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Chapter 4: Problem 91

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
Given the complex numbers \(z_1 = a_1 + b_1i\) and \(z_2 = a_2 + b_2i\), it has been successfully proven that the complex conjugate of their product \(\overline{z_1 z_2} = a_1a_2 - b_1b_2 - (a_1b_2 + a_2b_1)i\) is equal to the product of their individual complex conjugates \(\overline{z_1} \overline{z_2} = a_1a_2 - b_1b_2 - (a_1b_2 + a_2b_1)i\). Therefore, the assertion holds correct in the realm of complex numbers.

Step by step solution

01

Express the original complex numbers and their conjugates.

Let the two complex numbers be \(z_1 = a_1 + b_1i\) and \(z_2 = a_2 + b_2i\). The complex conjugates of \(z_1\) and \(z_2\) are \(\overline{z_1} = a_1 - b_1i\) and \(\overline{z_2} = a_2 - b_2i\) respectively.
02

Compute the product of the two complex numbers and its complex conjugate.

Compute the product of \(z_1\) and \(z_2\), written as \(z_1 z_2 = (a_1 + b_1i)(a_2 + b_2i) \). This simplifies to \(a_1a_2 - b_1b_2 + (a_1b_2 + a_2b_1)i\). The complex conjugate of the product is given as \(\overline{z_1 z_2} = a_1a_2 - b_1b_2 - (a_1b_2 + a_2b_1)i\).
03

Compute the product of the individual complex conjugates.

Compute the product of \(\overline{z_1}\) and \(\overline{z_2}\), written as \(\overline{z_1} \overline{z_2} = (a_1 - b_1i) (a_2 - b_2i)\). This simplifies to \(a_1a_2 - b_1b_2 - (a_1b_2 + a_2b_1)i\).
04

Compare Step 2 and Step 3 results.

Since both Step 2 and Step 3 produce the same result, namely \(a_1a_2 - b_1b_2 - (a_1b_2 + a_2b_1)i\), we can conclude that the complex conjugate of the product of two complex numbers is indeed equal to the product of their individual complex conjugates, hence proving the statement.

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Most popular questions from this chapter

In Exercises 89-96, use the dot product to find the magnitude of \(u\). $$ \mathbf{u}=\langle-9,40\rangle $$

In Exercises 45-60, use the theorem on page 356 to find all the solutions of the equation and represent the solutions graphically. $$ x^{5}-32=0 $$

In Exercises 1-24, use DeMoivre's Theorem to find the indicated power of the complex number. Write the result in standard form. $$ (2+2 i)^{6} $$

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