# Question regarding complex conjugates on the Argand plane

The following diagram appears in my textbook:

I am unsure of this visualization as I've learnt that the conjugate of a complex number in the form $$z=x+yi,x,y,\in\mathbb{R}$$ is simply $$z^*=x-yi.$$ This would mean, in polar form,

$$\DeclareMathOperator\cis{cis} z^*=r\cis(-\theta).$$

Is there something I am missing here? I'm genuinely confused as it seems that sources online disprove what is claimed in my textbook, yet my textbook continues to use this definition for later questions.

• This textbook is not using standard conventions: $z^*$ and $-z^*$ ought to be switched, assuming that you measure angles counterclockwise: from the positive real (horizontal) axis towards the positive imaginary (vertical) axis. Mar 10, 2021 at 3:55
• But also note that $-\theta$ and $2\pi - \theta$ differ by a complete rotation $(2\pi)$ so they're interchangeable as angles: $\operatorname{cis}(-\theta) = \operatorname{cis}(2\pi-\theta)$. Mar 10, 2021 at 3:58
• Thanks for the clarification! I take it that it is safe to say that $-z^*$ is not $r cis(-\theta)$? Mar 10, 2021 at 4:02

Among

• $$r\,\mathrm{cis}(\pi+\theta),$$
• $$r\,\mathrm{cis}(\pi-\theta),$$ and
• $$r\,\mathrm{cis}(2\pi-\theta),$$

only the third one refers to the complex conjugate of $$r\,\mathrm{cis}(\theta),$$ according to the standard definition.

Assuming that in this textbook $$\mathrm{Re}(z)$$ is indeed along the $$x$$-axis (i.e., $$z=x+iy$$), we can infer that the author

1. uses $$(0,2\pi]$$ as the principal argument, and
• either writes $$z$$'s conjugate as $$(-z^*),$$

• or writes $$z$$'s conjugate as $$(z^*)$$ but defines it as $$(-x+iy).$$

(1.) is fine (both $$(-\pi,\pi]$$ and $$(0,2\pi]$$ are conventional), but (2.) is strange and highly nonstandard.

That's taken from the IB Oxford AA HL textbook. In IB, the real scale is the x-axis and the imaginary scale is the y-axis. According to that, the book is wrong. It has gotten the conjugate and the conjugate of the opposite the wrong way around.

• That's correct, I was an IB student at the time and I was quite confused by the error. Hopefully, they've revised it in the next editions :) Jan 29 at 3:50