Editing William Rowan Hamilton (section)

==Mathematics==

Hamilton's [[mathematical]] studies seem to have been undertaken and carried to their full development without collaboration, and his writings do not belong to any particular school. He was intended by the university authorities who elected him to the Professorship of Astronomy to spend his time as he best could for the advancement of science, without restrictions.<ref name="EB1911"/>

===Quaternions===
{{main|History of quaternions}}
[[File:William Rowan Hamilton Plaque - geograph.org.uk - 347941.jpg|thumb|right|upright=1.4|Quaternion Plaque on [[Broom Bridge]] in Dublin]]
Hamilton made his discovery of the algebra of [[quaternion]]s in 1843.<ref name="Bruno2003"/>{{rp|210}} Among much prior related work, in 1840 [[Olinde Rodrigues|Benjamin Olinde Rodrigues]] had reached a result that amounted to their discovery in all but name.<ref>{{cite journal | author = Simon L. Altmann | title = Hamilton, Rodrigues and the quaternion scandal | journal = Mathematics Magazine | date = 1989 | volume = 62 | pages = 291–308 | doi = 10.2307/2689481 | jstor = 2689481 | issue = 5 }}</ref>

Hamilton was looking for ways of extending [[complex number]]s (which can be viewed as [[Point (geometry)|points]] on a 2-dimensional [[Argand diagram]]) to higher spatial dimensions. In working with four dimensions, rather than three, he created quaternion algebra. According to Hamilton, on 16 October he was out walking along the [[Royal Canal]] in Dublin with his wife when the solution in the form of the equation

:{{bigmath|1=''i''<sup>2</sup> = ''j''<sup>2</sup> = ''k''<sup>2</sup> = ''ijk'' = −1}}

occurred to him; Hamilton then carved this equation using his penknife into the side of the nearby [[Broom Bridge]] (which Hamilton called Brougham Bridge).<ref name="Bruno2003"/>{{rp|210}}

The quaternions involved abandoning the [[commutative law]], a radical step for the time. In the context of this prototype [[geometric algebra]], Hamilton also introduced the cross and dot products of vector algebra, the quaternion product being the [[cross product]] minus the [[dot product]] as [[Scalar (mathematics)|scalar]]. 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. He coined the [[neologism]]s "tensor" and "scalar", and was the first to use the word "vector" in the modern sense.<ref>{{Cite web |url=http://jeff560.tripod.com/v.html |title=Earliest Known Uses of Some of the Words of Mathematics (V) |access-date=15 June 2019 |archive-date=5 September 2015 |archive-url=https://web.archive.org/web/20150905141835/http://jeff560.tripod.com/v.html |url-status=live }}</ref>

===Other mathematical works===
Hamilton looked into the [[solution of the quintic]] in the [[theory of equations]], examining the results arrived at by [[Niels Henrik Abel]], [[George Jerrard]] and others in their respective research. There is Hamilton's paper on fluctuating functions in [[Fourier analysis]], and the invention of the [[hodograph]]. Of his investigations into the solutions, especially by [[numerical approximation]], of certain classes of physically important differential equations, only parts were published, at intervals, in the ''[[Philosophical Magazine]]''.<ref name="EB1911"/>

Hamilton also introduced the [[icosian game]] or ''Hamilton's puzzle''. It is based on the concept of a [[Hamiltonian path]] in [[graph theory]].<ref name="ODNB"/>