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

When dealing with complex numbers, it's essential to grasp the operations involving both their real and imaginary parts. Let's take two complex numbers, z1=a+biz1=a+bi and z2=c+diz2=c+di, where aa, bb, cc, and dd are real numbers and ii represents the imaginary unit. Now, to find the real part of the product z1z2z1z2, we perform the multiplication: z1z2=(a+bi)(c+di)z1z2=(a+bi)(c+di) Expanding this expression: z1z2=ac+adi+bci+bdi2z1z2=ac+adi+bci+bdi2 Remembering that i2=−1i2=−1, we simplify: z1z2=ac+adi+bci−bdz1z2=ac+adi+bci−bd Now, let's separate the real and imaginary parts: Re(z1z2)=ac−bdRe(z1z2)=ac−bd This is indeed the real part of the product of z1z1 and z2z2. Now, let's look at the right side of the equation: Re(z1)=aRe(z1)=a Re(z2)=cRe(z2)=c Im(z1)=bIm(z1)=b Im(z2)=dIm(z2)=d Now, substituting these into the expression Re(z1)Re(z2)−Im(z1)Im(z2)Re(z1)Re(z2)−Im(z1)Im(z2), we get: Re(z1)Re(z2)−Im(z1)Im(z2)=ac−bdRe(z1)Re(z2)−Im(z1)Im(z2)=ac−bd And this is exactly what we got for Re(z1z2)Re(z1z2) earlier. So, we've shown that Re(z1z2)=Re(z1)Re(z2)−Im(z1)Im(z2)Re(z1z2)=Re(z1)Re(z2)−Im(z1)Im(z2). This is a fundamental property when dealing with complex numbers, and understanding it thoroughly will assist you in various mathematical applications. If you have any further questions or need additional clarification, feel free to ask! read less