/*! 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} Q. 1TF The complex number i: If we defi... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 8: Q. 1TF (page 702)

The complex number i: If we define \(i=\sqrt{-1}\), show that \(i^{2}=-1,i^{3}=-i, and i^{4}=1\).

Short Answer

Expert verified

It is shown that \(i^{2}=-1,i^{3}=-i, and i^{4}=1\).

Step by step solution

01

Step 1. Given Information

It is given that \(i=\sqrt{-1}\). The 'i' is known as an imaginary number and it is used to represent a complex number that is non-real.

02

Step 2. Show

As it is given that \(i=\sqrt{-1}\) so if we square on both sides we get,

\(i=\sqrt{-1}\)

\(i^{2}=\left ( \sqrt{-1} \right )^{2}\)

\(i^{2}=-1\) ......(i)

The imaginary number, when multiplied by itself, gives a negative value.

If we multiplied \(i\) on both sides of equation (i) we get,

\(i^{2}=-1\)

\(i\times i^{2}=-1\times i\)

\(i^{3}=-i\).......(ii)

Now, if we multiplied \(i\) on both sides of equation (ii) we get,

\(i^{3}=-i\)

\(i\times i^{3}=-i\times i\)

\(i^{4}=-i^{2}\)

\(i^{4}=1\)

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

Show that the power series 鈭�k=0鈭�鈥�(鈭�1)k2k+1x2k+1converges conditionally when x=1and when x=-1. What does this behavior tell you about the interval of convergence for the series?

If f(x)=4x3-5x2+6x+1andP3(x) is the third Taylor polynomial for f at 鈭�1, what is the third remainder R3(x)? What is R4(x)? (Hint: You can answer this question without finding any derivatives.)

In Exercises 49鈥�56 find the Taylor series for the specified function and the given value of x0. Note: These are the same functions and values as in Exercises 41鈥�48.

53.lnx,3

Let 鈭�k=0鈭�鈥�akxkbe a power series in xwith a finite radius of convergence p. Prove that if the series converges absolutely at either 卤p, then the series converges absolutely at the other value as well.

What is the relationship between a Maclaurin series and a power series in x?
See all solutions

Recommended explanations on Math Textbooks

Discrete Mathematics

Read Explanation

Applied Mathematics

Read Explanation

Mechanics Maths

Read Explanation

Calculus

Read Explanation

Decision Maths

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.