Hamilton's father did not have a university education, and it is thought that Hamilton's genius came from his mother, Sarah Hutton. By the age of five, William had already learned Latin, Greek, and Hebrew. William soon mastered additional languages, but a turning point came in his life at the age of 12 when he met Zerah Colburn. Colburn could perform amazing mental arithmetical feats and Hamilton joined in competitions of arithmetical ability with him. It appears that losing to Colburn sparked Hamilton's interest in mathematics.

Hamilton's introduction to mathematics came at the age of 13 when he studied Clairaut's Algebra. At age 15 he started studying the works of Newton and Laplace. In 1822 Hamilton found an error in one of Laplace's works. He entered Trinity College, Dublin at the age of 18 and in his first year he obtained an optime in Classics, a distinction only awarded once in 20 years.

After love troubles, he became ill and at one point he even considered suicide. In this period he turned to poetry, which was a habit that he pursued for the rest of his life in times of anguish.

In 1827 he was appointed Andrews' Professor of Astronomy in Trinity College while he was still an undergraduate. The professorship carried the honorary title Royal Astronomer of Ireland. It turned out that Hamilton had made an poor choice as he lost interest in astronomy and spent all his time on mathematics.

Before beginning his duties in this prestigious position, Hamilton toured England and Scotland. He met the poet Wordsworth and they became friends. Wordsworth told Hamilton quite forcibly that his talents were in science rather than poetry.

In 1832 Hamilton published a treatise on the characteristic function applied to optics. Near the end of the work he applied the characteristic function to study Fresnel's wave surface. In 1833 Hamilton read a paper to the Royal Irish Academy expressing complex numbers as algebraic couples, or ordered pairs of real numbers. He used algebra in treating dynamics in On a General Method in Dynamics in 1834.

Hamilton was knighted in 1835. After the discovery of algebraic couples, he tried to extend the theory to triplets, and this became an obsession that plagued him for many years. His cousin Arthur died, and his wife temporarily left him with his children. At this point, William became depressed and started to have problems with alcohol so his sister came to live with him. Eventually he came up with the quaternions. Hamilton felt this discovery would revolutionize mathematical physics and he spent the rest of his life working on them. He wrote a poor book on the subject.

Determined to produce a work of lasting quality, Hamilton began to write another book, which ended up 800 pages and took 7 years to write. In fact the final chapter was incomplete when he died and the book was finally published with a preface by his son. Hamilton died from a severe attack of gout shortly after receiving the news that he had been elected the first foreign member of the National Academy of Sciences.