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 eix. This relation is called Euler’s formula.
This leads to the exponential representation of a complex number:
z= r e(iφ)