A quaternion extends complex numbers by introducing three imaginary basis elements $\mathbf i$, $\mathbf j$ and $\mathbf k$:

$$q=w+x\mathbf i+y\mathbf j+z\mathbf k.$$

They satisfy

$$\mathbf i^2=\mathbf j^2=\mathbf k^2=-1$$

and

$$\mathbf i\mathbf j=\mathbf k,\qquad \mathbf j\mathbf k=\mathbf i,\qquad \mathbf k\mathbf i=\mathbf j,$$

while reversing the order changes the sign.

Multiplication is not commutative

In general,

$$pq\ne qp.$$

This noncommutativity turns out to match the order-sensitive composition of three-dimensional rotations.

Conjugate and norm

The conjugate of

$$q=w+x\mathbf i+y\mathbf j+z\mathbf k$$

is

$$q^*=w-x\mathbf i-y\mathbf j-z\mathbf k.$$

The norm satisfies

$$|q|^2=qq^*=w^2+x^2+y^2+z^2.$$

For $q\ne0$,

$$q^{-1}=\frac{q^*}{|q|^2}.$$

Unit quaternions

Quaternions with norm one form a multiplicatively closed set. These unit quaternions provide the representation used for spatial rotations.