''right division'' or ''slash-division'' in this context. Note that with left and right division defined this way, is in general not the same as , nor is the same as . However, it holds that and .

### Pseudoinverse

To avoid problems when and/or do not exist, division can also be defined as multiplication by the
pseudoinverse In mathematics, and in particular, algebra, a generalized inverse (or, g-inverse) of an element ''x'' is an element ''y'' that has some properties of an inverse element but not necessarily all of them. The purpose of constructing a generalized inv ...
. That is, and , where and denote the pseudoinverses of ''A'' and ''B''.

## Abstract algebra

In
abstract algebra In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field. The ter ...
, given a
magma Magma () is the molten or semi-molten natural material from which all igneous rocks are formed. Magma is found beneath the surface of the Earth, and evidence of magmatism has also been discovered on other terrestrial planets and some natural ...
with binary operation ∗ (which could nominally be termed multiplication), left division of ''b'' by ''a'' (written ) is typically defined as the solution ''x'' to the equation , if this exists and is unique. Similarly, right division of ''b'' by ''a'' (written ) is the solution ''y'' to the equation . Division in this sense does not require ∗ to have any particular properties (such as commutativity, associativity, or an identity element). "Division" in the sense of "cancellation" can be done in any magma by an element with the cancellation property. Examples include matrix algebras and
quaternion In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by the Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. Hamilton defined a quat ...
algebras. A
quasigroup In mathematics, especially in abstract algebra, a quasigroup is an algebraic structure resembling a group in the sense that "division" is always possible. Quasigroups differ from groups mainly in that they need not be associative and need not hav ...
is a structure in which division is always possible, even without an identity element and hence inverses. In an
integral domain In mathematics, specifically abstract algebra, an integral domain is a nonzero commutative ring in which the product of any two nonzero elements is nonzero. Integral domains are generalizations of the ring of integers and provide a natural se ...
, where not every element need have an inverse, ''division'' by a cancellative element ''a'' can still be performed on elements of the form ''ab'' or ''ca'' by left or right cancellation, respectively. If a ring is finite and every nonzero element is cancellative, then by an application of the pigeonhole principle, every nonzero element of the ring is invertible, and ''division'' by any nonzero element is possible. To learn about when ''algebras'' (in the technical sense) have a division operation, refer to the page on
division algebra In the field of mathematics called abstract algebra, a division algebra is, roughly speaking, an algebra over a field in which division, except by zero, is always possible. Definitions Formally, we start with a non-zero algebra ''D'' over a fiel ...
s. In particular Bott periodicity can be used to show that any real
normed division algebra In mathematics, Hurwitz's theorem is a theorem of Adolf Hurwitz (1859–1919), published posthumously in 1923, solving the Hurwitz problem for finite-dimensional unital real non-associative algebras endowed with a positive-definite quadratic f ...
must be
isomorphic In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word i ...
to either the real numbers R, the
complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...
s C, the
quaternion In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by the Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. Hamilton defined a quat ...
s H, or the
octonion In mathematics, the octonions are a normed division algebra over the real numbers, a kind of hypercomplex number system. The octonions are usually represented by the capital letter O, using boldface or blackboard bold \mathbb O. Octonions have e ...
s O.

## Calculus

The
derivative In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. ...
of the quotient of two functions is given by the
quotient rule In calculus, the quotient rule is a method of finding the derivative of a function that is the ratio of two differentiable functions. Let h(x)=f(x)/g(x), where both and are differentiable and g(x)\neq 0. The quotient rule states that the deriv ...
: :$\text{'} = \frac.$

# Division by zero

Division of any number by
zero 0 (zero) is a number representing an empty quantity. In place-value notation such as the Hindu–Arabic numeral system, 0 also serves as a placeholder numerical digit, which works by multiplying digits to the left of 0 by the radix, usua ...
in most mathematical systems is undefined, because zero multiplied by any finite number always results in a product of zero. Entry of such an expression into most
calculator An electronic calculator is typically a portable electronic device used to perform calculations, ranging from basic arithmetic to complex mathematics. The first solid-state electronic calculator was created in the early 1960s. Pocket-sized ...
s produces an error message. However, in certain higher level mathematics division by zero is possible by the zero ring and algebras such as
wheels A wheel is a circular component that is intended to rotate on an axle bearing. The wheel is one of the key components of the wheel and axle which is one of the six simple machines. Wheels, in conjunction with axles, allow heavy objects to be ...
.Jesper Carlström
"On Division by Zero"
Retrieved October 23, 2018
In these algebras, the meaning of division is different from traditional definitions.

* 400AD Sunzi division algorithm *
Division by two In mathematics, division by two or halving has also been called mediation or dimidiation. The treatment of this as a different operation from multiplication and division by other numbers goes back to the ancient Egyptians, whose multiplication algor ...
* Galley division *
Inverse element In mathematics, the concept of an inverse element generalises the concepts of opposite () and reciprocal () of numbers. Given an operation denoted here , and an identity element denoted , if , one says that is a left inverse of , and that ...
*
Order of operations In mathematics and computer programming, the order of operations (or operator precedence) is a collection of rules that reflect conventions about which procedures to perform first in order to evaluate a given mathematical expression. For exampl ...
*
Repeating decimal A repeating decimal or recurring decimal is decimal representation of a number whose digits are periodic (repeating its values at regular intervals) and the infinitely repeated portion is not zero. It can be shown that a number is rational if ...