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For any two complex numbers z 1 and z 2 , show that Re(z 1 z 2 ) = Rez 1 Rez 2 – Imz 1 Imz 2

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

Grade:12

1 Answers

Harshit Singh
askIITians Faculty 5963 Points
4 years ago
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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

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