Let's take two complex numbers, z1 = 19 + 4i and z2 = 3 + 2i. Adding, subtracting or multiplying them presents no particular difficulty, provided we remember that i2 = −1. Division, however, can be trickier. One trick is to make good use of conjugation. Nothing to do with the pluperfect subjunctive! To “conjugate” a complex number, we simply change the sign of its imaginary part. Thus, the conjugate of z2, usually denoted by z2ˉ\bar{z_{2}}, is 3 − 2i. What use is that here? Look what happens when a complex number is multiplied by its conjugate:
(a+ib)×(a+ib)‾=(a+ib)×(a−ib)(a+ib)\times\overline{(a+ib)}=(a+ib)\times(a-ib)
=a2−i2b2=a^{2}-i^{2}b^{2}
=a2+b2=a^{2}+b^{2}
The result is always a real number!