Question

For any two complex numbers z₁ and z₂, prove that,
Re (z₁z₂) = Re z₁ Re z₂ - Im z₁ Im z₂

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Answer 1

Answer:

Let Z1,Z2 be two complex numbersz1= x + iyz2 = c + idz₁z₂ =(x + iy)*(c +id)=xc + ixd + icy -yd=(xc-yd) + i(xd + cy)Real Part of Z1Z2 - Re (Z1Z2) = xc - ydImaginary Part of Z1Z2 -Img (Z1Z2) = xd + cy

Explanation:

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Answer 2

Explanation:

Given :-

any two complex numbers z₁ and z₂

To prove:-

For any two complex numbers z₁ and z₂, prove that, Re (z₁z₂) = Re z₁ Re z₂ - Im z₁ Im z₂

Solution:-

For any two complex numbers z₁ and z₂

Used concept:-

The general form of a complex number=x+iy

Let the complex number z₁=a+ib and

let the other complex number z₂=c+id

Now

z₁z₂=(a+ib)(c+id)

=>z₁z₂=a(c+id)+ib(c+id)

=>z₁z₂=ac+aid+ibc+i²bd

we know that i²=-1

=>z₁z₂=ac+aid+ibc-bd

Rearranging the above

=>z₁z₂=(ac-bd)+(aid+ibc)

=>z₁z₂=(ac-bd)+i(ad+bc)

LHS:-

Re (z₁z₂) =ac-bd ------(1)

RHS:-

Re z₁ Re z₂ - Im z₁ Im z₂

= ac-bd ----------(2)

From (1/&(2)

LHS=RHS

Re (z₁z₂) = Re z₁ Re z₂ - Im z₁ Im z₂