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GeezerVison is showing mathworld.wolfram.com — Complex Division -- from Wolfram MathWorld

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Complex Division

The division of two complex numbers can be accomplished by multiplying the numerator and denominator by the complex conjugate of the denominator, for example, with z_1=a+bi and z_2=c+di, z=z_1/z_2 is given by

z=(a+bi)/(c+di)
(1)
=((a+bi)c+di^_)/((c+di)c+di^_)
(2)
(3)
=((a+bi)(c-di))/((c+di)(c-di))
(4)
=((ac+bd)+i(bc-ad))/(c^2+d^2),
(5)

where z^_ denotes the complex conjugate. In component notation with (x,y)=x+iy,

((a,b))/((c,d))=((ac+bd)/(c^2+d^2),(bc-ad)/(c^2+d^2)).
(6)

See also

Complex Addition, Complex Exponentiation, Complex Multiplication, Complex Number, Complex Subtraction, Division

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Cite this as:

Weisstein, Eric W. "Complex Division." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/ComplexDivision.html

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