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Chapter 2: Problem 101

Proof 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 the complex numbers \(a_{1} + b_{1}i\) and \(a_{2} + b_{2}i\) is equal to the sum of their complex conjugates.

Step by step solution

01

Identifying complex numbers and their conjugates

The given complex numbers are \(a_{1} + b_{1}i\) and \(a_{2} + b_{2}i\). The complex conjugate of a complex number \(a + bi\) is \(a 鈥� bi\) (the same real part a and the negation of its imaginary part). Therefore, the complex conjugates of \(a_{1} + b_{1}i\) and \(a_{2} + b_{2}i\) are respectively \(a_{1} - b_{1}i\) and \(a_{2} - b_{2}i\).
02

Compute the complex conjugate of the sum of two complex numbers

The sum of the two complex numbers is \(a_{1} + b_{1}i + a_{2} + b_{2}i = (a_{1} + a_{2}) + (b_{1} + b_{2})i\). The complex conjugate of this sum is \((a_{1} + a_{2}) - (b_{1} + b_{2})i\).
03

Compute the sum of the complex conjugates of the two complex numbers

The sum of the complex conjugates of the two complex numbers is \((a_{1} - b_{1}i) + (a_{2} - b_{2}i) = (a_{1} + a_{2}) - (b_{1} + b_{2})i\).
04

Conclusion

From Step 2 and Step 3, we see that the complex conjugate of the sum of the two complex numbers is equal to the sum of their complex conjugates. That is \((a_{1} + a_{2}) - (b_{1} + b_{2})i = (a_{1} + a_{2}) - (b_{1} + b_{2})i\). Therefore, it's proved that the complex conjugate of the sum of two complex numbers is equal to the sum of their complex conjugates.

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