Hamiltonian, the Quarternions and the Octonians

We’ve just extended the number system we can deal with by saying that we are perfectly at liberty (with certain important qualifications) to take the square root of a negative number and it gives you a multiple of i. We will see in the following sections that this is incredibly useful for being able to extend our mathematical machinery to new domains. In fact, this extension of the number system we are dealing with is not at all unique. The idea of irrational numbers was for a very long time believed not to be true. The possibility that you could have a number which was not a fraction seemed absurd. In fact even more basic than that, the idea of 0 as an important concept was not conceived for a long time in the history of mathematics. Zero was introduced by Indian mathematicians in the 9th century AD and the idea of irrational numbers was only dreamt up by the Greeks in around the 5th century BC.

So, we’ve just introduced a new dimension in numbers, but can we do any more than this? Well, it turns out that one can extend the system further and have an algebraic structure whereby there are three numbers, called i, j and k which each square to -1. The strange thing about these numbers however is that they do not commute. We certainly know that with normal numbers 3\times 4=4\times 3 and indeed with complex numbers z_1\times z_2=z_2\times z_1. This property of being able to swap the order is called commutation and these numbers are said to commute under multiplication. However, with this extension to i, j, k, known as the quaternionic numbers, this is no longer true and i\times j\ne j \times i. This alters the structure of such algebraic expressions greatly and makes mathematics with the quaternions very different. William Hamilton claimed to realise the necessary algebraic properties of the quaternions as he was crossing a bridge in Dublin and the rules are written there to this day. See figure \ref{quaternions}.

In fact it turns out that one can extend things still further, to a number system called the octonians. These have 7 numbers, all of which square to -1. In this case however, not only do they not commute, but multiplication within this number system is not associative. That is (o_1o_2)o_3\ne o_1(o_2o_3). The octonions are actually the largest possible ‘normed divisor algebra’ . We can’t define a larger number system without losing some very important properties.

If you want to know more about these algebraic structures, a good place to look is this interview with John Baez.

Taken from Wikipeida - William Rowan Hamilton Plaque - geograph.org.uk

William Hamilton’s plaque on the bridge on which he realised how the quaternionic algebra had to work.

Image from here.

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