Algebraic Operations on Complex Numbers

Fundamental Operations with complex numbers:

In performing operations with complex numbers we can proceed as in the algebra of real numbers replacing i² by -1 when it occurs.

1. Addition

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

2. Subtraction

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

3. Multiplication

(a+bi)(c+di) = ac+bc i+ bi²

  = (ac-bd)+(ad+bc)i

4. Division

(a+bi)/(c+di)=(a+bi)/(c+di)×(c-di)/(c-di)

=(ac-adi+bci-bdi²)/(c²-d²i² )

= (ac+bd+(bc-ad)i)/(c²-d²i² )

=(ac+bd)/(c²+d² )+(bc-ad)/(c²+d²) i

Inequalities in imaginary numbers are not defined. There is no validity if we say that imaginary is positive or negative.

     GC: z >0, 4 + zi < 2+4i are meaningless.

In real numbers if a² + b²=0 then a=b=0; however in complex numbers,

     z₁² + z₂²  does not imply z₁ = z₂ = 0

Equality in complex numbers

Two complex numbers z₁ = a₁ + ib, and  z₂ = a₂ + ib₂ are equal if their real and imaginary parts are equal.

        i.e. z₁ = z₂ = Rs(z₁) = Re(z₁)

                 and I_m(z₁)  =  I_m (z₂).