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Complex division [ (a+bi) / (a+bi) ]


Complex division [ (a+bi) / (a+bi) ]
bi:
bi:

The complex algorithm has: addition and subtraction, multiplication and division.

Division rule:

Let the complex number a+bi(a,b∈R) be divided by c+di(c,d∈R) whose quotient is x+yi(x,y∈R), ie \((a+bi)÷ (c+di)=x+yi\) ∵\((x+yi)(c+di)=(cx-dy)+(dx+cy)i\).

∴\((cx-dy)+(dx+cy)i=a+bi\).

Known by the plural definition \(cx-dy=a dx+cy=b\)

Solve this equation, get\( x=(ac+bd)/(c^2+d^2) y=(bc-ad)/(c^2+d^2)\)

Then there is:\((a+bi)/(c+di)=(ac+bd)/(c^2+d^2) +(bc-ad)/(c^2+d^2)i\)

 

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