the rules of complex number arithmetic include: addition, subtraction, multiplication and division.

the addition of complex numbers is carried out according to the following rules: assuming z1=a+bi and z2=c+di are any two complex numbers, then their sum is (a+bi)+(c+di)=(a+c) +(b+d)i.

it is stipulated that the multiplication of complex numbers is carried out according to the following rules: suppose z1=a+bi, z2=c+di (a, b, c, d∈r) are any two complex numbers, then their product (a+bi) (c +di)=(ac-bd)+(bc+ad)i.

division operation rules:

assume that the complex number a+bi(a, b∈r) is divided by c+di(c, d∈r), and its quotient is x+yi(x, y∈r), that is, (a+bi)÷

(c+di)=x+yi

∵(x+yi)(c+di)=(cx－dy)+(dx+cy)i.

∴(cx－dy)+(dx+cy)i=a+bi.

from the definition of equality of complex numbers, we know that cx-dy=a dx+cy=b

solving this system of equations, we get x=(ac+bd)/(c^2+d^2) y=(bc-ad)/(c^2+d^2)

so we have: (a+bi)/(c+di)=(ac+bd)/(c^2+d^2) +(bc-ad)/(c^2+d^2)i