Introduction

Consider the following questions:

1. Find all numbers $x$, such that $x^2 - 4=0$.

2. Find all numbers $x$, such that $x^2 +1 =0$.

Real numbers 2, and -2 will satisfy the first equation, but there is no real number such that $x^2 +1 =0$. We need a new number ( it is called imaginary number ) say $i$ whose square is -1. So we write $i = \sqrt {-1}$.

We want to be able to add this number to an other real number say $3$. the result will be $3+ i$. Or we might want to multiply $i$ with 5 then add the result to 3 . So we get $3+ 5i$.

In general every number of the form $a+ ib$ is called a complex number. Real numbers can be put in a one-to-one correspondence with the points on the X-axis. We can think of the correspondence of the complex numbers with the points on the XY-plane.