Imaginary numbers matematik 4 matematik 5

Alternative representations of complex numbers

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) 


 File:Complex conjugate picture.svg
matematik 1c matematik 4 matematik 5

Imaginary numbers

A solution to the simple second-degree equation

x2 + 1 =0

can not be found along the line of real-numbers.

Therefore it was necessary to invent a fictive number i such that i2=-1.

i.e. the imaginary numbers making it possible to take the square root of negative numbers.

They are represented along an axis horizontal to the axis of real numbers.

A combination of a real number and a imaginary number is called a complex number.

z = 2  + 3i.