William Rowan Hamilton is widely regarded as Ireland’s greatest scientist. Many people know of his most famous contribution to maths, quaternions. However, relatively few know just what quaternions are or what else Hamilton achieved.
Despite their arcane name, quaternions are actually quite simple. They are generalisations or extensions of the ordinary decimal numbers such as 17.3 or 3.1415 which we encounter every day.
Quaternions consist of four of these regular numbers bundled together in a particular way into a composite number, a kind of super-number, which gives them special properties. Hamilton took the name “quaternion” from the Bible, where it refers to a squad of four Roman soldiers.
Other contributions
While quaternions are the most widely known of Hamilton’s contributions to science, they are by no means his only contribution, nor are they his most important.
Hamilton also did extremely innovative work in the mathematical modelling of light propagation through crystals and predicted a new phenomenon, known as conic refraction, entirely on the basis of his theoretical model. This phenomenon caused a sensation in scientific circles when it was subsequently verified experimentally.
However, Hamilton’s greatest legacies are the important advances he made in the field of theoretical physics known as mechanics, which deals with the motion of objects under the influence of forces such as gravity.
Powerful
Although Newton’s Laws of Motion were known for 200 years, Hamilton developed a new approach to mechanics which, although equivalent to Newton’s Laws, is far more powerful. This approach is central to applications as diverse as navigating space probes through the solar system to designing drugs.
Hamilton’s work in mechanics took on even greater significance in the early part of the 20th century when it turned out to be critical to the development of quantum mechanics (QM). This is one of the twin pillars of our understanding of the physics of our universe (the other being Einstein’s Theory of General Relativity)
QM led directly to such technological breakthroughs as lasers and computer chips. QM also underpins emerging technologies such as nanotechnology.
Delving deeper
[This section provides a more detailed explanation of quaternions. It should be accessible to Leaving Certificate students or those with some familiarity with imaginary and complex numbers.]
The real or decimal numbers that we use on a daily basis, such as 3.1415, have their roots in India in the 6th century. However, they only entered mainstream use in Europe much later, in the 16th century. Complex numbers also emerged in the 16th century, as an extension to the real numbers.
The basic idea of complex numbers is to form a compound number consisting of an ordered pair of two real numbers such as (14, 3) or (13.5, 3.1). This is usually written as 13.5 + i 3.1 where i denotes the square root of -1.
Two of these complex numbers can be added in the obvious way to produce a third complex number:
(a + ib) + (c + id ) = (a + c) + i ( b + d ) .
However, the clever bit is the rule for multiplying them which involves cross-over between the first and second elements. The result of multiplying two complex numbers is determined by the rule
i2 = -1
and lead to
(a + ib )
