Summary

SUMMARY

In this chapter we studied

Rectangular form of a complex number is x + iy (or x + yi) , where x and y are real numbers.

Two complex numbers z₁ = x₁ + iy₁ and z₂ = x₂ + iy₂ are said to be equal if and only if

Re(z₁ ) = Re(z₂ )  and  Im(z₁ ) = Im(z₂ ) . That is  x₁ = x₂ and  y₁ = y₂ .

The conjugate of the complex number x + iy is defined as the complex number x - iy.

Properties of complex conjugates

If  z = x + iy, then  √[x² + y²] is called modulus of z . It is denoted by |z| .

Properties of Modulus of a complex number

Formula for finding square root of a complex number

Let r and θ be polar coordinates of the point P(x, y) that corresponds to a  non-zero complex number z = x + iy . The polar form or trigonometric form of a complex number P is

z = r(cosθ + i sinθ ) .

Properties of polar form

Property 1:

If z = r (cosθ + i sinθ ), then z^-1 = 1/r  (cosθ - i sinθ ) .

Property 2:

If z₁  = r₁ (cosθ₁  + i sinθ₁ ) and  z₂  = r₂ (cosθ₂  + i sinθ₂ ),

then   z₁ z₂  = r₁r₂ (cos(θ₁  + θ₂ ) + i sin(θ₁  + θ₂ )) .

Property 3:

If  z₁  = r₁ (cosθ₁  + i sinθ₁ ) and  z₂  = r₂ (cosθ₂  + i sinθ₂ ) ,

Then z₁/z₂ = r₁/r₂ [cos(θ₁- θ₂)  + i sin(θ₁- θ₂) ]

de Moivre’s Theorem

(a) Given any complex number cos θ + i sin θ and any integer n,

(cosθ + i sinθ )n = cos nθ + i sin nθ

(b) If x is rational, then  cos x θ + i sin x θ  in one of the values of (cos θ + i sin θ)x

The nth roots of complex number z = r (cosθ + i sinθ ) are