<P> To avoid problems when A and / or B do not exist, division can also be defined as multiplication with the pseudoinverse, i.e., A / B = AB and A \ B = A B, where A and B denote the pseudoinverse of A and B . </P> <P> In abstract algebra, given a magma with binary operation ∗ (which could nominally be termed multiplication), left division of b by a (written a \ b) is typically defined as the solution x to the equation a ∗ x = b, if this exists and is unique . Similarly, right division of b by a (written b / a) is the solution y to the equation y ∗ a = b . Division in this sense does not require ∗ to have any particular properties (such as commutativity, associativity, or an identity element). </P> <P> "Division" in the sense of "cancellation" can be done in any magma by an element with the cancellation property . Examples include matrix algebras and quaternion algebras . A quasigroup is a structure in which division is always possible, even without an identity element and hence inverses . 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 . </P> <P> The derivative of the quotient of two functions is given by the quotient rule: </P>

When does division of integers not have meaning and why