Complex Number Calculator
Complex multiplication [ (a+bi) × (a+bi) ]
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Complex number division [ (a+bi) / (a+bi) ]
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Complex square root: [ r1 = x+yi ; r2 = -x-yi ]
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Complex number operations include: addition, subtraction, multiplication and division.
Complex number addition is performed according to the following rules: Let z1=a+bi, z2=c+di be any two complex numbers, then their sum is (a+bi)+(c+di)=(a+c)+(b+d)i.
The multiplication of complex numbers is stipulated to be carried out according to the following rules: Let z1=a+bi, z2=c+di (a, b, c, d∈R) be any two complex numbers, then their product (a+bi)(c+di)=(ac-bd)+(bc+ad)i.
Division operation rules:
Let the complex number a+bi(a, b∈R), divided by c+di(c, d∈R), 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 complex number equality, we know that cx-dy=a dx+cy=b
Solve this system of equations and get x=(ac+bd)/(c^2+d^2) y=(bc-ad)/(c^2+d^2)
So: (a+bi)/(c+di)=(ac+bd)/(c^2+d^2) +(bc-ad)/(c^2+d^2)i