For any two complex numbers displaystyle - z - 1 - and displaystyle - z - 2 - prove that displaystyle Releft- - z - 1 - z - 2 - right- -Re- z - 1 -Re- z - 2 -Im- z - 1 -Im- z - 2 -

Question

Toppr Admin · Accepted Answer

Let z1-x1-iy1 and-xA0- z2-x2-iy2-x21D2-z1z2-x1-iy1-x2-iy2-x1x2-i-y1x2-x1y2-i2y1y2-x1x2-x2212-y1y2-i-y1x2-x1y2- since -i2-x2212-1-Hence Re-z1z2-Re-z1-Re-z2-x2212-Im-z1-Im-z2-

For any two complex numbers z1 and z2, prove that Re(z1z2)=Rez1Rez2−Imz1Imz2

Let z1=x1+iy1 and z2=x2+iy2⇒z1z2=(x1+iy1)(x2+iy2)=x1x2+i(y1x2+x1y2)+i2y1y2=(x1x2−y1y2)+i(y1x2+x1y2), since (i2=−1)Hence Re(z1z2)=Re(z1)Re(z2)−Im(z1)Im(z2)

For any two complex numbers $z_{1}$ and $z_{2}$ , prove that $R e (z_{1} z_{2}) = R e z_{1} R e z_{2} - I m z_{1} I m z_{2}$