Meaning of the sign of a complex number

Question

What is the meaning and the practical uses of the sign (signum function) of a complex number $z$ defined as $\frac{z}{|z|}$? Does it also extend to quaternions?

Answer 1

If we draw a line from the complex number $z$ to the origin and then we indicate where our line intersects with the unit circle we have identified the point $z/|z|$. Note that when $z$ is real this will simply be $1$ or $−1$.

Generally, we can use this expression $sign$ to mean the direction of the point from the origin. This idea intuitively goes from 1 dimension (+ or -) to $2$ dimensions (any point on the unit circle) to $3$ dimensions (any point on the unit sphere) we can extend it to the quaternians (visualizing the 4th dimension will make my analogy a little harder). People do look at something called unit quarternions and the verser which may be a helpful lead on how this idea extends to 4 dimensions.

Answer 2

You can think of a complex number $z=a+ib$ as a vector in the complex plane, and as such it has a direction and a length. The sign of $z$ gives the normalized direction of $z$ as a unit vector lying on the unit circle. We normalise $z$ by dividing by its modulus: $$\operatorname{sgn}(z)=\frac{z}{|z|}=\frac{a+bi}{\sqrt{a^2+b^2}}$$ This gives a unit complex number. For example take $z=5+3i$. Then $$\operatorname{sgn}(5+3i)=\frac{5+3i}{|5+3i|}=\frac{5}{\sqrt{34}}+\frac{3}{\sqrt{34}}i$$ where $$\left(\frac{5}{\sqrt{34}}\right)^2+\left(\frac{3}{\sqrt{34}}\right)=1$$

For a quaternion $q=a+bi+cj+dk$, the case is similar: $$\operatorname{sgn}(q)=\frac{a+bi+cj+dk}{|a+bi+cj+dk|}=\frac{a+bi+cj+dk}{\sqrt{a^2+b^2+c^2+d^2}}$$ For example take $q=1+2i+3j+5k$. Then $$\operatorname{sgn}(1+2i+3j+5k)=\frac{1+2i+3j+5k}{|1+2i+3j+5k|}=\frac{1}{\sqrt{39}}+\frac{2}{\sqrt{39}}i+\frac{3}{\sqrt{39}}j+\frac{5}{\sqrt{39}}k$$ where $$\left(\frac{1}{\sqrt{39}}\right)^2+\left(\frac{2}{\sqrt{39}}\right)+\left(\frac{3}{\sqrt{39}}\right)^2+\left(\frac{5}{\sqrt{39}}\right)^2=1$$

Note in the complex case we have the modulus $|z|=\sqrt{z\cdot\bar{z}}=\sqrt{(a+bi)\cdot(a-bi)}=\sqrt{a^2+b^2}$, where $\bar{z}$ is the complex conjugate. In the quaternion case we have $|q|=\sqrt{q\cdot\bar{q}}=\sqrt{(a+bi+cj+dk)\cdot(a-bi-cj-dk)}=\sqrt{a^2+b^2+c^2+d^2}$, where $\bar{q}$ is the quaternion conjugate.