For any two complex numbers z1 and z2, show that Re(z1z2) = Rez1 Rez2– Imz1Imz2

Welcome to AskIITians

Given: z1and z2are the two complex numbers

To prove: Re(z1z2) = Rez1Rez2– Imz1Imz2

Let z1= x1+iy1and z2= x2+iy2

Now, z1z2=(x1+iy1)(x2+iy2)

Now, split the real part and the imaginary part from the above equation:

⇒ x1(x2+iy2) +iy1(x2+iy2)

Now, multiply the terms:

= x1x2+ix1y2+ix2y1+i^2(y1y2)

We know that, i^2= -1, then we get

= x1x2+ix1y2+ix2y1-y1y2

Now, again seperate the real and the imaginary part:

= (x1x2-y1y2) +i (x1y2+x2y1)

From the above equation, take only the real part:

⇒ Re (z1z2) =(x1x2-y1y2)

It means that,

⇒ Re(z1z2) = Rez1Rez2– Imz1Imz2

Thanks