Complex Conjugates versus Quaternion Conjugates

The conjugate of a complex number

z = a + bi

is the complex number

z^* = a - bi

Taking the conjugate flips over a complex number, taking its reflection in the real axis.

Multiplication stretches and rotates complex numbers, and addition translates complex numbers. You can’t flip the complex plane over by any series of dilatations, rotations, and translations.

The situation is different for quaternions. The conjugate of a quaternion

q = a + bi + cj + dk


q^* = a - bi - cj - dk

You can flip four dimensional space over by a series of dilations, rotations, and translations. Namely

q^* = -\frac{1}{2}(q + iqi + jqj + kqk)

To prove this equation, let’s first see what happens when you multiply q on both sides by i:

i(a + bi + cj + dk)i = -a - bi + cj + dk

That is, the effect of multiplying on both sides by i is to flip the sign of the real component and the i component.

Multiplying on both sizes by j or k works analogously: it flips the sign of the real component and its component, and leaves the other two alone.

It follows that

\begin{align*} q +iqi + jqj + kqk &= \phantom{-}a + bi + cj +dk \\ & \,\phantom{=} - a - bi + cj + dk \\ & \,\phantom{=} - a + bi - cj + dk \\ & \,\phantom{=} - a + bi + cj - dk \\ &= -2a + 2bi + 2cj + 2dk\\ &= -2q^* \end{align*}

and so the result follows from dividing by −2.

Update: There’s an analogous theorem for octonions.

More on quaternions