"Imaginary" part of a quaternion

Question

With complex numbers, we write $Im(z)$ to denote the imaginary part of a complex number $z$. I was wondering what $Im(q)$, where $q$ is a quaternion, would show: would it be the $i$ part of the quaternion or all of the $i$, $j$, and $k$ parts? And if it shows all three parts, is there a notation for showing each of the individual parts?

In case this is unclear, if $q = 1 + 2i + 3j + 4k$, would $Im(q)$ be $2i + 3j + 4k$ (or possibly the vector $(2, 3, 4)$) or just $2$? And how would you the individual parts $3$ and $4$ separately?

Answer 1

Usually, the imaginary part of $a+bi+cj+dk$ is $bi+cj+dk$. If I want to talk about, say, $c$, then I mention the coefficient of $j$.

Answer 2

Real and imaginary part of quaternions is well-defined. Let's consider a quaternion $w+ix+jy+kz$. Often, we write the quaternion as a 4D vector $q=[w,x,y,z]^T$. Now we define the real part $$ \operatorname{Re}q = w \qquad\text{and}\qquad \operatorname{Im}q = \begin{bmatrix}x\\y\\z\end{bmatrix}. $$

This is really helpful, since you can represent quaternions $q$ in polar form $$ q = r\exp\biggl(\begin{bmatrix}0\\\mathbf{v}\end{bmatrix}\biggr), $$ where $r=\lVert q\rVert_2$ is multiplied by the exponential of the purely imaginary quaternion $[0,\mathbf{v}^T]^T$. This is very similar to $$ z = r\mathrm{e}^{i\phi},$$ where $z$ is a complex number, $r=|z|$ and $i\phi$ is a purely imaginary number.

If you just want the factor in front of one of $i,j,k$, refer to José's answer.