quaternions

Complex numbers extend the reals by adjoining a single square root of -1, and they describe rotations of the plane. The quaternions are the bolder next step: adjoin three square roots of -1, call them i, j, k, and demand they anticommute. The price is that multiplication stops commuting; the reward is a four-dimensional number system perfectly tuned to rotations in three-dimensional space. Hamilton famously carved the defining relation into Brougham Bridge in 1843.

Formally, H is the four-dimensional real algebra with basis 1, i, j, k subject to i^2 = j^2 = k^2 = i*j*k = -1, which forces i*j = k = -j*i, j*k = i = -k*j, k*i = j = -i*k. Every nonzero quaternion is invertible — H is a division ring — and its center is exactly the real scalars, so H is a central division algebra over R. Its dimension over its center is 4 = 2^2, as the theory of division algebras requires for a square dimension.

The quaternions of norm 1 form the group of unit quaternions, which double-covers the rotation group SO(3): conjugation q -> q*v*q^(-1) rotates a pure-imaginary vector v, and this is the workhorse behind smooth interpolation of orientations in graphics and robotics. As an algebra, H is the unique nontrivial element of the Brauer group of R, since the only division algebras finite over R are R, C, and H (Frobenius's theorem).

A caveat worth stating: there is no further commutative-friendly step. The octonions extend H to dimension eight but sacrifice associativity, and the Hurwitz theorem shows that R, C, H, and the octonions are the only normed division algebras over R. The quaternions are thus the last associative stop on that road.