Argand Diagrams

• Argand diagrams are a way to visually represent complex numbers. They provide a two-dimensional plot with the horizontal axis indicating the real component and the vertical axis indicating the imaginary component of a complex number.

• The number ‘a + bi’, where ‘a’ and ‘b’ are real numbers, is represented as a point (a, b) in the Argand plane.

• The modulus of a complex number is the distance from the origin to the point representing the complex number on the Argand diagram. It is calculated using Pythagoras’ theorem as √(a² + b²).

• The argument of a complex number refers to the angle which the line, joining the origin and the point representing the complex number, makes with the positive real axis. The argument is given by θ = arctan(b / a), though care must be taken to consider the correct quadrant of the complex number.

• Argand diagrams can be used to represent addition, subtraction, multiplication and division of complex numbers.

• When two complex numbers are added or subtracted, their representations on the Argand diagram are translated.

• Multiplication and division of complex numbers are interpreted as operations on the corresponding arguments and moduli.

• Conjugate of a complex number is the reflexion in the real axis on the Argand diagram. If a complex number is ‘a + bi’, its conjugate is ‘a - bi’.

• If z is a complex number, the segment from 0 to z is known as the line segment representation of z. The distance from 0 to z is equal to the modulus of z, and the angle from the positive x-axis to the segment is the argument of z.