Division is one of the four basic operations of arithmetic, the others
being addition, subtraction, and multiplication. The division of two
natural numbers is the process of calculating the number of times one
number is contained within another one.:7 For example, in the
picture on the right, the 20 apples are divided into four groups of
five apples, meaning that twenty divided by five gives four, or four
is the result of division of twenty by five. This is denoted as 20 / 5
= 4, 20 ÷ 5 = 4, or 20/5 = 4.
Division can be viewed either as quotition or as partition. In
quotition, 20 ÷ 5 means the number of 5s that must be added
to get 20. In partition, 20 ÷ 5 means the size of each
of 5 parts into which a set of size 20 is divided.
Division is the inverse of multiplication; if a × b = c, then a = c
÷ b, as long as b is not zero.
Division by zero
1 Notation 2 Computing
2.1 Manual methods 2.2 By computer or with computer assistance
3 Properties 4 Euclidean division 5 Of integers 6 Of rational numbers 7 Of real numbers 8 By zero 9 Of complex numbers 10 Of polynomials 11 Of matrices
11.1 Left and right division 11.2 Pseudoinverse
12 Abstract algebra 13 Calculus 14 See also 15 Notes 16 References 17 External links
v t e
addend (broad sense)
addend (broad sense)
addend (strict sense)
displaystyle scriptstyle left. begin matrix scriptstyle text summand ,+, text summand \scriptstyle text addend (broad sense) ,+, text addend (broad sense) \scriptstyle text augend ,+, text addend (strict sense) end matrix right ,=,
displaystyle scriptstyle text sum
displaystyle scriptstyle text minuend ,-, text subtrahend ,=,
displaystyle scriptstyle text difference
displaystyle scriptstyle left. begin matrix scriptstyle text factor ,times , text factor \scriptstyle text multiplier ,times , text multiplicand end matrix right ,=,
displaystyle scriptstyle text product
displaystyle scriptstyle left. begin matrix scriptstyle frac scriptstyle text dividend scriptstyle text divisor \scriptstyle text \scriptstyle frac scriptstyle text numerator scriptstyle text denominator end matrix right ,=,
displaystyle begin matrix scriptstyle text fraction \scriptstyle text quotient \scriptstyle text ratio end matrix
displaystyle scriptstyle text base ^ text exponent ,=,
displaystyle scriptstyle text power
nth root (√)
displaystyle scriptstyle sqrt[ text degree ] scriptstyle text radicand ,=,
displaystyle scriptstyle text root
displaystyle scriptstyle log _ text base ( text antilogarithm ),=,
displaystyle scriptstyle text logarithm
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 is written
displaystyle frac a b
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, then the divisor (or denominator), like this:
This is the usual way to specify division in most computer programming
languages since it can easily be typed as a simple sequence of ASCII
characters. Some mathematical software, such as
b ∖ a
displaystyle bbackslash a
A typographical variation halfway between these two forms uses a solidus (fraction slash) but elevates the dividend, and lowers the divisor:
displaystyle ^ a / _ b
Any of these forms can be used to display a fraction. A fraction is a division expression where both dividend and divisor are integers (typically called the numerator and denominator), and there is no implication that the division must be evaluated further. A second way to show division is to use the obelus (or division sign), common in arithmetic, in this manner:
a ÷ b
displaystyle adiv b
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. The obelus was introduced by Swiss mathematician Johann Rahn in 1659 in Teutsche Algebra.:211 In some non-English-speaking countries, "a divided by b" is written a : b. This notation was introduced by Gottfried Wilhelm Leibniz in his 1684 Acta eruditorum.:295 Leibniz disliked having separate symbols for ratio and division. However, in English usage the colon is restricted to expressing the related concept of ratios (then "a is to b"). Since the 19th century US textbooks have used
b ) a
displaystyle b overline )a
to denote a divided by b, especially when discussing long division.
The history of this notation is not entirely clear because it evolved
a + b
= ( a + b ) ÷ c =
displaystyle frac a+b c =(a+b)div c= frac a c + frac b c
in the same way as in multiplication
( a + b ) × c = a × c + b × c
displaystyle (a+b)times c=atimes c+btimes c
. But division is not left-distributive, i.e. we have
b + c
= a ÷ ( b + c ) =
displaystyle frac a b+c =adiv (b+c)=left( frac b a + frac c a right)^ -1 neq frac a b + frac a c
unlike multiplication. If there are multiple divisions in a row the order of operation goes from left to right, which is called left-associative:
a ÷ b ÷ c = ( a ÷ b ) ÷ c = a ÷ ( b × c ) = a ×
displaystyle adiv bdiv c=(adiv b)div c=adiv (btimes c)=atimes b^ -1 times c^ -1
Main article: Euclidean division
Say that 26 cannot be divided by 11; division becomes a partial function. Give an approximate answer as a decimal fraction or a mixed number, so
displaystyle tfrac 26 11 simeq 2.36
displaystyle tfrac 26 11 simeq 2 tfrac 36 100 .
This is the approach usually taken in numerical computation. Give the answer as a fraction representing a rational number, so the result of the division of 26 by 11 is
displaystyle tfrac 26 11 .
But, usually, the resulting fraction should be simplified: the result of the division of 52 by 22 is also
displaystyle tfrac 26 11
. This simplification may be done by factoring out the greatest common divisor. Give the answer as an integer quotient and a remainder, so
displaystyle tfrac 26 11 =2 mbox remainder 4.
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. Give the integer quotient as the answer, so
displaystyle tfrac 26 11 =2.
This is sometimes called integer division.
Dividing integers in a computer program requires special care. Some
programming languages, such as 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.
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 −∞ (F-division); rarer styles can
occur – see
displaystyle p/q over r/s = p over q times s over r = ps over qr .
All four quantities are integers, and only p may be 0. This definition ensures that division is the inverse operation of multiplication. Of real numbers Division of two real numbers 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. By zero Main article: Division by zero 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 product of zero. Entry of such an expression into most calculators produces an error message. Of complex numbers Dividing two complex numbers results in another complex number when the divisor is not 0, which is found using the conjugate of the denominator:
p + i q
r + i s
( p + i q ) ( r − i s )
( r + i s ) ( r − i s )
p r + q s + i ( q r − p s )
p r + q s
q r − p s
displaystyle 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 − i s
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:
− i s
− i s
i ( q − s )
displaystyle pe^ iq over re^ is = pe^ iq e^ -is over re^ is e^ -is = p over r e^ i(q-s) .
Again all four quantities p, q, r, s are real numbers, and r may not
One can define the division operation for polynomials in one variable
over a field. Then, as in the case of integers, one has a remainder.
g − f
displaystyle left( frac f g right) '= frac f'g-fg' g^ 2 .
400AD Sunzi division algorithm Division by two Field Fraction (mathematics) Galley division Group Inverse element Order of operations Quasigroup Repeating decimal
^ For example: limx→0 sin x/x = 1.
^ Blake, A. G. (1887). Arithmetic. Dublin, Ireland: Alexander Thom
^ a b Weisstein, Eric W. "Division". MathWorld.
^ Derbyshire, John (2004). Prime Obsession: Bernhard Riemann and the
Greatest Unsolved Problem in Mathematics. New York City: Penguin
Books. ISBN 978-0452285255.
^ a b Weisstein, Eric W. "Division by Zero". MathWorld.
^ Weisstein, Eric W. "
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