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 \tfrac{26}{11} (or as a
mixed 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 the
greatest common divisor. • Give the answer as an integer
quotient and a
remainder, 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 called
Euclidean division, because it is the basis of the
Euclidean algorithm. • Give the integer quotient as the answer, so \tfrac{26}{11} = 2. This is the
floor function applied to case 2 or 3. It is sometimes called
integer division, and denoted by "//". Dividing integers in a
computer program requires special care. Some
programming languages 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 , , , 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
modulo operation for the details.
Divisibility rules can sometimes be used to quickly determine whether one integer divides exactly into another.
Of rational numbers The result of dividing two
rational numbers 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 {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 nonzero). It is defined such that
a/
b =
c if and only if
a =
cb and
b ≠ 0.
Of complex numbers Dividing two
complex 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 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: {p e^{iq} \over r e^{is}} = {p e^{iq} e^{-is} \over r e^{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 be 0.
Of polynomials 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. See
Euclidean division of polynomials, and, for hand-written computation,
polynomial long division or
synthetic division.
Of matrices One can define a division operation for matrices. The usual way to do this is to define , where denotes the
inverse 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 is not
commutative, 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. 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. That is, and , where and denote the pseudoinverses of and .
Abstract algebra In
abstract algebra, given a
magma 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). A magma for which both and exist and are unique for all
a and all
b (the
Latin square property) is a
quasigroup. 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 the
cancellation property. Examples include
matrix algebras,
quaternion algebras, and quasigroups. 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 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 normed division algebra must be
isomorphic to either the real numbers
R, the
complex numbers
C, the
quaternions
H, or the
octonions
O.
Calculus The
derivative of the quotient of two functions is given by the
quotient rule: {\left(\frac fg\right)}' = \frac{f'g - fg'}{g^2}. == Division by zero ==