Division is 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 numbers is, among otherpossible interpretations, the process of calculating the number of times one number is contained within another.[1]: 7 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 remainder orEuclidean division of twonatural numbers provides 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 numbers orreal numbers. In these enlargednumber systems, division is the inverse operation to multiplication, that isa =c /b meansa ×b =c, as long asb is not zero. Ifb = 0, then this is adivision by zero, which is not defined.[a][4]: 246 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 domains and includepolynomial rings in 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 calledfields anddivision rings. In aring the 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.
The simplest way of viewing division is in terms ofquotition and partition: from the quotition perspective,20 / 5 means the number of 5s that must be added to get 20. In terms of partition,20 / 5 means the size of each of 5 parts into which a set of size 20 is divided. For example, 20 apples divide into five groups of four apples, meaning that "twenty divided by five is equal to four". This is denoted as20 / 5 = 4, or20/5 = 4.[2] 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 aremainder that will not go evenly into the dividend; for example,10 / 3 leaves a remainder of 1, as 10 is not a multiple of 3. Sometimes this remainder is added to the quotient as afractional part, so10 / 3 is equal to3+1/3 or3.33..., but in the context ofinteger division, where numbers have no fractional part, the remainder is kept separately (or exceptionally, discarded orrounded).[5] 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 /b is not always equal tob /a.[6] Division is also not, in general,associative, meaning that when dividing multiple times, the order of division can change the result.[7] 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:[8][9]
Division isright-distributive over addition and subtraction, in the sense that
Plus and minuses. Anobelus used 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 thedividend over thedivisor with a horizontal line, also called afraction bar, between them. For example, "a divided byb" can be written as:
which can also be read out loud as "dividea byb" or "a overb". A way to express division all on one line is to write thedividend (or numerator), then aslash, then thedivisor (or denominator), as follows:
A typographical variation halfway between these two forms uses asolidus (fraction slash), but elevates the dividend and lowers the divisor:
Any of these forms can be used to display afraction. A fraction is a division expression where both dividend and divisor areintegers (typically called thenumerator anddenominator), 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 asobelus though the term has additional meanings), common in arithmetic, in this manner:
This form is infrequent except in elementary arithmetic.ISO 80000-2-10.6 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 Rahn in 1659 inTeutsche Algebra.[10]: 211 The ÷ symbol is used to indicate subtraction in some European countries, so its use may be misunderstood.[11]
In some non-English-speaking countries, a colon is used to denote division:[12]
This notation was introduced byGottfried Wilhelm Leibniz in his 1684Acta eruditorum.[10]: 295 Leibniz disliked having separate symbols for ratio and division. However, in English usage thecolon is restricted to expressing the related concept ofratios.
Since the 19th century, US textbooks have used or to denotea divided byb, especially when discussinglong division. The history of this notation is not entirely clear because it evolved over time.[13]
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 afractional part (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).
Logarithm tables can be used to divide two numbers, by subtracting the two numbers' logarithms, then looking up theantilogarithm of the result.
Division can be calculated with aslide 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.
Inmodular arithmetic (modulo a prime number) and forreal numbers, nonzero numbers have amultiplicative inverse. In these cases, a division byx may be computed as the product by the multiplicative inverse ofx. This approach is often associated with the faster methods in computer arithmetic.
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 areunique integersq, thequotient, andr, the remainder, such thata =bq +r and 0 ≤r < |b|, where |b| denotes theabsolute value ofb.
Integers are notclosed 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 apartial function.
Give the answer as afraction representing arational number, so the result of the division of 26 by 11 is (or as amixed number, so) Usually the resulting fraction should be simplified: the result of the division of 52 by 22 is also. This simplification may be done by factoring out thegreatest common divisor.
Give the answer as an integerquotient and aremainder, so 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.
Give the integer quotient as the answer, so This is thefloor function applied to case 2 or 3. It is sometimes calledinteger division, and denoted by "//".
Dividing integers in acomputer program requires special care. Someprogramming languages treat integer division as in case 5 above, so the answer is an integer. Other languages, such asMATLAB and everycomputer 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 includediv,/,\, and%.[citation needed] Definitions vary regarding integer division when the dividend or the divisor is negative:rounding may be toward zero (so called T-division) or toward−∞ (F-division); rarer styles can occur – seemodulo operation for the details.
Divisibility rules can sometimes be used to quickly determine whether one integer divides exactly into another.
The result of dividing tworational numbers is another rational number when the divisor is not 0. The division of two rational numbersp/q andr/s can be computed as
All four quantities are integers, and onlyp may be 0. This definition ensures that division is the inverse operation ofmultiplication.
Dividing twocomplex numbers (when the divisor is nonzero) results in another complex number, which is found using the conjugate of the denominator:
This process of multiplying and dividing by is called 'realisation' or (by analogy)rationalisation. All four quantitiesp,q,r,s are real numbers, andr ands may not both be 0.
Division for complex numbers expressed in polar form is simpler than the definition above:
Again all four quantitiesp,q,r,s are real numbers, andr may not be 0.
One can define a division operation for matrices. The usual way to do this is to defineA /B =AB−1, whereB−1 denotes theinverse ofB, but it is far more common to write outAB−1 explicitly to avoid confusion. Anelementwise division can also be defined in terms of theHadamard product.
Becausematrix multiplication is notcommutative, one can also define aleft division or so-calledbackslash-division asA \B =A−1B. For this to be well defined,B−1 need not exist, howeverA−1 does need to exist. To avoid confusion, division as defined byA /B =AB−1 is sometimes calledright division orslash-division in 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) \C the same asA \ (B \C). However, it holds thatA / (BC) = (A /C) /B and(AB) \C =B \ (A \C).
To avoid problems whenA−1 and/orB−1 do 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 ofA andB.
Inabstract algebra, given amagma with binary operation ∗ (which could nominally be termed multiplication),left division ofb bya (writtena \b) is typically defined as the solutionx to the equationa ∗x =b, if this exists and is unique. Similarly,right division ofb bya (writtenb /a) is the solutiony to 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 \b andb /a exist and are unique for alla and 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 includematrix algebras,quaternion algebras, and quasigroups. In anintegral domain, where not every element need have an inverse,division by a cancellative elementa can still be performed on elements of the formab orca by left or right cancellation, respectively. If aring is finite and every nonzero element is cancellative, then by an application of thepigeonhole principle, every nonzero element of the ring is invertible, anddivision by 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 periodicity can be used to show that anyrealnormed division algebra must beisomorphic to either the real numbersR, thecomplex numbersC, thequaternionsH, or theoctonionsO.
Division of any number byzero in most mathematical systems is undefined, because zero multiplied by any finite number always results in theproduct being zero.[15] Entry of such an expression into mostcalculators produces an error message. However, in certain mathematical structures, division by zero is possible, such as in thezero ring and in algebraic structures such aswheels.[16] In these structures, the meaning of division is different from that of traditional definitions.
^Division by zero may be defined in some circumstances, either by extending the real numbers to theextended real number line or to theprojectively extended real line or when occurring as limit of divisions by numbers tending to 0. For example:limx→0sinx/x = 1.[2][3]
^Thomas Sonnabend (2010).Mathematics for Teachers: An Interactive Approach for Grades K–8. Brooks/Cole, Cengage Learning (Charles Van Wagner). p. 126.ISBN978-0-495-56166-8.
