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Quaternions

Hamilton made his discovery of the algebra of quaternions in 1843. Among much prior related work, in 1840 Benjamin Olinde Rodrigues had reached a result that amounted to their discovery in all but name.

Hamilton was looking for ways of extending complex numbers (which can be viewed as points on a 2-dimensional Argand diagram) to higher spatial dimensions. In working with four dimensions, rather than three, he created quaternion algebra. According to Hamilton, on 16 October he was out walking along the Royal Canal in Dublin with his wife when the solution in the form of the equation

occurred to him; Hamilton then carved this equation using his penknife into the side of the nearby Broom Bridge (which Hamilton called Brougham Bridge).

The quaternions involved abandoning the commutative law, a radical step for the time. In the context of this prototype geometric algebra, Hamilton also introduced the cross and dot products of vector algebra, the quaternion product being the cross product minus the dot product as scalar. Hamilton also described a quaternion as an ordered four-element multiple of real numbers, and described the first element as the "scalar" part, and the remaining three as the "vector" part. He coined the neologisms "tensor" and "scalar", and was the first to use the word "vector" in the modern sense.