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Please Note: I understand how the addition and multiplication of complex numbers work. I'm just little confused by the wording in the book.

I am looking at Visual Complex Analysis by Needham. It seems to me that everything is backwards. For example, this is the geometric definition of multiplication:

My problem is that we are rotating the plane. So suppose that I want to multiply $(2, 0)$ by $1 \angle \dfrac {\pi}{6}$. If I follow the directions and rotate the plane, this is what happens:

Here the black lines constitute the original plane, the blue dot is $(2, 0)$ in the original plane, and the red lines constitute the plane rotated by $\dfrac {\pi}{6}$. As we can see, the coordinates of the blue point with respect to the new axes are $\left(\dfrac {\sqrt3}{2}, -\dfrac 12 \right)$ when they should in fact be $\left(\dfrac {\sqrt3}{2}, \dfrac 12 \right)$

I have a similar problem with translations. The book says:

I start again with the number $(2, 0)$ and I want to add $\dfrac 12 + i \dfrac 12$ to it. So I shift the plane by $\dfrac 12 + i \dfrac 12$, and this is what I get:

So clearly, I am following these definitions "backwards". But going word by word I don't see how I am misinterpreting the text.

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  • $\begingroup$ This video may be relevant. $\endgroup$
    – Petch Puttichai
    Commented Mar 8, 2017 at 3:35
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    $\begingroup$ I agree that the author could have worded it better. What he means by a transformation of the plane is an action on each point of the plane, leaving the axes fixed. $\endgroup$
    – quasi
    Commented Mar 8, 2017 at 3:36
  • $\begingroup$ In your diagrams, you left the point fixed and applied the transformation to the axes instead. $\endgroup$
    – quasi
    Commented Mar 8, 2017 at 3:44
  • $\begingroup$ @quasi Ah ok, that makes more sense; the author did say something about an action on each point, but he never said anything about the axes. Please see the first passage again, I have edited the question to include the text above it. But then, when we are multiplying $z$ by $a$, why do we have to rotate every point in the plane by the angle of $a$? Why don't we just stop after rotating $z$? $\endgroup$
    – Ovi
    Commented Mar 8, 2017 at 3:45
  • $\begingroup$ As @quasi says, it’s not the axes that are rotated, but the individual points. So the point $1=1+0i$ gets rotated to $cos\pi/6+i\sin\pi/6=\sqrt3/2+i/2$. $\endgroup$
    – Lubin
    Commented Mar 8, 2017 at 3:46

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