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Eulerian Representation (Exponential Form)

Since we have e = cos θ + i sin θ and thus z can be expressed as z = re, where |z| = r and θ = arg(z).

Cube roots of unity

Let z = 11/3
z3 = 1
z3 – 1 = 0
(z – 1) (z2 + z + 1) = 0
z = 1 or, z = 59889.png
 
So, the cube roots of unity are 1, ω = 59882.png and ω2 = 59876.png.
 
Properties of cube roots of unity
  1. ω3 = 1.
  2. One of the cube roots of unity is real and the other two are conjugate complex numbers.
  3. Each complex cube root of unity is the square of the other.
  4. The sum of three cube roots of unity is zero, i.e., 1 + ω + ω2 = 0.
  5. The product of three cube roots of unity is 1.
  6. Each complex cube root of unity is the reciprocal of the other.
  7. The equation x2 + x + 1 = 0 has roots ω and ω2 and the equation x2 x + 1 = 0 has roots –ω and –ω2.
  8. Cube roots of –1 are –1, –ω, and –ω2.
  9. a3b3 = (ab)(aωb)(aω2b).
  10. a3 + b3 = (a + b)(a + ωb)(a + ω2b).
  11. a3 + b3 + c3 – 3abc = (a + b + c)(a + bω + cω2) (a + bω2 + cω).
  12. If 1, ω, and ω2 be cube roots of unity and n is a positive integer, then 1 + ωn + ω2n = 59869.png
  13. The idea of finding cube roots of 1 and –1 can be extended to find cube roots of any real number. If a is any positive real number, then a1/3 has values a1/3, a1/3 ω, and a1/3 ω2. If a is a negative real number, then a1/3 has values –|a|1/3, –|a|1/3 ω, and |a|1/3 ω2.
     
    For example, 81/3 has values 2, 2ω, and 2ω2 whereas (–8)1/3 attains values –2, –2ω, and –2ω2.




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