The Argand plane and geometric interpretation

Every complex number $z = a + bi$ can be drawn as the point $(a, b)$ in a plane , called the Argand plane (or complex plane). This single picture turns every algebraic identity into a geometric fact.

The Argand plane

Set up a Cartesian system with horizontal real axis and vertical imaginary axis. Plot $z = a + bi$ at the point $(a, b)$ . Then:

The point $(a, 0)$ is the real number $a$ .
The point $(0, b)$ is the purely imaginary number $bi$ .
The origin is $0 = 0 + 0i$ .

Two equivalent ways to think of a complex number:

as a point in the plane (positional);
as a vector from the origin to that point.

Geometric meaning of the basic operations

Addition is vector addition. $z + w$ is the vector sum, obtained by the parallelogram law. The point representing $z + w$ has coordinates equal to the sum of the coordinates.

Subtraction is a displacement. $z - w$ is the vector from $w$ to $z$ (or the point that you reach by translating $z$ by the negative of $w$ ).

Conjugation is reflection in the real axis. $\bar{z}$ has the same real part as $z$ but opposite imaginary part.

Multiplication by $-1$ is a $180^\circ$ rotation about the origin.

Multiplication by $i$ is a $90^\circ$ counterclockwise rotation about the origin. This is the geometric reason for the formula $i^2 = -1$ : rotating $90^\circ$ twice equals $180^\circ$ .

Modulus $|z|$ is the distance from $0$ to $z$ . And $|z - w|$ is the distance between $z$ and $w$ in the plane.

The modulus and triangle inequality, geometrically

The triangle inequality $|z + w| \le |z| + |w|$ becomes the familiar geometric fact: "in any triangle, the length of one side is at most the sum of the other two."

The locus $|z - z_0| = r$ in the Argand plane is a circle of centre $z_0$ and radius $r$ . So $|z - 1| = 2$ is the equation of a circle centred at $(1, 0)$ with radius $2$ .

Argument of a complex number

For $z \ne 0$ , the argument of $z$ , written $\arg(z)$ , is the angle (in radians) that the vector from $0$ to $z$ makes with the positive real axis, measured counterclockwise.

Argument is defined only up to adding $2\pi$ . The principal value $\text{Arg}(z)$ lies in $(-\pi, \pi]$ by convention.

If $z = a + bi$ with $z \ne 0$ , then $\tan(\arg z) = \frac{b}{a} \qquad \text{(with quadrant adjustment).}$

Worked examples

Example 1. Plot $z = 3 + 4i$ in the Argand plane. Compute $|z|$ and $\arg(z)$ .

The point is $(3, 4)$ in the first quadrant. $|z| = 5$ . $\tan\theta = 4/3$ , so $\arg(z) = \tan^{-1}(4/3) \approx 0.927$ rad ( $\approx 53.13^\circ$ ).

Example 2. Describe geometrically the set $|z - 2| = 3$ .

A circle centred at $(2, 0)$ with radius $3$ .

Example 3. Describe $|z - 1| = |z + 1|$ .

The set of points equidistant from $(1, 0)$ and $(-1, 0)$ , the perpendicular bisector of the segment, which is the imaginary axis (real part zero).

Example 4. If $z_1, z_2, z_3$ are the vertices of a triangle, where is the centroid?

Centroid $= \dfrac{z_1 + z_2 + z_3}{3}$ . The algebra inherits exactly from coordinate geometry.

Example 5 (harder). Show that $|z - 1| + |z + 1| = 4$ describes an ellipse with foci at $\pm 1$ and major axis $4$ .

This is the locus of points $P$ with $|PF_1| + |PF_2| = 4$ where $F_1 = (1, 0)$ and $F_2 = (-1, 0)$ . By the definition of an ellipse, this is exactly an ellipse with foci at $\pm 1$ and sum-of-distances $2a = 4$ , so semi-major axis $a = 2$ . Distance between foci is $2c = 2$ , so $c = 1$ and $b = \sqrt{a^2 - c^2} = \sqrt{3}$ . Equation: $\dfrac{x^2}{4} + \dfrac{y^2}{3} = 1$ .

Try it yourself

Plot $z_1 = 2 + 3i$ , $z_2 = -1 + i$ on the Argand plane.
Find the modulus and (principal) argument of $z = -1 + i$ , $z = -i$ , $z = -2 - 2i$ .
Describe geometrically: $|z + 3i| = 4$ .
Describe: $|z - 2| = |z - 4|$ .
Describe: $|z| = |z - 1|$ .
Find the distance between $z_1 = 3 + 4i$ and $z_2 = -2 + i$ .
If $z$ moves on the circle $|z - 1 - i| = 2$ , what is the maximum and minimum value of $|z|$ ?
Show $|z| = |\bar{z}|$ for any $z$ .
Describe: $\text{Re}(z) > \text{Im}(z)$ in the plane.
Find the locus: $|z - 2| + |z + 2| = 6$ (ellipse).
Find the locus: $|z - 1| - |z + 1| = 1$ (hyperbola).
Sketch the region $|z - 1| \le 1$ and $\text{Re}(z) \ge 1$ .

Pitfalls / Tricks

The principal argument $\text{Arg}(z)$ lies in $(-\pi, \pi]$ . Always check the quadrant: $\arg(-1 + i) = 3\pi/4$ , not $-\pi/4$ .
$|z - z_0| = r$ is a circle; $|z - z_1| = |z - z_2|$ is a perpendicular bisector; $|z - z_1| + |z - z_2| =$ const is an ellipse.
Argument is not defined for $z = 0$ (no direction).
Insight. Every geometric problem about points in $\mathbb{R}^2$ becomes a problem about complex numbers. Distance, rotation, circle, ellipse , all encoded directly.