Editing Division (mathematics) (section)

=== Abstract algebra ===
In [[abstract algebra]], given a [[Magma (algebra)|magma]] with binary operation ∗ (which could nominally be termed multiplication), '''left division''' of ''b'' by ''a'' (written {{math|''a'' \ ''b''}}) is typically defined as the solution ''x'' to the equation {{math|1=''a'' ∗ ''x'' = ''b''}}, if this exists and is unique.  Similarly, '''right division''' of ''b'' by ''a'' (written {{math|''b'' / ''a''}}) is the solution ''y'' to the equation {{math|1=''y'' ∗ ''a'' = ''b''}}.  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 {{math|''a'' \ ''b''}} and {{math|''b'' / ''a''}} 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 (mathematics)|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 (mathematics)|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 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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