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Chapter 4: Problem 92

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

Step by step solution

01

Representation of Complex Numbers

Let's represent the complex numbers as: Z1 = \(a_{1}+b_{1} i\) and Z2 = \(a_{2}+b_{2} i\). Here, \(a_{1}, b_{1}, a_{2}, b_{2}\) are real numbers.
02

Complex Conjugates

The complex conjugate of Z1 = \(a_{1}+b_{1} i\) is \(a_{1}-b_{1} i\) and the complex conjugate of Z2 = \(a_{2}+b_{2} i\) is \(a_{2}-b_{2} i\).
03

Sum of Complex Numbers

The sum of Z1 and Z2 is \(a_{1}+a_{2}+(b_{1}+b_{2}) i\).
04

Conjugate of the Sum

The complex conjugate of the sum of Z1 and Z2 is \((a_{1}+a_{2})-(b_{1}+b_{2}) i\).
05

Sum of Complex Conjugates

The sum of the conjugates is \((a_{1}-b_{1} i)+(a_{2}-b_{2} i) = (a_{1}+a_{2})-(b_{1}+b_{2}) i\).
06

Equality Check

From steps 4 and 5, it is clear that the complex conjugate of the sum of Z1 and Z2 is equal to the sum of their complex conjugates.

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Most popular questions from this chapter

In Exercises 45-60, use the theorem on page 356 to find all the solutions of the equation and represent the solutions graphically. $$ x^{4}-16 i=0 $$

In Exercises 67-70, find the slope and the \(y\)-intercept (if possible) of the equation of the line. Then sketch the line. $$ y-9=0 $$

In Exercises 1-24, use DeMoivre's Theorem to find the indicated power of the complex number. Write the result in standard form. $$ \left[3\left(\cos \frac{2 \pi}{3}+i \sin \frac{2 \pi}{3}\right)\right]^{3} $$

In Exercises 59-64, write the complex number in standard form. $$ \sqrt{-6} \cdot \sqrt{-2} $$

In Exercises 93-96, perform the operation and write the result in standard form. $$ \left(x^{3}-3 x^{2}\right)-\left(6-2 x-4 x^{2}\right) $$
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