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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
The complex conjugate of the sum of two complex numbers equals the sum of their complex conjugates.

Step by step solution

01

Understanding the Question

Let's denote the two complex numbers as \(z_1 = a_{1} + b_{1} i\) and \(z_2 = a_{2} + b_{2} i\). The question asks to prove that \(\overline{z_1 + z_2} = \overline{z_1} + \overline{z_2}\), where \(\overline{ () }\) denotes the complex conjugate operation.
02

Calculate left-hand side (lhs)

We first compute the left side of the equation. We start by calculating the sum \( z_1 + z_2\). This results in \( (a_{1} + a_{2}) + (b_{1} + b_{2})i \). Now we take complex conjugate of this sum: \(\overline{z_1 + z_2} = \overline{(a_{1} + a_{2}) + (b_{1} + b_{2})i} = (a_{1} + a_{2}) - (b_{1} + b_{2})i\).
03

Calculate right-hand side (rhs)

Now, it is time to compute the right side of the equation. We first compute the complex conjugate of \(z_1\) and \(z_2\) separately: \(\overline{z_1}= a_{1} - b_{1}i\) and \(\overline{z_2}= a_{2} - b_{2}i\). If you add these two up, you get \(\overline{z_1} + \overline{z_2} = (a_{1} - b_{1}i) + (a_{2} - b_{2}i) = (a_{1} + a_{2}) - (b_{1} + b_{2})i\).
04

Conclusion

After computing both sides of the equation, we can see that they are equal. Therefore, it is proven that the complex conjugate of the sum of two complex numbers is the sum of their complex conjugates. Hence, \(\overline{z_1 + z_2} = \overline{z_1} + \overline{z_2}\)

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