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Complex Fractions, Argand Diagrams, Magnitudes, Arguments and Products of Complex Numbers and Polar Forms

Complex Fractions

Typically we have to express a complex fractionin the formWe do this by multiplying top and bottom by the complex conjugate of the denominator, remembering thatThe complex conjugate of

Example: Expressin the form

Argand Diagrams

We may also have to plot complex numbers on an Argand diagram. This is a normal set of axes:is plotted as the pointIn the diagram below the complex numbers plotted as the point

Magnitudes, Arguments and the Polar Form of Complex Numbers

The magnitude of

the argument of

The polar form ofis written as

Multiplying Complex Numbers

Given two complex numbersandwe can find the product

We can express this in polar form as above,

then

Or we can expressandin polar form then using the normal rules for multiplying exponentials:

so

Dividing Complex numbers

We can use the method of the top of the page to express in cartesian form, or, if we require polar form, using the normal rule for dividing exponentials: