Grant says that multiplying any complex number represented by a point on the Argand plane by $cis(30^{\circ})$
i.e. $\cos30^{\circ} +i\sin(30^{\circ})$ rotates that point by $30$ degrees counterclockwise on the Argand plane. Maybe I did not pick on what he was trying to say in the video, but can someone elaborate on why this happens? Even people who haven't watched the video........

Here's a link by the way: https://www.youtube.com/watch?v=5PcpBw5Hbwo . Start watching after 32:10. I get why multiplying $cis(30^{\circ})$ to $1$, rotates $1$ by $30$ degrees counterclockwise, but can someone algebraically prove it for any general complex number (z)? It's easy to see with the Euler notation... but I am not well-worsed with it ( only know the basics ). I am requesting a proof strictly in $a + ib$ form and then elaborating on that geometrically to show why a point might rotate by $30$ degrees. I am sorry if I'm not able to articulate exactly what I mean, I'd be more than happy to clarify in the comments section.


1 Answer 1


Let $z = re^{i\theta}$ be a complex number where $r\gt 0$ and $-\pi\lt \theta \le \pi$. Consider another complex number which we want to multiply, $z_1 = r_1e^{i\theta_1}$ where $r_1\gt 0$ and $-\pi\lt \theta_1 \le \pi$. Then we have: $$z_p =zz_1 = (rr_1)(e^{i(\theta + \theta_1)})$$

So the product is another complex number and $|z_p| = rr_1 = |z||z_1|$. Also $Arg(z_p) = Arg(z) + Arg(z_1) +2k\pi$ and $k$ can be $0, 1$ or $-1$. For example if $\theta_1 = \frac{\pi}{6}$, then: $$-\pi\lt\theta \le \frac{5\pi}{6} \implies -\frac{5\pi}{6}\lt\theta + \theta_1\le \pi \implies Arg(z_p) = \theta + \theta_1$$ $$\frac{5\pi}{6}\lt\theta \le \pi \implies \pi \lt\theta + \theta_1\le \frac{7\pi}{6} \implies Arg(z_p) = \theta + \theta_1 - 2\pi$$

Note: It's straightforward to rewrite a complex number in exponential form using Euler's formula, see for example this link.


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