I am trying to define some code to convert a complex number in the form of $(a+bi)$ into one of the form $(b+ai)$. I guess I am trying to find some operand $x$ such that $(a+bi)x = (b+ai)$.
I can switch where the $i$ is located:
$$\begin{align} (a+bi)\times i &= (ai+bi^2)) \\ &= (ai+b(-1)) \\ &= (ai-b) \\ &= (-b+ai) \end{align}$$
But then I'm not sure how to change that $-b$ into a $b$. Is there some arithmetic I can do on a complex number to move the $i$ from $a$ to $b$?
How's 'bout this:
$a + bi \overset{\times i}{\longrightarrow} ia - b \overset{\times -1}{\longrightarrow} -ia + b \overset{\text{conjugate}}{\longrightarrow} b + ia, \tag 1$
or this:
$a + bi \overset{\text{conjugate}}{\longrightarrow} a - bi \overset{\times i}{\longrightarrow} b + ia; \tag 2$
that is, with
$z = a + bi, \tag 3$
$z \longrightarrow \overline {(-iz)} = i\bar z. \tag 3$