What is an Argand diagram?

An Argand diagram, also known as the complex plane or Argand plane, is a graphical representation used in complex analysis to visualize complex numbers. It consists of a cartesian coordinate system The Cartesian coordinate system is a mathematical system that uses perpendicular axes to represent points in space using numerical values., where the horizontal axis represents the real part of a complex number, and the vertical axis represents the imaginary part.

Complex numbers are plotted as points on the plane, allowing geometric interpretation of their properties, such as magnitude and phase. The Argand diagram provides a convenient way to understand and perform operations on complex numbers.

Graphical representation

Now we will see how a complex number will be represented in an Argand diagram.

For example, z represents the complex number, a is the real part, and b is the imaginary part. You can replace a and b with specific numerical values to represent a specific complex number.

The distance from the origin (0,0) to a specific point represents the magnitude or modulus of a complex number denoted by '$r$' .

The argument of a complex number $\theta$ is typically measured in radians and can take values from $0$ to $2\pi$, depending on the chosen range. It is a fundamental concept in complex analysis that describes properties such as the phase or rotation of complex numbers.

Calculating $\theta$ and $r$

To determine the values of magnitude ($r$) and argument ($\theta$) for a given complex number ($z$), we can utilize the properties of the complex number itself. These properties allow us to calculate the magnitude and argument using the provided complex number.

To calculate $\theta$ ,we need to first find the basic angle $\alpha$:

$\theta$ needs to be further evaluated based on which quadrant the point lies in.

• The point was in 1$^{st}$quadrant then $\theta = \alpha$

• The point was in 2 $^{nd}$quadrant then $\theta = \pi - \alpha$

• The point was in 3$^{rd}$quadrant then $\theta = \alpha + \pi$

• The point was in 4 $^{th}$quadrant then $\theta = 2\pi - \alpha$

Example

Plot the given complex numbers on an Argand diagram and determine their modulus (magnitude) and argument.

• $z_{1} = 5 + 4i$

• $z_{2} = -3 -2i$

• $z_{3} = 5 - 5_{i}$

As shown above, we have drawn three complex numbers $z_{1},z_{2}\text{ and } z_{3}$ on the Argand diagram and calculated their arguments and modulus from the formulas we just learned.

Conclusion

The Argand diagram finds application in various fields, such as electrical engineering, physics, and signal processing, where it helps visualize and analyze complex quantities and their relationships. The Argand diagram provides an intuitive representation that facilitates the understanding and manipulation of complex numbers, aiding in analyzing complex phenomena in different disciplines.