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### Lessons 4 and 5

#### The subject of the lessons

In these two lessons we are going to find a new way of representing complex numbers in the Gauss plane. Then, using Cabri, we'll find some interesting properties of this new representation.

#### The lessons begin

Exercise: use Cabri to represent in the Gauss plane the complex numbers 1+i and -1+2i. Get the sum of these two numbers with the usual rule and then represent this sum in the "Cabri plane". Show, using the software, that the "parallelogram rule" holds.

Exercise: move the points (1,1) and (-1,2) and see how the point (0,3) varies.

Exercise: now erase the sum, calculate the product of the two vectors (using Cabri Calculator), represent it in the plane, measure the length of all the vectors and the angles between them and the positive x axis (in the counterclockwise direction). By moving the original vectors can you see some simple relation between their lengths and angles and the lenght and angle of the product?

We can conclude with the following rules:

• Every complex number a+ib has a unique "length", that we call its absolute value, ρ, and is characterized by a unique angle between a vector and the positive x axis, that we call argument, θ: the complex number will be represented by this pair of numbers and we'll write: a+ib=[ρ,θ]. Square brackets are used to differentiate this couple of numbers from the cartesian coordinates.
• The product of two complex numbers [ρ11] and [ρ22] is simply [ρ1·ρ21 2].

#### The lessons end - Homework

Using Cabri  try to find a rule for the quotient of two complex numbers.

first published on april 04 2005 - last updated on april 04 2005