Complex Addition

Complex Addition:

With the geometric interpretation of the addition of two complex numbers, in Remark 5.3 Item iii, clarifies the notion of the “triangle inequality”. In Figure 5.1, the length of the edges of the parallelogram in blue are given by $|z_1|$ and $|z_2|$ , whereas the length of the diagonal in red is given by $|z_1 + z_2|$ . Hence, as the figure shows, the triangle inequality says that the length of the two shortest edges of a triangle is at least the length of the longest edge.

How to use the applet

Drag the two blue points to change the complex numbers

z_1 = x_1 + i y_1, \qquad z_2 = x_2 + i y_2.

The axes are the complex plane: horizontal $=\mathrm{Re}(z)$ , vertical $=\mathrm{Im}(z)$ .
The blue arrows from the origin represent the vectors $z_1$ and $z_2$ .
The red arrow represents the sum

z_1+z_2=(x_1+x_2)+i(y_1+y_2).

The dashed light-blue arrows show the parallelogram rule:
- the arrow $z_2\to z_1+z_2$ is a translated copy of $z_1$ ,
- the arrow $z_1\to z_1+z_2$ is a translated copy of $z_2$ . The picture forms a triangle (or parallelogram) with side lengths $|z_1|$ , $|z_2|$ , and $|z_1+z_2|$ .
  This illustrates the triangle inequality:

|z_1+z_2|\le |z_1|+|z_2|.

Equality happens when $z_1$ and $z_2$ point in the same direction (same argument).

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Points shaped like <> act as sliders

Description

Link to code.

Reference: Lecture Notes Calculus 1 ( May22,2022 ) Figure5.1. page 75

This applet visualizes addition of two complex numbers as vectors in the complex plane (horizontal axis $\mathrm{Re}(z)$ , vertical axis $\mathrm{Im}(z)$ ) and illustrates the geometric idea behind the triangle inequality. The origin is the point $0$ , and two draggable points represent the complex numbers $z_1$ and $z_2$ .

The two complex numbers are represented by blue arrows from the origin to the draggable points (board.create("arrow", [O, z1]) and board.create("arrow", [O, z2])). Their sum is represented by the red arrow from the origin to the point $z_3$ , where $z_3$ is defined dynamically as z3 = (z1.X() + z2.X(), z1.Y() + z2.Y()) in the code: const z3 = board.create("point", [() => z1.X() + z2.X(), () => z1.Y() + z2.Y()], ...).

To help students see the parallelogram (and hence the triangle inequality), the applet also draws two dashed light-azure translated arrows: z2 → z3 is a translated copy of $z_1$ , and it is labeled z₁; z1 → z3 is a translated copy of $z_2$ , and it is labeled z₂. These auxiliary arrows make it clear that the diagonal of the parallelogram corresponds to $z_1+z_2$ .