/*! 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 74 Determine whether each statement... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 74

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. In the complex number system, \(x^{2}+y^{2}\) (the sum of two squares) can be factored as \((x+y i)(x-y i)\)

Short Answer

Expert verified
The statement is false. The correct statement is: In the complex number system, \(x^{2}+y^{2}\) can be factored as \((x+ yi)(x- yi)\), where y is a real number and i is the imaginary unit.

Step by step solution

01

Understanding complex numbers

A complex number is a number that can be expressed in the form a + bi, where a and b are real numbers, and i is the imaginary unit, which satisfies the equation \(i^{2} = -1\). iota (i) can be thought of as the square root of -1. Therefore, when dealing with complex numbers, \(x^{2}+y^{2}\) doesn't factorize into \((x+y i)(x-y i)\) as there is no imaginary part in \(x^{2}+y^{2}\).
02

Comparison with true identity

However, considering the formula for multiplication of two complex numbers, if you multiply \((x+y i)\) by \((x-y i)\), which are essentially complex conjugates of each other, the product would be \(x^2 + y^2\). What this means is that the factors would actually be \(x + yi\) and \(x - yi\) multiplied, where y is also considered a real number and i is an imaginary unit, not \(x + y\) and \(x - y\). The given statement misinterpreted the i as part of the number instead of as an imaginary unit.
03

Correcting the Statement

Rewrite the expression in the corrected form. This makes the correct statement: In the complex number system, \(x^{2}+y^{2}\) can be factored as \((x+yi)(x-yi)\), where y is a real number and i is the imaginary unit.

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

Solve absolute value inequality. \(|x|>5\)

This will help you prepare for the material covered in the next section. $$\text { Solve: }-2 x-4-x+5$$

A bank offers two checking account plans. Plan A has a base service charge of \(\$ 4.00\) per month plus 10 莽 per check. Plan \(\mathrm{B}\) charges a base service charge of \(\$ 2.00\) per month plus \(15 \phi\) per check. a. Write models for the total monthly costs for each plan if \(x\) checks are written. b. Use a graphing utility to graph the models in the same \([0,50,10]\) by \([0,10,1]\) viewing rectangle. c. Use the graphs (and the intersection feature) to determine for what number of checks per month plan A. will be better than plan B. d. Verify the result of part (c) algebraically by solving an inequality.

Use interval notation to express solution sets and graph each solution set on a number line. Solve linear inequality. \(\frac{3 x}{10}+1 \geq \frac{1}{5}-\frac{x}{10}\)

Solve absolute value inequality. \(\left|3-\frac{3}{4} x\right|>9\)
See all solutions

Recommended explanations on Math Textbooks

Pure Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Probability and Statistics

Read Explanation

Geometry

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.