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:
- 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.
- 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.
- 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?
The lesson ends - Homework
Download the homework (pdf file).