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Chapter 2: Problem 101

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

Short Answer

Expert verified
By comparing the results we obtained from step 2 and step 4, we can see that they are identical. Therefore, we can conclude that the complex conjugate of the sum of two complex numbers is indeed the sum of their complex conjugates.

Step by step solution

01

Define the sum of the two complex numbers

The sum of two complex numbers \(a_{1}+b_{1} i\) and \(a_{2}+b_{2} i\) can be denoted as \(c = a_{1}+b_{1} i + a_{2}+b_{2} i\). Thus, \(c = (a_{1} + a_{2}) + (b_{1} + b_{2})i\).
02

Find the complex conjugate of the sum

The complex conjugate of \(c\) can be found by changing the sign of its imaginary part. Thus, the complex conjugate of \(c\), denoted as \(\overline{c}\), equals to \((a_{1} + a_{2}) - (b_{1} + b_{2})i\).
03

Define the complex conjugates of the two original complex numbers

The complex conjugate of \(a_{1}+b_{1} i\) and \(a_{2}+b_{2} i\) are \(\overline{z_{1}} = a_{1} - b_{1} i\) and \(\overline{z_{2}} = a_{2} - b_{2} i\), respectively.
04

Find the sum of the complex conjugates

The sum of the complex conjugates equals to \(\overline{z_{1}} + \overline{z_{2}} = (a_{1} - b_{1} i) + (a_{2} - b_{2} i)\). Thus, the result is \((a_{1} + a_{2}) - (b_{1} + b_{2})i\).

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