## William Rowan Hamilton (1805 - 1865)

Hamilton was something of a child prodigy and was one of Ireland’s greatest mathematicians, despite suffering from depression throughout his life.

Born in Dublin, he was orphaned at an early age and went to live with his uncle. There he learnt to read English, Latin, Greek and Hebrew - by the age of 5!

At 10 he could read Sanscrit and speak French! He obviously had a natural aptitude for languages, which helped him greatly when at 13 he studied a book on algebra written in French.

He went on to read Newton and Laplace, and amazingly found an error in Laplace’s Mechanique Celeste, on which he wrote a paper in 1823, causing a bit of a stir!

Whilst only 21 - and still an undergraduate at Trinity College in Dublin - he was appointed the Royal Astronomer of Ireland, Director of the Dunsink Observatory, and Professor of Astronomy at the University!

During his first year there he fell into a deep depression and took to writing poetry. On leaving college, he toured England, where he met the poet Wordsworth. Wordsworth advised Hamilton to stick to science!

He eventually married, and although he and his wife had three children, he did not hold his wife in high regard. After the birth of his daughter, he became depressed again.

He continued to work, but sadly died from gout after hearing that he was the first foreign mathematician to be elected as a member of the National Academy of Science in America.

## Hamilton’s mathematics

Normally, algebra involves numbers and how they behave under certain operations - usually addition, multiplication etc. So for example we write,

x.x.x.x=x^4 \mbox{and}\ x+x+x+x=4x

Hamilton had the ingenious idea of looking at pairs of numbers, and their algebra. There were good reasons to do this, as vectors have two parts - a direction and a size; vectors are things such as speed, acceleration etc and these are vitally important in almost all branches of Science.

He found that the algebras he investigated had many different properties to the ordinary algebra of single numbers, but that they shed important light on the physical things from which they were derived.

He showed that complex numbers could also be regarded as a pair of numbers and that this was a powerful new way of thinking about them.

Let us explore an example. Suppose that (a, b) are (c, d) are two such pairs of numbers and that their multiplication works like this:

Then we have,

(0,1)(0,1)=(0-1,0+0)=(-1,0)

i.e

(0,1)^2=({-1},0)

and that, to Hamilton, suggested something squared is negative - which is complex numbers!

In 1833 he sent a paper to the Irish Academy where the algebra of complex numbers appears as an algebra of ordered pairs of numbers.

This led to his work on the algebra of triples and quadruples - particularly on their multiplication - and eventually led to his development of the algebra of quaternions, the first non-commutative algebra.

This occupied him for the rest of his life, writing the Treatise on Quaternions, and the Elements of Quaternions (this was published after his death).

Hamilton also contributed to the toy industry. Based on his work on graph theory, he developed a game that consisted of a regular dodecahedron (made from wood), with each of the 20 vertices labelled with the name of a famous city.

The aim of the game was to find a path along the edges of the dodecahedron so that each city was only passed once. Such a winning path is called a Hamilton Cycle.