16 Geometrical Effects of Dividing by a Complex Number

Dividing one complex number by another has a scaling and rotating effect. The scale factor is the multiplicative inverse (reciprocal) of the modulus and the angle of rotation is the additive inverse (negative) of the principal value of the argument. Animations (using GeoGebra) are used to illustrate these ideas.
Detailed solutions of the following examples are given:
(a) Write a complex number in Polar form that has the effect of undoing the rotation and scaling when multiplying by 1+i .
(b) Convert your part (a) answer to Cartesian form.
(c) Derive your part (b) answer without resorting to the use of any Polar form representations.
The viewer is encouraged to attempt these questions before watching the solutions.
Previous videos in this series are:
01 What is a Complex Number?
02 Adding, Subtracting and Multiplying Complex Numbers
03 Dividing Complex Numbers
04 Complex Conjugates
05 The Field of Complex Numbers
06 The Complex Plane
07 The Modulus of a Complex Number
08 Distance on the Complex Plane
09 Properties of the Modulus of a Complex Number
10 Complex Numbers and the Unit Circle
11 The Polar Form of a Complex Number
12 The Principal Argument of a Complex Number
13 The Geometrical Effects of Multiplying by a Complex Number
14 Multiplication of Complex Numbers in Polar Form
15 The Principal Argument when Multiplying Complex Numbers

Key words: Complex division, multiplicative inverse, additive inverse, scaling, origin, rotation, modulus, argument, principal value, Arg(z), polar form, cartesian form.
M337, Open University, Unit A1, complex analysis