The Strange Numbers That Birthed Modern Algebra

Everything you could do with the real and complex numbers, you could do with the quaternions, except for one jarring difference. Whereas 2 × 3 and 3 × 2 both equal 6, order matters for quaternion multiplication. Mathematicians had never encountered this behavior in numbers before, even though it reflects how everyday objects rotate. Place your phone face-up on a flat surface, for example. Spin it 90 degrees to the left, and then flip it away from you. Note which way the camera points. Returning to the original position, flip it away from you first and then turn it to the left second. See how the camera points to the right instead? This initially alarming property, known as non-commutativity, turns out to be a feature the quaternions share with reality.

But a bug lurked within the new number system too. While a phone or arrow turns all the way around in 360 degrees, the quaternion describing this 360-degree rotation only turns 180 degrees up in four-dimensional space. You need two full rotations of the phone or arrow to bring the associated quaternion back to its initial state. (Stopping after one turn leaves the quaternion inverted, because of the way imaginary numbers square to –1.) For a bit of intuition about how this works, take a look at the rotating cube above. One turn puts a twist in the attached belts while the second smooths them out again. Quaternions behave somewhat similarly.

Upside-down arrows produce spurious negative signs that can wreak havoc in physics, so nearly 40 years after Hamilton’s bridge vandalism, physicists went to war with one another to keep the quaternion system from becoming standard. Hostilities broke out when a Yale professor named Josiah Gibbs defined the modern vector. Deciding the fourth dimension was entirely too much trouble, Gibbs decapitated Hamilton’s creation by lopping off the a term altogether: Gibbs’ quaternion-spinoff kept the i, j, k notation, but split the unwieldy rule for multiplying quaternions into separate operations for multiplying vectors that every math and physics undergraduate learns today: the dot product and the cross product. Hamilton’s disciples labeled the new system a “monster,” while vector fans disparaged the quaternions as “vexatious” and an “unmixed evil.” The debate raged for years in the pages of journals and pamphlets, but ease of use eventually carried vectors to victory.