Division (mathematics)

"Divided" redirects here. For other uses, seeDivided (disambiguation).

20 / 4 = 5, illustrated here with apples. This is said verbally, "Twenty divided by four equals five."

Divisionis one of the four basic operations ofarithmetic.The other operations areaddition,subtraction,andmultiplication.What is being divided is called thedividend,which is divided by thedivisor,and the result is called thequotient.

At an elementary level the division of twonatural numbersis, among otherpossible interpretations,the process of calculating the number of times one number is contained within another.For example, if 20 apples are divided evenly between 4 people, everyone receives 5 apples (see picture). However, this number of times or the number contained (divisor) need not beintegers.

Thedivision with remainderorEuclidean divisionof twonatural numbersprovides an integerquotient,which is the number of times the second number is completely contained in the first number, and aremainder,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 an integer quotient plus a remainder, the natural numbers must be extended torational numbersorreal numbers.In these enlargednumber systems,division is the inverse operation to multiplication, that isa=c/bmeansa×b=c,as long asbis not zero. Ifb= 0,then this is adivision by zero,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 variousalgebraic structures,different ways of defining mathematical structure. Those in which a Euclidean division (with remainder) is defined are calledEuclidean domainsand includepolynomial ringsin oneindeterminate(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 calledfieldsanddivision rings.In aringthe elements by which division is always possible are called theunits(for example, 1 and −1 in the ring of integers). Another generalization of division to algebraic structures is thequotient group,in which the result of "division" is a group rather than a number.

Introduction

The simplest way of viewing division is in terms ofquotition and partition:from the quotition perspective,20 / 5means the number of 5s that must be added to get 20. In terms of partition,20 / 5means 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 as20 / 5 = 4,or⁠20/5⁠= 4.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 aremainderthat will not go evenly into the dividend; for example,10 / 3leaves a remainder of 1, as 10 is not a multiple of 3. Sometimes this remainder is added to the quotient as afractional part,so10 / 3is equal to⁠3+1/3⁠or3.33...,but in the context ofintegerdivision, where numbers have no fractional part, the remainder is kept separately (or exceptionally, discarded orrounded).When the remainder is kept as a fraction, it leads to arational number.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 notcommutative,meaning thata/bis not always equal tob/a.Division is also not, in general,associative,meaning that when dividing multiple times, the order of division can change the result.For example,(24 / 6) / 2 = 2,but24 / (6 / 2) = 8(where the use of parentheses indicates that the operations inside parentheses are performed before the operations outside parentheses).

Division is traditionally considered asleft-associative.That is, if there are multiple divisions in a row, the order of calculation goes from left to right:

$a/b/c=(a/b)/c=a/(b\times c)\;\neq \;a/(b/c)=(a\times c)/b.$

Division isright-distributiveover addition and subtraction, in the sense that

${\frac {a\pm b}{c}}=(a\pm b)/c=(a/c)\pm (b/c)={\frac {a}{c}}\pm {\frac {b}{c}}.$

This is the same formultiplication,as$(a+b)\times c=a\times c+b\times c$.However, division isnotleft-distributive,as

${\frac {a}{b+c}}=a/(b+c)\;\neq \;(a/b)+(a/c)={\frac {ac+ab}{bc}}.$For example${\frac {12}{2+4}}={\frac {12}{6}}=2,$but${\frac {12}{2}}+{\frac {12}{4}}=6+3=9.$

This is unlike the case in multiplication, which is both left-distributive and right-distributive, and thusdistributive.

Notation

Further information:Division sign

Plus and minuses. Anobelusused as a variant of the minus sign in an excerpt from an official Norwegian trading statement form called «Næringsoppgave 1» for the taxation year 2010.

Division is often shown in algebra and science by placing thedividendover thedivisorwith a horizontal line, also called afraction bar,between them. For example, "adivided byb"can be written as:

${\frac {a}{b}}$

which can also be read out loud as "divideabyb"or"aoverb".A way to express division all on one line is to write thedividend(or numerator), then aslash,then thedivisor(or denominator), as follows:

$a/b$

This is the usual way of specifying division in most computerprogramming languages,since it can easily be typed as a simple sequence ofASCIIcharacters. (It is also the only notation used forquotient objectsinabstract algebra.) Somemathematical software,such asMATLABandGNU Octave,allows the operands to be written in the reverse order by using thebackslashas the division operator:

$b\backslash a$

A typographical variation halfway between these two forms uses asolidus(fraction slash), but elevates the dividend and lowers the divisor:

${}^{a}\!/{}_{b}$

Any of these forms can be used to display afraction.A fraction is a division expression where both dividend and divisor areintegers(typically called thenumeratoranddenominator), and there is no implication that the division must be evaluated further. A second way to show division is to use thedivision sign(÷, also known asobelusthough 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 acalculator.The obelus was introduced by Swiss mathematicianJohann Rahnin 1659 inTeutsche Algebra.The ÷ symbol is used to indicate subtraction in some European countries, so its use may be misunderstood.

In some non-English-speaking countries, a colon is used to denote division:

$a:b$

This notation was introduced byGottfried Wilhelm Leibnizin his 1684Acta eruditorum.Leibniz disliked having separate symbols for ratio and division. However, in English usage thecolonis restricted to expressing the related concept ofratios.

Since the 19th century, US textbooks have used$b)a$or$b{\overline {)a}}$to denoteadivided byb,especially when discussinglong division.The history of this notation is not entirely clear because it evolved over time.

Computing

Main articles:Long divisionandDivision algorithm

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 ofshort division,if the divisor is small, orlong division,if the divisor is larger. If the dividend has afractionalpart (expressed as adecimal fraction), 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 anabacus.

Logarithm tablescan be used to divide two numbers, by subtracting the two numbers' logarithms, then looking up theantilogarithmof the result.

Division can be calculated with aslide ruleby 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

Moderncalculatorsandcomputerscompute division either by methods similar to long division, or by faster methods; seeDivision algorithm.

Inmodular arithmetic(modulo a prime number) and forreal numbers,nonzero numbers have amultiplicative inverse.In these cases, a division byxmay be computed as the product by the multiplicative inverse ofx.This approach is often associated with the faster methods in computer arithmetic.

Division in different contexts

Euclidean division

Main article: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,thedividend,andb,thedivisor,such thatb≠ 0, there areuniqueintegersq,thequotient,andr,the remainder, such thata=bq+rand 0 ≤r< |b|, where |b| denotes theabsolute valueofb.

Of integers

Integers are notclosedunder 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:

1. Say that 26 cannot be divided by 11; division becomes apartial function.
2. Give an approximate answer as afloating-point number.This is the approach usually taken innumerical computation.
3. Give the answer as afractionrepresenting arational number,so the result of the division of 26 by 11 is${\tfrac {26}{11}}$(or as amixed number,so${\tfrac {26}{11}}=2{\tfrac {4}{11}}.$) Usually the resulting fraction should be simplified: the result of the division of 52 by 22 is also${\tfrac {26}{11}}$.This simplification may be done by factoring out thegreatest common divisor.
4. Give the answer as an integerquotientand aremainder,so${\tfrac {26}{11}}=2{\mbox{ remainder }}4.$To make the distinction with the previous case, this division, with two integers as result, is sometimes calledEuclidean division,because it is the basis of theEuclidean algorithm.
5. Give the integer quotient as the answer, so${\tfrac {26}{11}}=2.$This is thefloor functionapplied to case 2 or 3. It is sometimes calledinteger division,and denoted by "//".

Dividing integers in acomputer programrequires special care. Someprogramming languagestreat integer division as in case 5 above, so the answer is an integer. Other languages, such asMATLABand everycomputer algebra systemreturn 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:roundingmay be toward zero (so called T-division) or toward−∞(F-division); rarer styles can occur – seemodulo operationfor the details.

Divisibility rulescan sometimes be used to quickly determine whether one integer divides exactly into another.

Of rational numbers

The result of dividing tworational numbersis another rational number when the divisor is not 0. The division of two rational numbersp/qandr/scan be computed as $${p/q \over r/s}={p \over q}\times {s \over r}={ps \over qr}.$$

All four quantities are integers, and onlypmay be 0. This definition ensures that division is the inverse operation ofmultiplication.

Of real numbers

Division of tworeal numbersresults in another real number (when the divisor is nonzero). It is defined such thata/b=cif and only ifa=cbandb≠ 0.

Of complex numbers

Dividing twocomplex numbers(when the divisor is nonzero) results in another complex number, which is found using the conjugate of the denominator: $${p+iq \over r+is}={(p+iq)(r-is) \over (r+is)(r-is)}={pr+qs+i(qr-ps) \over r^{2}+s^{2}}={pr+qs \over r^{2}+s^{2}}+i{qr-ps \over r^{2}+s^{2}}.$$

This process of multiplying and dividing by$r-is$is called 'realisation' or (by analogy)rationalisation.All four quantitiesp,q,r,sare real numbers, andrandsmay not both be 0.

Division for complex numbers expressed in polar form is simpler than the definition above: $${pe^{iq} \over re^{is}}={pe^{iq}e^{-is} \over re^{is}e^{-is}}={p \over r}e^{i(q-s)}.$$

Again all four quantitiesp,q,r,sare real numbers, andrmay not be 0.

Of polynomials

One can define the division operation forpolynomialsin one variable over afield.Then, as in the case of integers, one has a remainder. SeeEuclidean division of polynomials,and, for hand-written computation,polynomial long divisionorsynthetic division.

Of matrices

One can define a division operation for matrices. The usual way to do this is to defineA/B=AB⁻¹,whereB⁻¹denotes theinverseofB,but it is far more common to write outAB⁻¹explicitly to avoid confusion. Anelementwise divisioncan also be defined in terms of theHadamard product.

Left and right division

Becausematrix multiplicationis notcommutative,one can also define aleft divisionor so-calledbackslash-divisionasA\B=A⁻¹B.For this to be well defined,B⁻¹need not exist, howeverA⁻¹does need to exist. To avoid confusion, division as defined byA/B=AB⁻¹is sometimes calledright divisionorslash-divisionin this context.

With left and right division defined this way,A/ (BC)is in general not the same as(A/B) /C,nor is(AB) \Cthe same asA\ (B\C).However, it holds thatA/ (BC) = (A/C) /Band(AB) \C=B\ (A\C).

Pseudoinverse

To avoid problems whenA⁻¹and/orB⁻¹do not exist, division can also be defined as multiplication by thepseudoinverse.That is,A/B=AB⁺andA\B=A⁺B,whereA⁺andB⁺denote the pseudoinverses ofAandB.

Abstract algebra

Inabstract algebra,given amagmawith binary operation ∗ (which could nominally be termed multiplication),left divisionofbbya(writtena\b) is typically defined as the solutionxto the equationa∗x=b,if this exists and is unique. Similarly,right divisionofbbya(writtenb/a) is the solutionyto the equationy∗a=b.Division in this sense does not require ∗ to have any particular properties (such as commutativity, associativity, or an identity element). A magma for which botha\bandb/aexist and are unique for allaand allb(theLatin square property) is aquasigroup.In a quasigroup, division in this sense is always possible, even without an identity element and hence without inverses.

"Division" in the sense of "cancellation" can be done in any magma by an element with thecancellation property.Examples includematrixalgebras,quaternionalgebras, and quasigroups. In anintegral domain,where not every element need have an inverse,divisionby a cancellative elementacan still be performed on elements of the formaborcaby left or right cancellation, respectively. If aringis finite and every nonzero element is cancellative, then by an application of thepigeonhole principle,every nonzero element of the ring is invertible, anddivisionby any nonzero element is possible. To learn about whenalgebras(in the technical sense) have a division operation, refer to the page ondivision algebras.In particularBott periodicitycan be used to show that anyrealnormed division algebramust beisomorphicto either the real numbersR,thecomplex numbersC,thequaternionsH,or theoctonionsO.

Calculus

Thederivativeof the quotient of two functions is given by thequotient rule: $${\left({\frac {f}{g}}\right)}'={\frac {f'g-fg'}{g^{2}}}.$$

Division by zero

Main article:Division by zero

Division of any number byzeroin most mathematical systems is undefined, because zero multiplied by any finite number always results in aproductof zero.Entry of such an expression into mostcalculatorsproduces an error message. However, in certain higher level mathematics division by zero is possible by thezero ringand algebras such aswheels.In these algebras, the meaning of division is different from traditional definitions.

See also

• 400AD Sunzi division algorithm
• Division by two
• Galley division
• Inverse element
• Order of operations
• Repeating decimal

External links

• Planetmath division
• Division on a Japanese abacusselected fromAbacus: Mystery of the Bead
• Chinese Short Division Techniques on a Suan Pan