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### Lesson 3

#### The subject of the lesson

In this lesson we're going to summarize the results obtained so far. Then we'll be able to give a formal definition of Complex Number and, in the end, we'll introduce a useful geometric representation of these numbers.

#### The lesson begins

The most important outcomes of last lesson are:

1. Firstly we introduced a new object, i, with the the property that i2=-1. From now on this new object will be called imaginary unit.
2. Mixing the imaginary unit with the "old" real numbers we have obtained expressions of the form a+ib, with a and b real numbers. These expressions are the Complex Numbers. The set of all complex numbers is usually denoted by C.
3. We have seen that it is possible to perform the ordinary operations with these numbers, and the operations have the same properties they had in R: associative and commutative properties of addition and multiplication, distribuitive property of multiplication over addition.

While real numbers are usually denoted by x, complex numbers are usually denoted by z: we write z = a+ib; a is the real part of z, while b is its imaginary part.

Now let's observe that a complex number is, substantially, a pair of real numbers: we could write z=(a,b). We already know a standard way of representing couples of real numbers, using Cartesian coordinates. We can conclude that complex numbers may be represented as points in a plane: when a plane is used to represent complex numbers it is usually called a Gauss or Argand-Gauss plane.

Exercise: Represent in the plane the numbers z1=2+i, z2=3-2i, z3=-2-i.

Exercise: Represent in the plane the numbers z1=1+i and z2=-1+2i. Calculate the sum of these two numbers and represent it in the plane. Do you remember something similar in the theory of vectors?