• Properties of Cube Roots of Unity
  
  
  • Cube Roots of Unity are in G.P.
  
  
  • Each complex cube root of unity is the square of the other complex cube root of unity.
    
    Example : w=−1+3√i2, w2=−1−3√i2
  
  
  
  • 1+w+w2=0
  
  
  • Product of all cube roots of unity = 1. i.e., w3=1
  
  
  • 1w=w2 and 1w2=w
• Fourth Roots of Unity , (1)1/4 are +1, -1, +i, -i



• Polar and Exponential Forms of Complex Numbers   



Polar and Exponential Forms are very useful in dealing with the multiplication, division, power etc. of complex numbers.
Polar Form of a Complex Number


A complex number z=x+iy can be expressed in polar form as

z=r∠θ=r cisθ=r(cosθ+isinθ) (Please not that θ can be in degrees or radians)

where r=|z|=x2+y2−−−−−−√ (note that r ≥ 0 and and r = modulus or absolute value or magnitude of the complex number)

θ=arg z=tan−1(yx)(θ denotes the angle measured counterclockwise from the positive real axis.) 

θ is called the argument of z. it should be noted that 2π n +θ is also an argument of z where n=⋯−3,−2,−1,0,1,2,3,⋯. Note that while there can be many values for the argument, we will normally select the smallest positive value)

Please note that we need to make sure that θ is in the correct quadrant. i.e., θ should be in the same quadrant where the complex number is located in the complex plane. This will be clear from the next topic where we will go through various examples to convert complex numbers between polar form and rectangular form. It is strongly recommended to go through those examples to get the concept clear.

x=r cosθ

y=r sinθ


If −π<θ≤π,θ is called as principal argument of z(In this statement, θ is expressed in radian)
Exponential form of a Complex Number

We have already seen that in polar form , a complex number can be expressed as z=r(cosθ+isinθ). By Euler's Formula, we have eiθ=cosθ+isinθ

Hence, we can express a complex number in Exponential form as z=reiθ (Note that θ is in radians)

While there may be many values of θ satisfying this, we will normally select the smallest positive value.
From Cartesian form, complex form and exponential form, a complex number can be expressed as

z=r cis θ=r∠θ=r(cosθ+isinθ)=reiθ
Note that radians and degrees are two units for measuring angles.

360°=2π radian



• How to Convert Complex Numbers from Rectangular Form to Polar Form and Exponential Form