TOPICS
Miscellaneous

Example-1 :-  complex number

Solution :-
        complex number
   

Example-2 :-  Find the modulus and argument of the complex numbers:
complex number

Solution :-
(i) We have, (1+i)/(1-i)
= (1+i)/(1-i) x (1+i)/(1+i)
= (1+i²+2i)/(1-i²)
= (1-1+2i)/(1+1)
= 2i/2
= i 
= 0 + i

  Hence, z = 0 + i
  Now, 0 = r cos θ, 1 = r sin θ
  By squaring and adding, we get 
  r2 cos2 θ + r2 sin2 θ = 02 + 12
  r2 (cos2 θ + sin2 θ) = 0 + 1
  r2 (cos2 θ + sin2 θ) = 1
  r x 1 = √1
  r = 1
  Modulus = 1

  Therefore, 0 = r cos θ and 1 = r sin θ
  cos θ = 0 and sin θ = 1, which gives θ = π/2
  Argument = π/2
(ii) We have, 1/(1+i)
= 1/(1+i) x (1-i)/(1-i)
= (1-i)/(1-i²)
= (1-i)/(1+1)
= (1-i)/2
= 1/2 - i/2

  We have, z = 1/2 - i/2
  Now, 1/2 = r cos θ, -1/2 = r sin θ
  By squaring and adding, we get 
  r2 cos2 θ + r2 sin2 θ = (1/2)2 + (-1/2)2
  r2 (cos2 θ + sin2 θ) = 1/4 + 1/4
  r2 (cos2 θ + sin2 θ) = 1/2
  r x 1 = 1/√2
  r = 1/√2
  Modulus = 1/√2

  Therefore, 1/2 = r cos θ and -1/2 = r sin θ
  cos θ = 1/√2 and sin θ = -1/√2, which gives θ = -π/4
  Argument = -π/4
   

Example-3 :-  complex number

Solution :-
        complex number
    

Example-4 :-  complex number

Solution :-
        complex number
   

Example-5 :-  Convert the complex number complex number

Solution :-
        complex number
   
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