Back when the geometric interpretation of complex numbers as a plane was reasonably fresh, Sir William Hamilton became interested in finding a system of algebra that would allow him to express three dimensional space in the same way. To do this, Hamilton needed a way to add and multiply points in 3 dimensional space together.

Addition came easy. Picking some arbitrary origin and axes

*1*,

**, and**

*i***to work with, Hamilton just defined**

*j**(a + b*.

**i**+c**j**) + (d + e**i**+f**j**) = ( a + d ) + ( b + e )**i**+ ( c + f )**j**Multiplication, though, was a problem. Assuming that these new quantities were distributive, Hamilton needed to define

**in such a way that various other properties still held. Despite his best efforts, he couldn't do it.**

*ij*