Geometric Representation and the Complex Plane
The complex plane represents the complex number z = a + bi as the point (a, b), with the real part on the horizontal axis and the imaginary part on the vertical axis. In brief: addition of complex numbers corresponds to vector addition, the distance of z from the origin is |z| = √(a2 + b2), and the conjugate is the mirror image in the real axis. For z = 3 + 4i, the point is (3, 4) and |z| = 5.
The geometric picture, introduced by Wessel, Argand and Gauss around 1800, made complex numbers intuitive: they are points or vectors in the plane, and operations are geometric transformations. This guide explains the complex plane, how to plot numbers and find distances, how addition and subtraction work geometrically, how the conjugate and the negative appear and how to describe simple sets of points, with a figure and a table.
What is the complex plane?
The complex plane (also the Argand plane or Gauss plane) is a Cartesian plane in which the horizontal axis is the real axis and the vertical axis is the imaginary axis. The complex number z = a + bi corresponds to the point P(a, b) or to the vector from the origin to P. Real numbers lie on the real axis, and purely imaginary numbers lie on the imaginary axis.
What does a complex number look like in the plane?
The modulus |z| = √(a2 + b2) is the length of the vector: for 3 + 4i it is 5 by the Pythagorean theorem. The angle from the positive real axis to the vector is the argument.
How do addition and subtraction look geometrically?
Complex numbers add like vectors: z + w is the fourth vertex of the parallelogram with vertices 0, z and w. The difference z − w is the vector from w to z. For z = 3 + 4i and w = 1 − 2i: z + w = 4 + 2i, and z − w = 2 + 6i. The triangle inequality follows from the geometry: |z + w| ≤ |z| + |w|, as one side of a triangle is at most the sum of the other two. Here |z + w| = √20 ≈ 4.47 and |z| + |w| = 5 + 2.24 = 7.24.
- The negative −z is the reflection in the origin (a rotation by 180°).
- The conjugate z̄ = a − bi is the reflection in the real axis.
- Multiplication by i rotates the vector by 90° counterclockwise: i(a + bi) = −b + ai. For example, i(3 + 4i) = −4 + 3i.
- Multiplication by a real number k scales the vector by k.
How do you find distances and midpoints?
The distance between the points z and w is |z − w|. For z = 1 + 2i and w = 4 + 6i: z − w = −3 − 4i and |z − w| = 5. The midpoint of z and w is (z + w)/2 = (5 + 8i)/2 = 2.5 + 4i. The point dividing the segment in the ratio m : n is (n z + m w)/(m + n).
Example 1: a triangle
The points A = 1 + i, B = 5 + i and C = 3 + 4i. The sides: |B − A| = 4, |C − B| = |−2 + 3i| = √13, |A − C| = |−2 − 3i| = √13. The triangle is isosceles with the apex at C. The perimeter is 4 + 2√13 ≈ 11.21.
Example 2: a parallelogram
Three vertices of a parallelogram are A = 0, B = 4 + i and C = 6 + 5i. The fourth vertex D with AB and DC parallel and equal: D = C − B + A = 2 + 4i. Check: the midpoints of AC (3 + 2.5i) and BD (3 + 2.5i) coincide ✓.
Which simple sets can you describe?
- |z| = r: a circle with center 0 and radius r.
- |z − z0| = r: a circle with center z0.
- |z − z0| < r: the open disc.
- Re z = c: a vertical line. Im z = c: a horizontal line.
- |z − p| = |z − q|: the perpendicular bisector of the segment from p to q.
Example 3
|z − 2 + i| = 3 means |z − (2 − i)| = 3: a circle with center (2, −1) and radius 3. The point 2 + 2i satisfies it, because |2 + 2i − 2 + i| = |3i| = 3.
Table of points and their images
| Operation | Geometric meaning | Result | Point |
|---|---|---|---|
| z | original | 3 + 4i | (3, 4) |
| −z | rotation by 180° | −3 − 4i | (−3, −4) |
| z̄ | reflection in the real axis | 3 − 4i | (3, −4) |
| iz | rotation by 90° | −4 + 3i | (−4, 3) |
| 2z | scaling by 2 | 6 + 8i | (6, 8) |
| z + (1 + i) | translation | 4 + 5i | (4, 5) |
| |z| | distance from 0 | 5 | – |
Where does the complex plane come from?
Caspar Wessel, a Norwegian-Danish surveyor, presented the geometric interpretation in 1797 (published in 1799). Jean-Robert Argand published a similar idea in 1806, and Gauss used it and gave it authority, so that the plane is called the Gauss plane or Argand plane. Before this, imaginary numbers were considered by many mathematicians to be unreal tools. The picture made them geometric objects.
Where is it used today?
Engineers represent AC voltages as rotating vectors (phasors) in the complex plane. In computer graphics, rotation by an angle θ is multiplication by cos θ + i sin θ. In dynamics and fractals, the Mandelbrot set is a subset of the complex plane. Control engineers study the location of poles in the complex plane to decide whether a system is stable.
Which mistakes are common?
Swapping the axes
The real part goes on the horizontal axis and the imaginary part on the vertical axis: 3 + 4i is (3, 4), not (4, 3).
Plotting bi as a point on the real axis
Purely imaginary numbers lie on the vertical axis.
Adding the moduli
|z + w| is usually smaller than |z| + |w|. Only for numbers pointing in the same direction are they equal.
Confusing −z and the conjugate
−z is the reflection in the origin, and the conjugate is the reflection in the real axis.