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| {{about|the arithmetical operation||Division (disambiguation)}}
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| {{redirect|Divided}}
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| [[Image:Divide20by4.svg|right|thumb|200px|<math>20 \div 4=5</math>]]
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| [[Image:Division chart.png|thumb|Division of numbers 0-10. Line labels = dividend. X axis = divisor. Y axis = quotient.]]
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| In [[mathematics]], especially in [[elementary arithmetic]], '''division''' (÷) is an arithmetic operation.
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| Specifically, if ''b'' times ''c'' equals ''a'', written:
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| :''a'' = ''b'' × ''c''
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| where ''b'' is not [[0 (number)|zero]], then ''a'' divided by ''b'' equals ''c'', written:
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| :''a'' ÷ ''b'' = ''c''
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| For instance,
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| :6 ÷ 3 = 2
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| since
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| :3 x 2 = 6
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| In the expression a ÷ b = c, ''a'' is called the '''dividend''' or '''numerator''', ''b'' the '''divisor''' or '''denominator''' and the result ''c'' is called the '''[[quotient]]'''.
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| Conceptually, division describes two distinct but related settings. ''Partitioning'' involves taking a set of size ''a'' and forming ''b'' groups that are equal in size. The size of each group formed, ''c'', is the quotient of ''a'' and ''b''. ''Quotative'' division involves taking a set of size ''a'' and forming groups of size ''c''. The number of groups of this size that can be formed, ''b'', is the quotient of ''a'' and ''c''.<ref>Fosnot and Dolk 2001. ''Young Mathematicians at Work: Constructing Multiplication and Division''. Portsmouth, NH: Heinemann.</ref>
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| Teaching division usually leads to the concept of [[Fraction (mathematics)|fraction]]s being introduced to students. Unlike [[addition]], [[subtraction]], and [[multiplication]], the set of all [[integer]]s is not [[Closure (mathematics)|closed]] under division. Dividing two integers may result in a [[remainder]]. To complete the division of the remainder, the [[number system]] is extended to include fractions or [[rational number]]s as they are more generally called.
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| == Notation ==
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| Division is often shown in algebra and science by placing the ''dividend'' over the ''divisor'' with a horizontal line, also called a [[Vinculum (symbol)|vinculum]] or fraction bar, between them. For example, ''a'' divided by ''b'' is written
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| :<math>\frac ab</math>
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| This can be read out loud as "a divided by b", "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 (punctuation)|slash]], then the ''divisor'' (or denominator), like this:
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| :<math>a/b\,</math>
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| This is the usual way to specify division in most computer [[programming language]]s since it can easily be typed as a simple sequence of [[ASCII]] characters.
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| A typographical variation halfway between these two forms uses a [[solidus (punctuation)|solidus]] (fraction slash) but elevates the dividend, and lowers the divisor:
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| :{{frac|''a''|''b''}}
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| Any of these forms can be used to display a [[fraction (mathematics)|fraction]]. A fraction is a division expression where both dividend and divisor are [[integer]]s (although 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 [[obelus]] (or division sign), common in arithmetic, in this manner:
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| :<math>a \div b</math>
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| This form is infrequent except in elementary arithmetic. [[ISO 80000-2]]-9.6 states it should not be used. The obelus is also used alone to represent the division operation itself, as for instance as a label on a key of a [[calculator]].
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| In some non-[[English language|English]]-speaking cultures, "a divided by b" is written ''a'' : ''b''. This notation was introduced in 1631 by [[William Oughtred]] in his ''Clavis Mathematicae'' and later popularized by [[Gottfried Wilhelm Leibniz]].<ref name="first_symbol_use">[http://jeff560.tripod.com/operation.html Earliest Uses of Symbols of Operation], Jeff MIller</ref> However, in English usage the [[colon (punctuation)|colon]] is restricted to expressing the related concept of [[ratio]]s (then "a is to b").
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| In elementary mathematics the notation <math>b)~a</math> or <math>b )\overline{~a~}</math> is used to denote ''a'' divided by ''b''. This notation was first introduced by [[Michael Stifel]] in ''Arithmetica integra'', published in 1544.<ref name="first_symbol_use"/>
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| == Computing ==
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| {{main|Division algorithm}}
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| ===Manual methods===
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| Division is often introduced through the notion of "sharing out" a set of objects, for example a pile of sweets, 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 (division)|chunking]]", i.e., division by repeated subtraction.
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| More systematic and more efficient (but also more formalised and more rule-based, and more removed from an overall holistic picture of what division is achieving), a person who knows the [[multiplication tables]] can divide two integers using pencil and paper using the method of [[short division]], if the divisor is simple. [[Long division]] is used for larger integer divisors. If the dividend has a [[fraction (mathematics)|fractional]] part (expressed as a [[decimal fraction]]), one can continue the algorithm past the ones place as far as desired. If the divisor has a fractional part, we can restate the problem by moving the decimal to the right in both numbers until the divisor has no fraction.
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| A person can calculate division with an [[abacus]] by repeatedly placing the dividend on the abacus, and then subtracting the divisor the offset of each digit in the result, counting the number of divisions possible at each offset.
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| A person can use [[logarithm tables]] to divide two numbers, by subtracting the two numbers' logarithms, then looking up the antilogarithm of the result.
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| A person can calculate division with a [[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.
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| ===By computer or with computer assistance===
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| Modern computers compute division by methods that are faster than long division: see [[Division algorithm]].
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| In [[modular arithmetic]], some numbers have a [[modular multiplicative inverse|multiplicative inverse]] with respect to the modulus. We can calculate division by multiplication in such a case. This approach is useful in computers that do not have a fast division instruction.
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| == Algorithm ==
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| {{main|Euclidean division}}
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| The division algorithm is a mathematical theorem that precisely expresses the outcome of the usual process of division of integers. In particular, the theorem asserts that integers called the quotient ''q'' and remainder ''r'' always exist and that they are uniquely determined by the dividend ''a'' and divisor ''d'', with ''d'' ≠ 0. Formally, the theorem is stated as follows: There exist [[Uniqueness quantification|unique]] integers ''q'' and ''r'' such that ''a'' = ''qd'' + ''r'' and 0 ≤ ''r'' < | ''d'' |, where | ''d'' | denotes the [[absolute value]] of ''d''.
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| == Of integers ==
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| Division of integers is not [[Closure (mathematics)|closed]]. 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:
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| # Say that 26 cannot be divided by 11; division becomes a [[partial function]].
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| # Give an approximate answer as a [[decimal fraction]] or a [[mixed number]], so <math>\tfrac{26}{11} \simeq 2.36</math> or <math>\tfrac{26}{11} \simeq 2 \tfrac {36}{100}.</math> This is the approach usually taken in [[numerical computation]].
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| # Give the answer as a [[fraction (mathematics)|fraction]] representing a [[rational number]], so the result of the division of 26 by 11 is <math>\tfrac{26}{11}.</math> But, usually, the resulting fraction should be simplified: the result of the division of 52 by 22 is also <math>\tfrac{26}{11}</math>. This simplification may be done by factoring out the [[greatest common divisor]].
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| # Give the answer as an integer ''[[quotient]]'' and a ''[[remainder]]'', so <math>\tfrac{26}{11} = 2 \mbox{ remainder } 4.</math> To make the distinction with the previous case, this division, with two integers as result, is sometimes called ''[[Euclidean division]]'', because it is the basis of the [[Euclidean algorithm]].
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| # Give the integer quotient as the answer, so <math>\tfrac{26}{11} = 2.</math> This is sometimes called ''integer division''.
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| Dividing integers in a [[computer program]] requires special care. Some [[programming language]]s, such as [[C (programming language)|C]], treat integer division as in case 5 above, so the answer is an integer. Other languages, such as [[MATLAB]] and every [[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.
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| Names and symbols used for integer division include div, /, \, and %. Definitions vary regarding integer division when the dividend or the divisor is negative: rounding may be toward zero (so called T-division) or toward [[Extended real number line|−∞]] (F-division); rarer styles can occur – see [[Modulo operation]] for the details.
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| [[Divisibility rule]]s can sometimes be used to quickly determine whether one integer divides exactly into another. | |
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| == Of rational numbers ==
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| The result of dividing two [[rational number]]s is another rational number when the divisor is not 0. We may define division of two rational numbers ''p''/''q'' and ''r''/''s'' by
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| :<math>{p/q \over r/s} = {p \over q} \times {s \over r} = {ps \over qr}.</math> | |
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| All four quantities are integers, and only ''p'' may be 0. This definition ensures that division is the inverse operation of [[multiplication]].
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| == Of real numbers ==
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| Division of two [[real number]]s results in another real number when the divisor is not 0. It is defined such ''a''/''b'' = ''c'' if and only if ''a'' = ''cb'' and ''b'' ≠ 0.
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| == By zero == | |
| {{main|Division by zero}}
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| Division of any number by [[zero]] (where the divisor is zero) is undefined. This is because zero multiplied by any finite number always results in a [[multiplication|product]] of zero. Entry of such an expression into most [[calculator]]s produces an error message.
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| == Of complex numbers ==
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| Dividing two [[complex number]]s results in another complex number when the divisor is not 0, defined thus:
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| :<math>{p + iq \over r + is} = {p r + q s \over r^2 + s^2} + i{q r - p s \over r^2 + s^2}.</math>
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| All four quantities are real numbers. ''r'' and ''s'' may not both be 0.
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| Division for complex numbers expressed in polar form is simpler than the definition above:
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| :<math>{p e^{iq} \over r e^{is}} = {p \over r}e^{i(q - s)}.</math>
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| Again all four quantities are real numbers. ''r'' may not be 0.
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| == Of polynomials ==
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| One can define the division operation for [[polynomial]]s in one variable over a [[field (mathematics)|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]] or [[synthetic division]].
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| == Of matrices ==
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| One can define a division operation for matrices. The usual way to do this is to define {{nowrap|1=''A'' / ''B'' = ''AB''<sup>−1</sup>}}, where {{nowrap|''B''<sup>−1</sup>}} denotes the [[inverse matrix|inverse]] of ''B'', but it is far more common to write out {{nowrap|''AB''<sup>−1</sup>}} explicitly to avoid confusion.
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| === Left and right division ===
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| Because [[matrix multiplication]] is not [[commutative]], one can also define a [[left division]] or so-called ''backslash-division'' as {{nowrap|1=''A'' \ ''B'' = ''A''<sup>−1</sup>''B''}}. For this to be well defined, {{nowrap|''B''<sup>−1</sup>}} need not exist, however {{nowrap|''A''<sup>−1</sup>}} does need to exist. To avoid confusion, division as defined by {{nowrap|1=''A'' / ''B'' = ''AB''<sup>−1</sup>}} is sometimes called ''right division'' or ''slash-division'' in this context.
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| Note that with left and right division defined this way, {{nowrap|''A''/(''BC'')}} is in general not the same as {{nowrap|(''A''/''B'')/''C''}} and nor is {{nowrap|(''AB'')\''C''}} the same as {{nowrap|''A''\(''B''\''C'')}}, but {{nowrap|1=''A''/(''BC'') = (''A''/''C'')/''B''}} and {{nowrap|1=(''AB'')\''C'' = ''B''\(''A''\''C'')}}.
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| === Pseudoinverse ===
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| To avoid problems when {{nowrap|''A''<sup>−1</sup>}} and/or {{nowrap|''B''<sup>−1</sup>}} do not exist, division can also be defined as multiplication with the [[Moore–Penrose pseudoinverse|pseudoinverse]], i.e., {{nowrap|1=''A'' / ''B'' = ''AB''<sup>+</sup>}} and {{nowrap|1=''A'' \ ''B'' = ''A''<sup>+</sup>''B''}}, where {{nowrap|''A''<sup>+</sup>}} and {{nowrap|''B''<sup>+</sup>}} denote the pseudoinverse of ''A'' and ''B''.
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| == In abstract algebra ==
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| In [[abstract algebra]]s such as [[Matrix (mathematics)|matrix]] algebras and [[quaternion]] algebras, fractions such as <math>{a \over b}</math> are typically defined as <math>a \cdot {1 \over b}</math> or <math>a \cdot b^{-1}</math> where <math>b</math> is presumed an invertible element (i.e., there exists a [[multiplicative inverse]] <math>b^{-1}</math> such that <math>bb^{-1} = b^{-1}b = 1</math> where <math>1</math> is the multiplicative identity). In an [[integral domain]] where such elements may not exist, ''division'' can still be performed on equations of the form <math>ab = ac</math> or <math>ba = ca</math> by left or right cancellation, respectively. More generally "division" in the sense of "cancellation" can be done in any [[ring (mathematics)|ring]] with the aforementioned cancellation properties. If such a ring is finite, then by an application of the [[pigeonhole principle]], every nonzero element of the ring is invertible, so ''division'' by any nonzero element is possible in such a ring. To learn about when ''algebras'' (in the technical sense) have a division operation, refer to the page on [[division algebra]]s. 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]]s '''C''', the [[quaternion]]s '''H''', or the [[octonion]]s '''O'''.
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| <!-- Left vs right, definition of quasigroup, relationship to inverse elements in presence of associativity, examples: groups, octonions -->
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| == Calculus ==
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| The [[derivative]] of the quotient of two functions is given by the [[quotient rule]]:
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| :<math>{\left(\frac fg\right)}' = \frac{f'g - fg'}{g^2}.</math>
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| There is no general method to [[integral|integrate]] the quotient of two functions.
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| == See also ==
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| * [[Rod calculus#Division|400AD Sunzi division algorithm]]
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| * [[Fraction (mathematics)]]
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| * [[Division by two]]
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| * [[Field (mathematics)|Field]]
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| * [[Galley division]]
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| * [[Group (mathematics)|Group]]
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| * [[Inverse element]]
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| * [[Order of operations]]
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| * [[Quasigroup]] (left division)
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| * [[Repeating decimal]]
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| == References ==
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| {{Reflist}}
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| == External links ==
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| {{Commons category|Division (mathematics)}}
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| * {{planetmath reference|id=6148|title=Division}}
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| * [http://webhome.idirect.com/~totton/abacus/pages.htm#Division1 Division on a Japanese abacus] selected from [http://webhome.idirect.com/~totton/abacus/ Abacus: Mystery of the Bead]
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| * [http://webhome.idirect.com/~totton/suanpan/sh_div/ Chinese Short Division Techniques on a Suan Pan]
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| * [http://www.math.wichita.edu/history/topics/arithmetic.html#div Rules of divisibility]
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| {{More footnotes|date=February 2008}}
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| {{Elementary arithmetic}}
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| [[Category:Elementary arithmetic]]
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| [[Category:Binary operations]]
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| [[Category:Division| ]]
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