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# What contributions to math did Sir William Rowan Hamilton make?

I have to write a school report on a mathematician and what contribution they made. Mine is on Sir William Rowan Hamilton. All the websites are confusing and do not say exactly what he helped with. Help?

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http://www.hamilton.tcd.ie/

Sir William Rowan Hamilton (August 4, 1805 – September 2, 1865) was an Irish mathematician, physicist, and astronomer who made important contributions to the development of optics, dynamics, and algebra. His discovery of quaternions is perhaps his best known investigation. Hamilton's work was also significant in the later development of quantum mechanics. Hamilton is said to have showed immense talent at a very early age, prompting astronomer Bishop Dr. John Brinkley to remark in 1823 of Hamilton at the age of eighteen: “This young man, I do not say will be, but is, the first mathematician of his age.”

Life

William Rowan Hamilton's mathematical career included the study of geometrical optics, adaptation of dynamic methods in optical systems, applying quaternion and vector methods to problems in mechanics and in geometry, development of theories of conjugate algebraic couple functions (in which complex numbers are constructed as ordered pairs of real numbers), solvability of polynomial equations and general quintic polynomial solvable by radicals, the analysis on Fluctuating Functions (and the ideas from Fourier analysis), linear operators on quaternions and proving a result for linear operators on the space of quaternions (which is a special case of the general theorem which today is known as the Cayley-Hamilton Theorem). Hamilton also invented "Icosian Calculus", which he used to investigate closed edge paths on a dodecahedron that visit each vertex exactly once.

Quaternions

Quaternion Plaque on Broome BridgeThe other great contribution Hamilton made to mathematical science was his discovery of quaternions in 1843.

Hamilton was looking for ways of extending complex numbers (which can be viewed as points on a 2-dimensional plane) to higher spatial dimensions. He could not do so for 3 dimensions, and in fact it was later shown that it is impossible. Eventually Hamilton tried 4 dimensions and created quaternions. According to Hamilton, on October 16 he was out walking along the Royal Canal in Dublin with his wife when the solution in the form of the equation i2 = j2 = k2 = ijk = − 1

suddenly occurred to him; Hamilton then promptly carved this equation using his penknife into the side of the nearby Broom Bridge (which Hamilton called Brougham Bridge), for fear he would forget it. Since 1989, the National University of Ireland, Maynooth has organized a pilgrimage, where mathematicians take a walk from Dunsink observatory to the bridge where, unfortunately, no trace of the carving remains, though a stone plaque does commemorate the discovery.

The quaternion involved abandoning commutativity, a radical step for the time. Not only this, but Hamilton had in a sense invented the cross and dot products of vector algebra. Hamilton also described a quaternion as an ordered four-element multiple of real numbers, and described the first element as the 'scalar' part, and the remaining three as the 'vector' part.

In 1852, Hamilton introduced quaternions as a method of analysis. His first great work is Lectures on Quaternions (Dublin, 1852). Hamilton confidently declared that quaternions would be found to have a powerful influence as an instrument of research. He popularized quaternions with several books, the last of which, Elements of Quaternions, had 800 pages and was published shortly after his death.

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William Rowan Hamilton

• 5 years ago