Complex Multiplication

Complex Multiplication:

Using the polar form of complex numbers we can find a geometric interpretation of the multiplication of two complex numbers. For $z_1, z_2 \in \mathbb{C}$ with
$z_1 = r_1 \cdot e^{i\varphi_1}$ and $z_2 = r_2 \cdot e^{i\varphi_2}$ with
$r_1, r_2 \in (0,\infty)$ and $\varphi_1, \varphi_2 \in (-\pi,\pi]$ , it holds that

z := z_1 z_2 = r_1 r_2 e^{i(\varphi_1 + \varphi_2)} = r e^{i\varphi},

i.e., we obtain the absolute value of $z$ by multiplying the absolute values of $z_1$ and $z_2$ , and the angle $\varphi$ by adding the angles $\varphi_1$ and $\varphi_2$ . This is illustrated in Figure 5.4.

Note that we may have $\varphi \notin (-\pi,\pi]$ , so $\varphi$ may not be the principal value of $\arg(z)$ .

How to use the applet

Drag the points $z_1$ (green) and $z_2$ (red) in the complex plane.
The arrows from the origin represent $z_1$ , $z_2$ , and the product $z=z_1z_2$ (black).
The colored angle sectors show the arguments (angles with the positive real axis):

\varphi_1=\arg(z_1),\qquad \varphi_2=\arg(z_2),\qquad \varphi=\arg(z).

z_1=r_1 e^{i\varphi_1},\qquad z_2=r_2 e^{i\varphi_2},

then

z=z_1z_2=(r_1r_2)e^{i(\varphi_1+\varphi_2)}.

So, when you drag $z_1$ and $z_2$ :

the length of the product vector changes like $|z|=|z_1|\,|z_2|$ ;
the angle of the product changes like $\varphi \approx \varphi_1+\varphi_2$ (modulo $2\pi$ , so it may “wrap around”).

Loading graph…

Hold Shift + scroll to zoom
Hold Shift + drag to move
Points shaped like <> act as sliders

Description

Link to code.

Reference: Lecture Notes Calculus 1 ( May22,2022 ) Figure 5.4 page 88

This applet visualizes complex multiplication using the polar interpretation. The axes represent the complex plane: the horizontal axis is $\mathrm{Re}(z)$ and the vertical axis is $\mathrm{Im}(z)$ . Two draggable points represent the complex numbers $z_1$ and $z_2$ , and the product $z=z_1z_2$ is shown as a third vector.

The points $z_1$ and $z_2$ are draggable. The product $z$ is defined from their Cartesian coordinates using the standard algebraic rule $(a_1+ib_1)(a_2+ib_2)=(a_1a_2-b_1b_2)+i(a_1b_2+b_1a_2)$ . In code this is kept explicitly as Re(z) = z1.X()*z2.X() - z1.Y()*z2.Y() and Im(z) = z1.X()*z2.Y() + z1.Y()*z2.X().

        const z = createPoint(board,
          [
            () => z1.X() * z2.X() - z1.Y() * z2.Y(),
            () => z1.X() * z2.Y() + z1.Y() * z2.X(),
          ],
          {
            name: "",
            fixed: true,
            withLabel: false,
          },
          Z_COLOR
        );

The three arrows from the origin (created with board.create("arrow", [O, ...])) represent the vectors $z_1$ (green), $z_2$ (red), and $z=z_1z_2$ (black). The colored angle sectors labeled $\varphi_1$ , $\varphi_2$ , and $\varphi$ (created with board.create("angle", [U, O, ...])) visualize their arguments.

To avoid an unstable argument near the origin, the applet constrains the draggable points so that $|z_1|$ and $|z_2|$ stay above a small minimum radius (MIN_RADIUS), enforced in enforceMinRadius during dragging.