Quaternions 


Multiplication of quaternions does not necessarily commute  note that $ij=ji$ . The algebra of Quaternions was one of the first noncommutative algebras to be devised.
For 13 years Hamilton had tried to find a way to extend the complex numbers into three dimensions, but every attempt to multiply triples (with associated division) failed. Finally he realised that the additional dimension would work. This final revelation occurring in a sudden moment of inspiration whilst on a walk along a canal in Dublin. A plaque now commemorates the location.