Example-1 :-
Example-2 :- Find the modulus and argument of the complex numbers:
(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 :-
Example-4 :-
Example-5 :- Convert the complex number