/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 101 Prove that the complex conjugate... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 2: Problem 101

Prove that the complex conjugate of the sum of two complex numbers \( a_1 + a_1i \) and \( a_2 + b_2i \) is the sum of their complex conjugates.

Short Answer

Expert verified
The proof has been successfully completed by applying the definition of complex conjugates and the respective properties of addition, resulting in both sides of the equation being established as identical.

Step by step solution

01

Identifying the problem

First, recognize the need to prove the equation: \((a_1 + b_1i) + (a_2 + b_2i)\)' = (a_1 + b_1i)' + (a_2 + b_2i)'\), where \((a_1 + b_1i)'\) and \((a_2 + b_2i)'\) are conjugates of the respective complex numbers.
02

Defining complex conjugates

Recognize that the complex conjugate of a complex number \(a + bi\) is \(a - bi\). So, \((a_1 + b_1i)'\) is \(a_1 - b_1i\) and \((a_2 + b_2i)'\) is \(a_2 - b_2i\).
03

Applying the conjugate to the sum of complex numbers

Applying the definition of complex conjugate to the left side of our equation we get \((a_1 + a_2 + (b_1 + b_2)i)'\) which becomes \(a_1 + a_2 - (b_1 + b_2)i\).
04

Evaluating the right side of the equation

The right side of our equation, which is adding the conjugates of the two complex numbers, becomes \(a_1 - b_1i + a_2 - b_2i = a_1 + a_2 - (b_1 + b_2)i\).
05

Comparing both sides

Upon comparing, both sides of the equation are identical: \(a_1 + a_2 - (b_1 + b_2)i = a_1 + a_2 - (b_1 + b_2)i\). Therefore, this proves that the complex conjugate of the sum of two complex numbers is equal to the sum of the complex conjugates of these two numbers. This completes the proof.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

In Exercises 73 - 76, (a) use the graph to determine any \( x \)-intercepts of the graph of the rational function and (b) set \( y = 0 \) and solve the resulting equation to confirm your result in part (a). \( y = \dfrac{2x}{x -3} \)

In Exercises 13 - 30, solve the inequality and graph the solution on the real number line. \( x^2 \le 16 \)

In Exercises 83 - 86, (a) find the interval(s) for \( b \) such that the equation has at least one real solution and (b) write a conjecture about the interval(s) based on the values of the coefficients. \( x^2 + bx - 4 = 0 \)

In Exercises 13 - 30, solve the inequality and graph the solution on the real number line. \( x^2 + 2x > 3 \)

Fill in the blanks. Between two consecutive zeros, a polynomial must be entirely ________ or entirely ________.
See all solutions

Recommended explanations on Math Textbooks

Pure Maths

Read Explanation

Geometry

Read Explanation

Mechanics Maths

Read Explanation

Decision Maths

Read Explanation

Calculus

Read Explanation

Statistics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.