For any two complex numbers z 1 and z 2 , prove that Re (z 1 z 2 ) = Re z 1 Re z 2 – Im z 1 Im z 2

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Solution

Let,

z 1 = x 1 +i y 1 and z 2 = x 2 +i y 2

Therefore,

z 1 z = 2 ( x 1 +i y 1 )( x 2 +i y 2 ) = x 1 x 2 +i x 1 y 2 +i x 2 y 1 + i 2 y 1 y 2 =( x 1 x 2 − y 1 y 2 )+i( x 1 y 2 − x 2 y 1 )

Now,

Re( z 1 z 2 )= real part of z 1 z 2

=( x 1 x 2 − y 1 y 2 )

=Re z 1 ·Re z 2 −Im z 1 ·Im z 2

Hence, Re( z 1 z 2 )=Re z 1 ·Re z 2 −Im z 1 ·Im z 2 has been verified.

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