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Every multiplication you have done in school obeys one quiet rule: a × b = b × a. William Rowan Hamilton spent years stuck on a problem, and solved it on a walk in Dublin by giving that rule up. His quaternions were the first number system studied in which order matters. This post tells the story from his own letters and shows the rule-breaking in two lines of algebra.
Who was William Rowan Hamilton?
According to his MacTutor biography, Hamilton was born in Dublin on 4 August 1805 and died there on 2 September 1865. He learned languages early, met mathematics properly at 13 through Clairaut's algebra, and in 1822 found an error in Laplace's work on celestial mechanics, which brought him to the attention of John Brinkley, the Royal Astronomer of Ireland.
He entered Trinity College Dublin at 18. In 1827, still an undergraduate and aged 21, he was appointed Andrews' Professor of Astronomy, a post that carried the title Royal Astronomer of Ireland. He was knighted in 1835.
What had he done before the quaternions?
His early fame came from optics. In 1832, working from his mathematical theory of rays, he predicted that light entering certain crystals in a particular direction would spread out into a hollow cone, an effect called conical refraction. He asked his colleague Humphrey Lloyd to test it, and Lloyd observed it two months later. A theory had predicted something nobody had seen.
He then turned the same methods on mechanics. In "On a General Method in Dynamics" (1834) and a second paper the next year, he applied the same characteristic-function method to the motion of bodies. This work is why a central quantity of advanced mechanics, the energy function written H, is called the Hamiltonian.
Why couldn't he multiply triplets?
In 1833 Hamilton presented complex numbers as pairs of real numbers, (a, b) for a + bi. Multiplying complex numbers has a geometric meaning: it rotates and stretches points in a plane. Hamilton wanted the same thing for three-dimensional space, using triples a + bi + cj. He could add and subtract triples easily. He could not find a multiplication that worked.
The problem became a family joke. In a letter to his son Archibald dated 5 August 1865, kept in the Trinity College Dublin collection of Hamilton's letters, he recalled that every morning in October 1843 his sons asked, "Well, Papa, can you multiply triplets?" His answer: "No, I can only add and subtract them."
What happened on 16 October 1843?
The same letter describes the day. It was a Monday, and Hamilton was walking along the Royal Canal with his wife to preside at a council meeting of the Royal Irish Academy. "An electric circuit seemed to close; and a spark flashed forth," he wrote. He could not resist cutting the key formula into a stone of Brougham Bridge as they passed:
i² = j² = k² = ijk = −1
The carving has long since worn away, and in 1958 the Royal Irish Academy put up a plaque at the bridge. Hamilton read his first paper on quaternions to the Academy on 13 November 1843, and published "On quaternions; or on a new system of imaginaries in algebra" in the Philosophical Magazine in 1844.
What is a quaternion, and which rule does it break?
A quaternion has four parts: q = a + bi + cj + dk, where a, b, c and d are real numbers and i, j and k are three different square roots of −1. Hamilton's two insights were that triples cannot work but quadruples can, and that the price is commutativity.
You can see the price with the bridge formula. Start from ijk = −1 and multiply both sides on the right by k:
ijk × k = −k, so ij × k² = −k, so ij × (−1) = −k, so ij = k.
The full rules are ij = k, jk = i and ki = j, and reversing any pair flips the sign: ji = −k, kj = −i, ik = −j. Now compare two products:
(1 + i)(1 + j) = 1 + j + i + ij = 1 + i + j + k
(1 + j)(1 + i) = 1 + i + j + ji = 1 + i + j − k
Same two quaternions, different order, different answer. MacTutor calls quaternions the first noncommutative algebra to be studied. You have already met a cousin of this idea: the cross product of vectors also changes sign when you swap the order.
Where are quaternions used today?
Hamilton spent the rest of his life on quaternions. He published Lectures on Quaternions in 1853, and his larger Elements of Quaternions was unfinished when he died; it was published with a preface by his son.
The idea outlived the fashion for it. In a 1985 computer graphics paper, Ken Shoemake argued that quaternions are the natural way to describe the rotations of tumbling objects and moving cameras, and gave a method for smoothly interpolating between rotations. Quaternions are now a standard tool wherever software has to turn objects in three dimensions.
What can a student take from Hamilton's way of working?
Keep a hard problem alive. Hamilton returned to triplets for years. The breakthrough came on a walk, but only because the problem was already loaded in his mind.
Question the rule you take for granted. He found the answer when he dropped two assumptions: that three numbers were enough, and that order could not matter.
Write it down at once. In an 1858 letter to P. G. Tait in the same collection, he said he pulled out a pocket-book on the spot and made an entry.
Check consequences. The bridge formula is short, but every other rule follows from it, as the ij = k derivation shows.
Quaternions are not in the CBSE or JEE syllabus, but the ideas they grow from are. If complex numbers or the cross product still feel shaky, the Fermi Concept Library covers every concept in physics, chemistry, biology, mathematics and computing, so you can revise each one on its own. Our guides to complex numbers for JEE and vector algebra are the natural next step.
Before you close this page, work out (1 + i)(1 + j) and (1 + j)(1 + i) yourself and confirm that they differ.
