As mentioned in the latest post any complex number may be represented by an arrow in the complex plane. This number is unambiguously described by two numbers: its real part x and its imaginary part y. z= x+iy. This is called the Cartesian representation. (Rene Descartes)

From the figure below it is evident that

x = r cosφ

y = r sin φ  and thus z = r( cosφ + i sinφ) . This is called polar representation of the complex number z. r is the modulus of the vector z (i.e. its length) and φ is called the argument of z.    The modulus can be  computed by multiplying the  complex number z with its conjugated complex number z

By adding the  Taylor series for cos(x) and  i sin(x) we get the series expansion for the exponential function e^ix. This relation is called Euler’s formula.

This leads to the exponential representation of a complex number:

z= r e^(iφ) 

This