Complex Numbers

Natural number : n

Imaginary unit : i

Complex number : z

Real part : a,c

Imaginary part : bi,di

Modulus of complex number : r, r1, r2

Argument of a complex number : φ, φ1, φ2

i1

  • z = a + bi
  • Complex plane

img

  • a+bi
  • c+di
  • a.b
  • divide

  • Conjugate complex numbers     conjugate

  • a = rcosφ, b = rsinφ

    Fig7

  • Polar presentation of complex numbers : a+bi = r(cosφ+isinφ)
  • Modulus and Argument of a complex number. If a+bi is a complex number, then       Modulus,   modulus              Argument , argument
  • Product in Polar Representation,  z1.z2
  • Conjugate numbers in polar representation,   rbar.JPG
  • Inverse of a complex number in polar representation,

inv

  • Quotient in polar representation,

quo

  • Power of a complex number,

pow

  • Formulae “De Moivre”,

de

  • N’th root of a complex number,

nth

where k=0,1,2……. (n-1)

 

  • Euler’s Formulae,

eul