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Chapter 2: Problem 100

Prove that the complex conjugate of the product of two complex numbers \(a_{1}+b_{1} i\) and \(a_{2}+b_{2} i\) is the product of their complex conjugates.

Short Answer

Expert verified
The proof shows that the complex conjugate of the product of two complex numbers is the same as the product of their complex conjugates.

Step by step solution

01

- Definition of complex conjugate

For any complex number \(z = a + bi\), where \(a\) and \(b\) are real numbers, and \(i\) is the imaginary unit, the complex conjugate of \(z\) is defined as \(\bar{z} = a - bi\).
02

- Definition of the product of complex numbers

According to the multiplication rule for complex numbers, the product of two complex numbers \(z_{1} = a_{1} + b_{1}i\) and \(z_{2} = a_{2} + b_{2}i\) is given by \(z_{1}.z_{2} = (a_{1} a_{2} - b_{1} b_{2}) + (a_{1} b_{2} + a_{2} b_{1})i \).
03

- Calculate the complex conjugate of the product

Find the complex conjugate of the product \( z_{1} . z_{2} \). To do this, replace \(i\) by \(-i\) in the product. Therefore, \(\overline{z_{1} . z_{2}} = (a_{1} a_{2} - b_{1} b_{2}) - (a_{1} b_{2} + a_{2} b_{1})i \).
04

- Compute the product of complex conjugates

Now, let us calculate the product of the complex conjugates of \(z_{1}\) and \(z_{2}\), which are \(\bar{z}_{1}= a_{1} - b_{1}i \), and \(\bar{z}_{2}= a_{2} - b_{2}i \) respectively. Therefore, by applying the multiplication rule for complex numbers from Step 2, \(\bar{z}_{1}.\bar{z}_{2} = (a_{1} a_{2} - b_{1} b_{2}) - (a_{1} b_{2} + a_{2} b_{1})i \).
05

- Compare the two expressions

A close look at the results from Step 3 and Step 4 shows that they are exactly identical. Therefore, it is correct to say that the complex conjugate of the product of two complex numbers is equal to the product of their complex conjugates.

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