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# Complex Numbers, Quaternions and Beyond

## Sam Evington, Undergraduate Mathematics Seminar

Week 2, Wednesday 25 January HT 2012
8:15pm

In 1835 Hamilton gave a rigorous definition of a complex number as simply an order pair of real numbers $(a,b)$. He defined addition coordinatewise and multiplication by $(a,b)(c,d) = (ac-bd,ad+bc)$. He then went on to show that the expected properties follow from these definitions and nothing else. There’s no need to pluck a square root of $-1$ out of thin air! This leads one naturally to ask if there is a similar rule for multiplying triples of reals $(a,b,c)$. Given the many applications of complex numbers to 2-dimensional geometry and physics a generalisation to the 3 dimensions of space would be immensely useful.

Alas we shall see that no such system exists. All is not lost however. If one ventures into a fourth dimension (and gives up the notion that $A$ times $B$ should equal $B$ times $A$ then a generalisation of complex numbers does exist. These are the Quaternions, and we shall establish their basic theory then look at their applications in 3-dimensional geometry, the invention of vector and modern physics.

If time permits we shall look beyond Quaternions and discover why (with perhaps 1 exception) there’s nothing else worthy of being called a number system.