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  Octonions & Sedenions

 
. THE BEGINNING - How did octonions came about ?
         Here is a brief account based on the most widely accepted sequence of events.
William Rowan Hamilton, an Irish mathematician also known for his contribution in mechanics, was trying to generalize complex numbers (which he considered as an ordered pair of two real numbers) to a hypercomplex number which may be viewed as an ordered set of more than two numbers. As he walked along the Royal Canal, on his way to the Royal Irish Academy in october 1843, it came clear to him in a "flash" what this new number system should be. He carved on a stone of Brougham Bridge its multiplication rule for the number system with four ordered real numbers and called them quaternions.  A commemorative plaque now marks the place.  Immediately after his publication on quaternions J.T.Graves (in U.K.) wrote to Hamilton of the 8-dimensional extension and called them octaves. Unfortunately, Graves's discovery was not published at that time.  Arthur Cayley (in U.K.) published a paper in 1845 which outlined how to construct a theory for octonion with seven imaginary roots of -1. 
Here are some of my old snapshots of the plaque and the Brougham Bridge, just outside of Dublin in Ireland.
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The Plaque reads:
Here as he walked by
on the 16th of October 1843
Sir William Rowan Hamilton
in a flash of genius discovered 
the fundamental formula for
quaternion multiplication
i 2  =  j 2  =  k 2  =  i j k  =  - 1
and cut it on a stone of this bridge.

By the way that is not the author  in the photograph.