"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 .