OPERATIONS WITH COMPLEX NUMBERS IN RECTANGULAR AND POLAR FORM

Polar form of a complex number will be

z = r(cos θ + i sin θ)

Rectangular form will be

z = x + iy

Let two complex numbers represented in rectangular form,

a + ib and c + id

Let z₁ = a + ib and z₂ = c + id

Adding complex numbers in rectangular form :

z₁ + z₂ =  a + ib + c + id

= (a + c) + i(b + d)

Subtracting complex numbers in rectangular form :

z₁ - z₂ =  (a + ib) - (c + id)

= (a - c) + i(b - d)

Multiplying complex numbers in rectangular form :

z₁ x  z₂ =  (a + ib) x (c + id)

= ac + iad + ibc + i²bd

= (ac - bd) + i(ad + bc)

Dividing complex numbers in rectangular form :

z_{1 /}  z₂ =  (a + ib) / (c + id)

Multiply both numerator and denominator by the conjugate of the denominator.

Let two complex numbers represented in polar form,

 r₁(cos θ₁ + i sin θ₁) and r₂(cos θ₂ + i sin θ₂)

Let z₁ = r₁(cos θ₁ + i sin θ₁) and z₂ = r₂(cos θ₂ + i sin θ₂)

Multiplying complex numbers in polar  form :

z₁ x  z₂ =  r₁(cos θ₁ + i sin θ₁) r₂(cos θ₂ + i sin θ₂)

= r₁ r₂ [cos (θ₁ + θ₂) + i sin (θ₁ + θ₂)]

Dividing complex numbers in rectangular form :

z_{1 /}  z₂ =  r₁(cos θ₁ + i sin θ₁) / r₂(cos θ₂ + i sin θ₂)

= (r₁ / r₂) [cos (θ₁ - θ₂) + i sin (θ₁ - θ₂)]