/*! 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 61 What is the conjugate of \(a+b i... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 10: Problem 61

What is the conjugate of \(a+b i ?\)

Short Answer

Expert verified
The conjugate of \(a + bi\) is \(a - bi\).

Step by step solution

01

- Identify the complex number

The given complex number is in the form of \(a + bi\), where \(a\) is the real part and \(b\) is the imaginary part.
02

- Understand the concept of a conjugate

The conjugate of a complex number \(a + bi\) is \(a - bi\). This means that you change the sign of the imaginary part.
03

- Apply the concept

To find the conjugate of \(a + bi\), change \(bi\) to \(-bi\). Thus, the conjugate of \(a + bi\) is \(a - bi\).

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!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

conjugate of complex numbers
In the realm of complex numbers, the conjugate holds a significant position. The conjugate of a complex number can be thought of as a mirror reflection. To find the conjugate of a complex number, you change the sign of the imaginary part. For example, if you have the complex number denoted as \(a + bi\), the conjugate would be \(a - bi\). This switch in sign from + to - for the imaginary part is what defines the conjugate.
This operation is crucial in various mathematical scenarios:
  • When adding or subtracting complex numbers
  • When multiplying or dividing complex numbers
  • In complex number division, conjugates are used to rationalize the denominator
Remembering this simple rule of changing the sign of the imaginary part can greatly simplify working with complex numbers.
real part
The real part of a complex number refers to the non-imaginary component. In a complex number of the form \(a + bi\), the real part is represented by \(a\). Unlike the imaginary part, the real part does not involve the imaginary unit \(i\).
Here's how the real part plays an essential role:
  • It helps in comparing the magnitude of different complex numbers
  • When graphing complex numbers on an Argand diagram, the real part is plotted along the horizontal axis
  • During addition or subtraction of complex numbers, you only combine the real parts together
Understanding the real part helps in foundational algebra and more advanced fields including complex analysis.
imaginary part
The imaginary part of a complex number is what gives it complexity. It involves the imaginary unit \(i\), where \(i\) is defined as \(\sqrt{-1}\). In a complex number formatted as \(a + bi\), the term \(bi\) is the imaginary part.\
Properties and uses of the imaginary part include:
  • In polar form representation of complex numbers, where magnitude and angle come into play
  • Enabling solutions to equations that do not have real solutions
  • When multiplying complex numbers, the imaginary parts play a crucial role in deriving the result
Effectively distinguishing and knowing how to handle the imaginary part is key to mastering complex arithmetic and application in engineering fields.

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

Even though the root indexes of the terms are not equal, the sum \(\sqrt{64}+\sqrt[3]{125}+\sqrt[4]{16}\) can be simplified quite easily. What is this sum? Why can we add these terms so easily?

Add or subtract as indicated. Write your answers in the form \(a+b i\) $$ (-1+i)+(2+5 i)+(3+2 i) $$

Solve each formula for the indicated variable. $$ V=\sqrt{\frac{2 K}{m}} \text { for } m $$

Solve each equation. $$ -6 x^{2}+7 x=-10 $$

Find each power of \(i\) $$ i^{83} $$
See all solutions

Recommended explanations on Math Textbooks

Decision Maths

Read Explanation

Statistics

Read Explanation

Calculus

Read Explanation

Logic and Functions

Read Explanation

Applied Mathematics

Read Explanation

Probability and 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.