Using complex numbers to represent regions on an Argand diagram

Using complex numbers to represent regions on an Argand diagram

Argand Diagrams and Complex Numbers

Basic Concepts

  • An Argand diagram is a graphical representation of complex numbers. It is essentially a Cartesian coordinate system with the real part of a complex number represented along the x-axis and the imaginary part along the y-axis.
  • A complex number a + bi is represented as a point (a, b) in the Argand diagram.

Representing Regions on Argand Diagram

  • The set of points represented by a particular condition in the complex plane (i.e. Argand diagram) forms a region.
  • The origin is represented as 0 in the Argand diagram.
  • The modulus of a complex number, represented by ** z ** (where z is a complex number), is the distance from the origin to the point representing the complex number.
  • Re(z) and Im(z) represent the real and imaginary parts of the complex number z respectively.

Inequality Conditions

  • Inequality conditions can be used to define regions in the Argand diagram. For example, the inequality ** z - a < r** (where a is a real number, and r is the radius) represents all points inside a circle with centre a and radius r.
  • Similarly, the inequality Re(z) < a represents all points to the left of the vertical line x = a on the complex plane.
  • Im(z) > b represents all points above the horizontal line y = b on the complex plane.

Complex Conjugate

  • The complex conjugate of a complex number is mirrored across the x-axis on the Argand diagram. If z = a + bi, then its complex conjugate z* = a - bi.

Loci on the Argand Diagram

  • Loci is a set of points satisfying a certain condition. Loci in the complex plane can represent various geometric shapes including lines, circles, and regions thereof.
  • A locus on the Argand diagram could represent a circle (for example, an equation of the form ** z - a = r), a **line (for example, Re(z) = a, or Im(z) = b), or a half-plane (for example, Re(z) > a, or Im(z) < b).

By understanding these principles and employing them effectively, complex number regions can be visualised, helping bridge the conceptual gap between algebra and geometry.