Complex numbers bolt one imaginary unit i onto the reals and, as a reward, multiplication becomes rotation of the plane. The obvious dream is to do the same in three dimensions. William Rowan Hamilton chased it for years — and discovered you cannot do it with two imaginary units, but you can with three. The result is the quaternions, the four-dimensional numbers that run essentially every 3-D rotation in graphics, robotics, and aerospace.

Three imaginary units, one rule

A quaternion extends a + bi with three imaginary units i, j, k:

Hamilton's entire theory rests on a single line linking the three units, the one he is said to have carved into a bridge:

Step 1 — derive the products of pairs. From ijk = -1, multiply on the right by k and use k^2 = -1: ij\,k^2 = -k, so -ij = -k, giving ij = k. The same trick on the others yields the cyclic rule:

Step 2 — discover that order matters. Reverse a product. Since ji = -ij (check: jiijk \to bookkeeping with i^2 = -1 forces the sign flip), we get the anti-cyclic rule:

So ij = k but ji = -k: quaternion multiplication is non-commutative. Swapping the order of a product can flip its sign — which is exactly the behaviour you want from rotations, where doing A then B differs from B then A.

Scalar plus vector

The four numbers split naturally into a real scalar and a 3-D vector of the imaginary parts. Write

treating (i, j, k) as the basis directions (\hat{x}, \hat{y}, \hat{z}). This is the form that makes everything click, because the multiplication rule turns out to be built from the two vector products you already know.

Step 1 — multiply two quaternions as (s_1 + \mathbf{v}_1)(s_2 + \mathbf{v}_2). Distribute the four pieces:

Step 2 — expand the vector-times-vector term. For two pure imaginary quaternions the product splits, by the i,j,k rules above, into a scalar piece and a vector piece. The scalar piece is exactly the negative dot product, and the vector piece is exactly the cross product:

(The -\mathbf{v}_1\cdot\mathbf{v}_2 comes from the i^2 = j^2 = k^2 = -1 diagonal terms; the \mathbf{v}_1\times\mathbf{v}_2 comes from the cyclic ij = k off-diagonal terms.)

Step 3 — assemble the full product. Collect the scalar pieces and the vector pieces:

Both classical vector products — the dot and the cross — fall straight out of one quaternion multiplication. And because the cross product obeys \mathbf{v}_1\times\mathbf{v}_2 = -\,\mathbf{v}_2\times\mathbf{v}_1, the product is non-commutative exactly when the two vector parts are not parallel.

Conjugate, norm, inverse

Step 1 — the conjugate flips the vector part:

Step 2 — multiply \mathbf{q}\,\mathbf{q}^{*}. Use the product rule with s_2 = w and \mathbf{v}_2 = -\mathbf{v}. The vector part is w(-\mathbf{v}) + w\mathbf{v} + \mathbf{v}\times(-\mathbf{v}) = \mathbf{0}, and the scalar part is w^2 - \mathbf{v}\cdot(-\mathbf{v}) = w^2 + \mathbf{v}\cdot\mathbf{v}. So the imaginary parts vanish and we are left with a pure real number, the squared norm:

Step 3 — read off the inverse. Divide \mathbf{q}\,\mathbf{q}^{*} = \lVert\mathbf{q}\rVert^2 by the scalar \lVert\mathbf{q}\rVert^2:

For a unit quaternion (\lVert\mathbf{q}\rVert = 1) this collapses to the bargain that makes them so cheap in code: \mathbf{q}^{-1} = \mathbf{q}^{*} — to undo the rotation, just negate the vector part. Those unit quaternions are precisely the ones that rotate 3-D space.

Multiply two quaternions live

Set the eight coefficients of \mathbf{q}_1 = w_1 + x_1 i + y_1 j + z_1 k and \mathbf{q}_2 = w_2 + x_2 i + y_2 j + z_2 k. The diagram works the rule (s_1 s_2 - \mathbf{v}_1\cdot\mathbf{v}_2,\ s_1\mathbf{v}_2 + s_2\mathbf{v}_1 + \mathbf{v}_1\times\mathbf{v}_2) live, showing each of the four output components — and the running norm — as you move the knobs.