Editing Hurwitz quaternion (section)

==Division with remainder==
The ordinary integers and the [[Gaussian integer]]s allow a division with remainder or [[Euclidean division]].

For positive integers ''N'' and ''D'', there is always a quotient ''Q'' and a nonnegative remainder ''R'' such that
* ''N'' = ''QD'' + ''R'' where ''R'' < ''D''.
For complex or Gaussian integers ''N'' = ''a'' + i''b'' and ''D'' = ''c'' + i''d'', with the norm N(''D'') > 0, there always exist ''Q'' = ''p'' + i''q'' and ''R'' = ''r'' + i''s'' such that
* ''N'' = ''QD'' + ''R'', where N(''R'') < N(''D'').
However, for Lipschitz integers ''N'' = (''a'', ''b'', ''c'', ''d'') and ''D'' = (''e'', ''f'', ''g'', ''h'') it can happen that N(''R'') = N(''D''). This motivated a switch to Hurwitz integers, for which the condition N(''R'') < N(''D'') is guaranteed.<ref>{{harvnb|Conway|Smith|2003|p=56}}</ref>

Many algorithms depend on division with remainder, for example, [[Euclidean algorithm|Euclid's algorithm]] for the [[greatest common divisor]].