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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 complex conjugate of the product \( a_{1}a_{2} - b_{1}b_{2} - i(a_{1}b_{2}+ a_{2}b_{1}) \) of two complex numbers \( (a_{1}+b_{1} i) \) and \( (a_{2}+b_{2} i) \) is the same as the product \( a_{1}a_{2} - b_{1}b_{2} - i(a_{1}b_{2}+ a_{2}b_{1}) \) of their complex conjugates. Thus, this identity is true and therefore proved.

Step by step solution

01

Define a complex number and complex conjugate

A complex number is typically in the form of \(a+bi\), where \(a\) and \(b\) are real numbers, and \(i\) is the imaginary unit (\(i^2 = -1\)). The complex conjugate of a complex number is formed by changing the sign of its imaginary part. So, the complex conjugate of \(a + bi\) is \(a - bi\)
02

Multiply the two complex numbers

Let's multiply the two complex numbers \( (a_{1}+b_{1} i) \) and \( (a_{2}+b_{2} i) \). This gives\((a_{1}+b_{1} i)(a_{2}+b_{2} i)\) = \(a_{1}a_{2} - b_{1}b_{2} + i(a_{1}b_{2}+ a_{2}b_{1})\).
03

Find the complex conjugate of the product

Now we find the complex conjugate of the previous result which is \(a_{1}a_{2} - b_{1}b_{2} - i(a_{1}b_{2}+ a_{2}b_{1})\).
04

Find the product of the complex conjugates

Next, we calculate the product of the complex conjugates of the original two numbers, which becomes \((a_{1}-b_{1} i)(a_{2}-b_{2} i)\) = \(a_{1}a_{2} - b_{1}b_{2} - i(a_{1}b_{2}+ a_{2}b_{1})\).
05

Compare the results

Now that we have the complex conjugate of the product (from step 3) and the product of the complex conjugates (from step 4), we can compare them. As you can see, they are identical, which confirms the initial assertion.

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