Complex Numbers

A complex number

To construct a complex number, we associate with each real number a second real number.
A complex number is then an ordered pair of real numbers (a,b).
We write that new number as

a + bi

The '+' and the i are just symbols for now.

We call 'a' the real part and 'bi' the imaginary part of the complex number.
 

Ex :

(2 , 4.6) or 2 + 4.6i ;
(0 , 5) or 0 + 5i ;
(-5 , 36/7) or -5 + (36/7)i ;
 

Instead of 0 + bi, we write 5i.
Instead of a + 0i, we write a.
Instead of 0 + 1i, we write i.
 

The set of all complex numbers is C.

 representation of a complex number

A complex number has a representation in a plane.
Simply take an x-axis and an y-axis (orthonormal) and give the complex number a + bi the representation-point P with coordinates (a,b).
The point P is the image-point of the complex number (a,b).
 

The plane with all the representations of the complex numbers is called the Gauss-plane.
With the complex number a + bi corresponds just one vector OP or P.

The image points of the real numbers 'a' are on the x-axis. Therefore we say that the x-axis is the real axis.

The image points of the 'pure imaginary numbers' 'bi' are on the y-axis. Therefore we say that the y-axis is the imaginary axis.