# Eulerian Representation (Exponential Form)

Since we have

*e*= cos^{iÎ¸}*Î¸*+*i*sin*Î¸*and thus*z*can be expressed as*z*=*re*, where |^{iÎ¸}*z*| =*r*and*Î¸*= arg(*z*).# Cube roots of unity

Let

*z*= 1^{1/3}â‡’

*z*^{3}= 1â‡’

*z*^{3}â€“ 1 = 0â‡’ (

*z*â€“ 1) (*z*^{2}+*z*+ 1) = 0â‡’

*z*= 1 or,*z*=So, the cube roots of unity are 1,

*Ï‰*= and*Ï‰*^{2}= .**Properties of cube roots of unity**

*Ï‰*^{3}= 1.- One of the cube roots of unity is real and the other two are conjugate complex numbers.
- Each complex cube root of unity is the square of the other.
- The sum of three cube roots of unity is
*z*ero, i.e., 1 +*Ï‰*+*Ï‰*^{2}= 0. - The product of three cube roots of unity is 1.
- Each complex cube root of unity is the reciprocal of the other.
- The equation
*x*^{2}+*x*+ 1 = 0 has roots*Ï‰*and*Ï‰*^{2}and the equation*x*^{2}â€“*x*+ 1 = 0 has roots â€“*Ï‰*and â€“*Ï‰*^{2}. - Cube roots of â€“1 are â€“1, â€“
*Ï‰*, and â€“*Ï‰*^{2}. *a*^{3}â€“*b*^{3}= (*a*â€“*b*)(*a*â€“*Ï‰b*)(*a*â€“*Ï‰*^{2}*b*).*a*^{3}+*b*^{3}= (*a*+*b*)(*a*+*Ï‰b*)(*a*+*Ï‰*^{2}*b*).*a*^{3}+*b*^{3}+ c^{3}â€“ 3*abc*= (*a*+*b*+*c*)(*a*+ b*Ï‰*+ c*Ï‰*^{2}) (*a*+ b*Ï‰*^{2}+ c*Ï‰*).- If 1,
*Ï‰*, and*Ï‰*^{2}be cube roots of unity and*n*is*Ï‰*+^{n}*Ï‰*^{2n}= - The idea of finding cube roots of 1 and â€“1 can be extended to find cube roots of any real number. If
*a*is any positive real number, then*a*^{1/3 }has values*a*^{1/3},*a*^{1/3}*Ï‰*, and*a*^{1/3}*Ï‰*^{2}. If*a*is*a*^{1/3}has values â€“|*a*|^{1/3}, â€“|*a*|^{1/3}*Ï‰*, and |*a*|^{1/3}*Ï‰*^{2}.^{1/3}has values 2, 2*Ï‰*, and 2*Ï‰*^{2}whereas (â€“8)^{1/3}attains values â€“2, â€“2*Ï‰*, and â€“2*Ï‰*^{2}.