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Division is one of the four basic operations of
arithmetic Arithmetic (from the Ancient Greek, Greek wikt:en:ἀριθμός#Ancient Greek, ἀριθμός ''arithmos'', 'number' and wikt:en:τική#Ancient Greek, τική wikt:en:τέχνη#Ancient Greek, έχνη ''tiké échne', 'art' or 'cr ...
, the ways that numbers are combined to make new numbers. The other operations are
addition Addition (usually signified by the plus symbol The plus and minus signs, and , are mathematical symbol A mathematical symbol is a figure or a combination of figures that is used to represent a mathematical object A mathematical object is an ...

addition
,
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 ...

subtraction
, 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 arithmetic, elementary Operation (mathematics), mathematical operations ...

multiplication
. 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 total order, ordering (as in "this is the ''third'' largest city in the country"). In common mathematical terminology, w ...
s is, among other possible interpretations, the process of calculating the number of times one number is contained within another. This number of times is not always an
integer An integer (from the Latin Latin (, or , ) is a classical language A classical language is a language A language is a structured system of communication Communication (from Latin ''communicare'', meaning "to share" or "to ...
(a number that can be obtained using the other arithmetic operations on the natural numbers). The
division with remainder In arithmetic, Euclidean division – or division with remainder – is the process of division (mathematics), dividing one integer (the dividend) by another (the divisor), in a way that produces a quotient and a remainder smaller than the divisor ...
or
Euclidean division In arithmetic, Euclidean division – or division with remainder – is the process of division (mathematics), dividing one integer (the dividend) by another (the divisor), in a way that produces a quotient and a remainder smaller than the divisor ...
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 total order, ordering (as in "this is the ''third'' largest city in the country"). In common mathematical terminology, w ...

natural numbers
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 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 (mathematics), fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ) ...
s (the numbers that can be obtained by using arithmetic on natural numbers) or
real number In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
s. In these enlarged
number system A number is a mathematical object A mathematical object is an abstract concept arising in mathematics. In the usual language of mathematics, an ''object'' is anything that has been (or could be) formally defined, and with which one may do deduc ...
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 Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...
, which is not defined. Both forms of division appear in various
algebraic structure In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
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 #Definition, Euclidean function which allows a suitable generalization of the Euclidean division of ...
s and include
polynomial ring In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
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 File:A NASA Delta IV Heavy rocket launches the Parker Solar Probe (29097299447).jpg, FIELDS heads into space in August 2018 as part of the ''Parker Solar Probe'' FIELDS is a science instrument on the ''Parker Solar Probe'' (PSP), designed to mea ...
and
division ringIn algebra, a division ring, also called a skew field, is a ring (mathematics), ring in which division (mathematics), division is possible. Specifically, it is a zero ring, nonzero ring in which every nonzero element has a multiplicative inverse, th ...
s. In a ring the elements by which division is always possible are called the
units Unit may refer to: Arts and entertainment * UNIT Unit may refer to: Arts and entertainment * UNIT, a fictional military organization in the science fiction television series ''Doctor Who'' * Unit of action, a discrete piece of action (or beat) in ...
(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 (mathematics), 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 "factore ...
, 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 expression ...
: 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 Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
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 Real may refer to: * Reality, the state of things as they exist, rather than as they may appear or may be thought to be Currencies * Brazilian real (R$) * Central American Republi ...
, so is equal to or , but in the context of
integer An integer (from the Latin Latin (, or , ) is a classical language A classical language is a language A language is a structured system of communication Communication (from Latin ''communicare'', meaning "to share" or "to ...
division, where numbers have no fractional part, the remainder is kept separately (or exceptionally, discarded or
rounded
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 (mathematics), 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 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 Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
, meaning that is not always equal to . Division is also not, in general,
associative In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
, 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 = (a / b) / c = a / (b \times c) \ne a/(b/c)= (a\times c)/b. Division is right-distributive over addition and subtraction, in the sense that : \frac = (a \pm b) / c = (a/c)\pm (b/c) =\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 arithmetic, elementary Operation (mathematics), mathematical operations ...

multiplication
, as (a + b) \times c = a \times c + b \times c. However, division is ''not'' left-distributive, as : \frac = a / (b + c) \ne (a/b) + (a/c) = \frac. 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 Slash may refer to: * Slash (punctuation), the "/" character Arts and entertainment Fictional characters * Slash (Marvel Comics) * Slash (Teenage Mutant Ninja Turtles), Slash (''Teenage Mutant Ninja Turtles'') Music * Slash (musician), stage n ...
, 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 formal language In logic, mathematics, computer science, and linguistics, a formal language consists of string (computer science), words whose symbol (formal), letters are taken from an alphabet (computer science) ...

programming language
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 Character encoding is the process of assigning numbers to graphical Graphics (from Greek Greek may refer to: Greece Anything of, ...
characters. (It is also the only notation used for
quotient objectIn category theory Category theory formalizes mathematical structure and its concepts in terms of a Graph labeling, labeled directed graph called a ''Category (mathematics), category'', whose nodes are called ''objects'', and whose labelled dire ...
s in
abstract algebra In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), rings, field (mathema ...
.) Some
mathematical software Mathematical software is software Software is a collection of instructions Instruction or instructions may refer to: Computing * Instruction, one operation of a processor within a computer architecture instruction set * Computer program, a c ...
, such as
MATLAB MATLAB (an abbreviation of "MATrix LABoratory") is a and environment developed by . MATLAB allows manipulations, plotting of and data, implementation of s, creation of s, and interfacing with programs written in other languages. Althoug ...
and
GNU Octave GNU Octave is software featuring a high-level programming language In computer science Computer science deals with the theoretical foundations of information, algorithms and the architectures of its computation as well as practical techn ...
, allows the operands to be written in the reverse order by using the
backslash The backslash is a typographical mark used mainly in computing Computing is any goal-oriented activity requiring, benefiting from, or creating computing machinery. It includes the study and experimentation of algorithmic processes and develop ...
as the division operator: :b\backslash a A typographical variation halfway between these two forms uses a
solidus Solidus (Latin Latin (, or , ) is a classical language belonging to the Italic languages, Italic branch of the Indo-European languages. Latin was originally spoken in the area around Rome, known as Latium. Through the power of the Roman Republi ...
(fraction slash), but elevates the dividend and lowers the divisor: :^\!/_ Any of these forms can be used to display a
fraction A fraction (from Latin ', "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-fifths ...
. A fraction is a division expression where both dividend and divisor are
integer An integer (from the Latin Latin (, or , ) is a classical language A classical language is a language A language is a structured system of communication Communication (from Latin ''communicare'', meaning "to share" or "to ...
s (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 to indicate Division (mathematics), mathematical division. However, this usage, though widespread in Anglophone countries, is no ...
(÷, also known as
obelus An obelus (plural: obeluses or obeli) is a term in typography Typography is the art and technique of arranging type to make written language A written language is the representation of a spoken or gestural language A language ...

obelus
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 ISO 80000 or IEC 80000 is an international standard introducing the International System of Quantities (ISQ). It was developed and promulgated jointly by the International Organization for Standardization The International Organization for ...
-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 device used to perform s, ranging from basic to complex . The first calculator was created in the early 1960s. Pocket-sized devices became available in the 1970s, especially after the , the f ...

calculator
. 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, ∴.Such as on pg 53; Teutsche Algeb ...
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 is a West Germanic languages, West Germanic language first spoken in History of Anglo-Saxon England, early medieval England, which has eventually become the World language, leading lan ...

English
-speaking countries, a colon is used to denote division: :a : b This notation was introduced by
Gottfried Wilhelm Leibniz Gottfried Wilhelm (von) Leibniz ; see inscription of the engraving depicted in the " 1666–1676" section. ( – 14 November 1716) was a German polymath A polymath ( el, πολυμαθής, , "having learned much"; la, homo universalis, " ...

Gottfried Wilhelm Leibniz
in his 1684 ''Acta eruditorum''. Leibniz disliked having separate symbols for ratio and division. However, in English usage the
colon Colon commonly refers to: * Colon (punctuation) (:), a punctuation mark * Major part of large intestine, the final section of the digestive system Colon may also refer to: Places * Colon, Michigan, US * Colon, Nebraska, US * Kowloon, Hong Kong, s ...

colon
is restricted to expressing the related concept of
ratio In mathematics, a ratio indicates 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 ...

ratio
s. Since the 19th century, US textbooks have used b)a or b \overline to denote ''a'' divided by ''b'', especially when discussing
long division In arithmetic Arithmetic (from the Ancient Greek, Greek wikt:en:ἀριθμός#Ancient Greek, ἀριθμός ''arithmos'', 'number' and wikt:en:τική#Ancient Greek, τική wikt:en:τέχνη#Ancient Greek, έχνη ''tiké échne', 'ar ...

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 In arithmetic, short division is a division algorithm which breaks down a division (mathematics), division problem into a series of easier steps. It is an abbreviated form of long division — whereby the products are omitted and the partial remaind ...
, if the divisor is small, or
long division In arithmetic Arithmetic (from the Ancient Greek, Greek wikt:en:ἀριθμός#Ancient Greek, ἀριθμός ''arithmos'', 'number' and wikt:en:τική#Ancient Greek, τική wikt:en:τέχνη#Ancient Greek, έχνη ''tiké échne', 'ar ...

long division
, if the divisor is larger. If the dividend has a fractional part (expressed as a
decimal fraction The decimal numeral system A numeral system (or system of numeration) is a writing system A writing system is a method of visually representing verbal communication Communication (from Latin ''communicare'', meaning "to share") is t ...
), 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 that has been in use since ancient times and is still in use today. It was used in the ancient Near East The ancient Near East was the home of e ...

abacus
. Logarithm tables can be used to divide two numbers, by subtracting the two numbers' logarithms, then looking up the
antilogarithm In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...
of the result. Division can be calculated with a
slide rule The slide rule is a mechanical . The slide rule is used primarily for and and for functions such as , , s, and . They are not designed for addition or subtraction which was usually performed manually, with used to keep track of the magnitude ...

slide rule
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 device used to perform s, ranging from basic to complex . The first calculator was created in the early 1960s. Pocket-sized devices became available in the 1970s, especially after the , the f ...

calculator
s and
computer A computer is a machine that can be programmed to Execution (computing), carry out sequences of arithmetic or logical operations automatically. Modern computers can perform generic sets of operations known as Computer program, programs. These ...

computer
s compute division either by methods similar to long division, or by faster methods; see
Division algorithm A division algorithm is an algorithm which, given two integers N and D, computes their quotient and/or remainder, the result of Euclidean division. Some are applied by hand, while others are employed by digital circuit designs and software. Divisi ...
. In
modular arithmetic #REDIRECT Modular arithmetic #REDIRECT Modular arithmetic#REDIRECT Modular arithmetic In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure ( ...
(modulo a prime number) and for
real numbers Real may refer to: * Reality, the state of things as they exist, rather than as they may appear or may be thought to be Currencies * Brazilian real (R$) * Central American Republic real * Mexican real * Portuguese real * Spanish real * Spanish col ...

real numbers
, nonzero numbers have a
multiplicative inverse Image:Hyperbola one over x.svg, thumbnail, 300px, alt=Graph showing the diagrammatic representation of limits approaching infinity, The reciprocal function: . For every ''x'' except 0, ''y'' represents its multiplicative inverse. The graph forms a r ...
. 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 Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...

absolute value
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 Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...

partial function
. # Give an approximate answer as a "
real Real may refer to: * Reality Reality is the sum or aggregate of all that is real or existent within a system, as opposed to that which is only Object of the mind, imaginary. The term is also used to refer to the ontological status of things, ind ...
" number. This is the approach usually taken in
numerical computation Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic computation, symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics). Numerical analysis ...
. # Give the answer as a
fraction A fraction (from Latin ', "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-fifths ...
representing a
rational number In mathematics, a rational number is a number that can be expressed as the quotient or fraction (mathematics), fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ) ...
, so the result of the division of 26 by 11 is \tfrac (or as a mixed number, 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 ...

greatest common divisor
. # Give the answer as an integer ''
quotient In arithmetic Arithmetic (from the Ancient Greek, Greek wikt:en:ἀριθμός#Ancient Greek, ἀριθμός ''arithmos'', 'number' and wikt:en:τική#Ancient Greek, τική wikt:en:τέχνη#Ancient Greek, έχνη ''tiké échne' ...
'' and a ''
remainder In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
'', 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 division (mathematics), dividing one integer (the dividend) by another (the divisor), in a way that produces a quotient and a remainder smaller than the divisor ...
'', 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 effi ...
. # 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 (mathematics), 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 ...

floor function
'', also sometimes called ''integer division'' at an elementary level. Dividing integers in a
computer program In imperative programming, a computer program is a sequence of instructions in a programming language that a computer can execute or interpret. In declarative programming, a ''computer program'' is a Set (mathematics), set of instructions. A comp ...
requires special care. Some
programming language A programming language is a formal language In logic, mathematics, computer science, and linguistics, a formal language consists of string (computer science), words whose symbol (formal), letters are taken from an alphabet (computer science) ...

programming language
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 and environment developed by . MATLAB allows manipulations, plotting of and data, implementation of s, creation of s, and interfacing with programs written in other languages. Althoug ...
and every
computer algebra system A computer algebra system (CAS) or symbolic algebra system (SAS) is any mathematical software Mathematical software is software used to mathematical model, model, analyze or calculate numeric, symbolic or geometric data. It is a type of applica ...

computer algebra system
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 A number is a mathematical object A mathematical object is an abstract concept arising in mathematics. In the usual language of mathematics, an ''object'' is anything that has been (or could be) formally d ...

rounding
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 (mathematics), division, after one number is divided by another (called the ''modular arithmetic, modulus'' of the operation). Given two positive numbers a ...
for the details.
Divisibility rule A divisibility rule is a shorthand way of determining whether a given integer An integer (from the Latin wikt:integer#Latin, ''integer'' meaning "whole") is colloquially defined as a number that can be written without a Fraction (mathematics), ...
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 (mathematics), fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ) ...
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 arithmetic, elementary Operation (mathematics), mathematical operations ...

multiplication
.


Of real numbers

Division of two
real number In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
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 contains the real numbers and a specific element denoted , called the imaginary unit, and satisfying the equation . Moreover, every complex number can be expressed in the for ...

complex number
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 Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

polynomial
s in one variable over a
field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grassl ...
. Then, as in the case of integers, one has a remainder. See
Euclidean division of polynomials In algebra, the greatest common divisor (frequently abbreviated as GCD) of two polynomials is a polynomial, of the highest possible degree, that is a factor of both the two original polynomials. This concept is analogous to the greatest common ...
, and, for hand-written computation,
polynomial long division In algebra Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. In ...
or
synthetic division In algebra Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. In ...
.


Of matrices

One can define a division operation for matrices. The usual way to do this is to define , where denotes the
inverse Inverse or invert may refer to: Science and mathematics * Inverse (logic), a type of conditional sentence which is an immediate inference made from another conditional sentence * Additive inverse (negation), the inverse of a number that, when add ...
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 Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...

matrix multiplication
is not
commutative In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
, 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 Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
. That is, and , where and denote the pseudoinverses of ''A'' and ''B''.


Abstract algebra

In
abstract algebra In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), rings, field (mathema ...
, given a
magma Magma () is the molten or semi-molten natural material from which all igneous rock Igneous rock (derived from the Latin word ''ignis'' meaning fire), or magmatic rock, is one of the three main The three types of rocks, rock types, the others ...
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 (mathematics), matrix algebras and quaternion algebras. A quasigroup is a structure in which division is always possible, even without an identity element and hence inverses. In an integral domain, 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 (mathematics), 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 algebras. In particular Bott periodicity can be used to show that any real number, real normed division algebra must be isomorphic to either the real numbers R, the
complex number In mathematics, a complex number is an element of a number system that contains the real numbers and a specific element denoted , called the imaginary unit, and satisfying the equation . Moreover, every complex number can be expressed in the for ...

complex number
s C, the quaternions H, or the octonions O.


Calculus

The derivative of the quotient of two functions is given by the quotient rule: :' = \frac.


Division by zero

Division of any number by zero in most mathematical systems is undefined, because zero multiplied by any finite number always results in a multiplication, product of zero. Entry of such an expression into most
calculator An electronic calculator is typically a portable device used to perform s, ranging from basic to complex . The first calculator was created in the early 1960s. Pocket-sized devices became available in the 1970s, especially after the , the f ...

calculator
s produces an error message. However, in certain higher level mathematics division by zero is possible by the zero ring and algebras such as Wheel theory, wheels.Jesper Carlström
"On Division by Zero"
Retrieved October 23, 2018
In these algebras, the meaning of division is different from traditional definitions.


See also

* Rod calculus#Division, 400AD Sunzi division algorithm * Division by two * Galley division * Inverse element * Order of operations * Repeating decimal


Notes


References


External links


Planetmath division


selected fro
Abacus: Mystery of the Bead

Chinese Short Division Techniques on a Suan Pan


{{Authority control Division (mathematics), Elementary arithmetic