This is not my post. I copied it from John Baez' site @ http://math.ucr.edu/home/baez/dublin/index.html#hawking. This describes how Hamilton invented the quaternions.
In 1835, at the age of 30, Hamilton had discovered how to treat complex numbers as pairs of real numbers. Fascinated by the relation between complex numbers and 2-dimensional geometry, he tried for many years to invent a bigger algebra that would play a similar role in 3-dimensional geometry. In modern language, it seems he was looking for a 3-dimensional normed division algebra. His quest built to its climax in October 1843. He later wrote to his son:
Every morning in the early part of the above-cited month, on my coming down to breakfast, your (then) little brother William Edwin, and yourself, used to ask me: "Well, Papa, can you multiply triplets?" Whereto I was always obliged to reply, with a sad shake of the head: `No, I can only add and subtract them".The problem was that there exists no 3-dimensional normed division algebra. He really needed a 4-dimensional algebra.
That is to say, I then and there felt the galvanic circuit of thought close; and the sparks which fell from it were the fundamental equations between i,j,k; exactly such as I have used them ever since.And in a famous act of mathematical vandalism, he carved these equations into the stone of the Brougham Bridge:
He spent the rest of his life working on quaternions.
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