Hamilton was a child prodigy, and not just in mathematics; he also managed to learn an extraordinary number of languages, some of them very obscure. In 1823 he entered Trinity College in his native city of Dublin, and four years later at the age of 22 was appointed professor of astronomy and Astronomer Royal for Ireland; these posts were given to him in order that he could continue to research unhampered by teaching commitments.

In 1827 he produced his first original work, in the theory of optics, expounded in his paper A Theory of Systems of Rays. In 1832 he did further theoretical work on rays, and predicted conical refraction under certain conditions in biaxial crystals. This was soon confirmed experimentally. In dynamics he introduced Hamilton's equations – a set of equations (similar to equations of Joseph Lagrange) describing the positions and momenta of a collection of particles. The equations involve the Hamiltonian function, which is used extensively in quantum mechanics. Hamilton's principle is the principle that the integral with respect to time of the kinetic energy minus the potential energy of a system is a minimum.

One of Hamilton's most famous discoveries was that of quaternions. These are a generalization of complex numbers with the property that the commutative law does not hold for them (i.e., A × B does not equal B × A). Hamilton's discovery of such an algebraic system was important for the development of abstract algebra; for instance, the introduction of matrices. Hamilton spent the last 20 years of his life trying to apply quaternions to problems in applied mathematics, although the more limited theory of vector analysis of Josiah Willard Gibbs was eventually preferred. Toward the end of his life Hamilton drank increasingly, eventually dying of gout.