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Complex Numbers

  • A number of the form x + iy is called a complex number. x is called the real part and y is called the imaginary part. x and y belong to R. The value of i = √(-1). x = Re(x + iy) and y = Im(x + iy)
  • If z = x + iy and x = 0 then we call z purely imaginary and if y = 0 then we call z purely real.
  • If z = x + iy then complex conjugate of z is con(z)and con(z) = x - iy. Re(con(z)) = Re(z) and Im(con(z)) = - Im(z)
  • Polar representation of z = x + iy is z = r (cos θ + i sin θ).
    r = √(x2 + y2), r is called the modulus of z.
    θ = tan-1 (y/x), θ is called the argument of z. We denote it as Arg z.

De Moivre's Theorem

  • De Moivre's Theorem states einθ = cos (nθ) + i sin (nθ) for any n∈Z and any angle θ.
  • We can conclude using De Moivre's theorem that [r (cos θ + i sin θ)]n = rn [cos (nθ) + i sin (nθ)]
  • Uses of De Moivre's Theorem
    1. It can be used to find and prove trigonometrical identities.
    2. It can be used to find roots of a complex number.
    3. The three cube roots of unity are 1,ω,ω2 where ω = (-1 + i√3)/2