How can we describe complex numbers ?

The simplest way to describe a complex number is by its real and imaginary part, z=x+yi, this may be wrote as Re(z)=x and Im(z)=y. These complex numbers follow the same rules as normal algebra, that is we can add, subtract and multiply normally. However be careful with division, as you will want to 'Rationalize the denominator'.Another way to describe a complex number is in polar coordinate, this means we convert the point into a distance, called a modulus in complex numbers and an angle, called an argument in complex numbers.
A quick reminder of the rules is that the modulus, denoted |z|=sqrt(x^2 + y^2) and the argument, denoted Arg(z)=arctan(y/x). As with polar co-ordinates, when x<0 , care needs to be taken with using arctan (tan inverse), as we will need to account for our calculators giving us the wrong answer. The easy correction is -pi if y<0 and +pi if y>0, but using an argand diagram (a coordinate plot of z as (x,y) points)