Agrand Diagrams

Argand Diagrams

Understanding Argand Diagrams

Comprehend that an Argand diagram is a way to graphically represent complex numbers. The horizontal axis represents the real part and the vertical axis represents the imaginary part.
Learn that a complex number, represented by z = a + bi, corresponds to a point (a, b) on the Argand diagram.

Complex Conjugates and Argand Diagrams

Recognise that the complex conjugate of a number z = a + bi is denoted by z* = a - bi. On the Argand diagram, they are mirrored across the real axis.
Understand that the product of a complex number z and its conjugate z* is always a real, positive number, corresponding to the square of the modulus of z.

Modulus and Argument

Realise that the modulus of a complex number represented on an Argand diagram is the distance between the origin and the point representing the complex number.
Note that the argument (also known as phase or amplitude) of a complex number is the angle that line joining the origin to the point makes with the positive direction of the real (x) axis.
Remember that the modulus and argument together provide another way to represent complex numbers, known as polar form or mod-arg form.

Operations on Complex Numbers

Understand that addition and subtraction of complex numbers corresponds to vector addition and subtraction on the Argand diagram.
Be aware that multiplication of complex numbers in mod-arg form corresponds to multiplying the moduli and adding the arguments.
Similarly, division of complex numbers in mod-arg form corresponds to dividing the moduli and subtracting the arguments.

Mastering these concepts of Argand diagrams and the operations on complex numbers helps in visual understanding of complex numbers which further simplifies many complex number related tasks like finding roots of complex numbers, etc. It is very versatile and plays a important role in various fields like physics, engineering, and computer science.