Upload File

introduction

19
Introduction Recall that the imaginary unit i is equal to . A fraction with i in the denominator does not have a rational denominator, since is not a rational number. Similar to rationalizing a fraction with an irrational square root in the denominator, fractions with i in the denominator can also have the denominator rationalized. 1 4.3.4: Dividing Complex Numbers

Upload: clare

Post on 24-Feb-2016

19 views

Category:

Documents


0 download

Report

Tags:

DESCRIPTION

Introduction Recall that the imaginary unit i is equal to . A fraction with i in the denominator does not have a rational denominator , since is not a rational number. - PowerPoint PPT Presentation

TRANSCRIPT

Page 1: Introduction

IntroductionRecall that the imaginary unit i is equal to . A fraction with i in the denominator does not have a rational denominator, since is not a rational number. Similar to rationalizing a fraction with an irrational square root in the denominator, fractions with i in the denominator can also have the denominator rationalized.

1

4.3.4: Dividing Complex Numbers

Page 2: Introduction

Key Concepts• Any powers of i should be simplified before dividing

complex numbers. • After simplifying any powers of i, rewrite the division of

two complex numbers in the form a + bi as a fraction. • To divide two complex numbers of the form a + bi and

c + di, where a, b, c and d are real numbers, rewrite the quotient as a fraction.

2

4.3.4: Dividing Complex Numbers

Page 3: Introduction

Key Concepts, continued• Rationalize the denominator of a complex fraction by

using multiplication to remove the imaginary unit i from the denominator.

• The product of a complex number and its conjugate is a real number, which does not contain i.

• Multiply both the numerator and denominator of the fraction by the complex number in the denominator.

• Simplify the rationalized fraction to find the result of the division.

3

4.3.4: Dividing Complex Numbers

Page 4: Introduction

Key Concepts, continued• In the following equation, let a, b, c, and d be real

numbers.

4

4.3.4: Dividing Complex Numbers

Page 5: Introduction

Common Errors/Misconceptions• multiplying only the denominator by the complex

conjugate • incorrectly determining the complex conjugate of the

denominator

5

4.3.4: Dividing Complex Numbers

Page 6: Introduction

Guided PracticeExample 2Find the result of (10 + 6i ) ÷ (2 – i ).

6

4.3.4: Dividing Complex Numbers

Page 7: Introduction

Guided Practice: Example 2, continued1. Rewrite the expression as a fraction.

7

4.3.4: Dividing Complex Numbers

Page 8: Introduction

Guided Practice: Example 2, continued2. Find the complex conjugate of the

denominator. The complex conjugate of a – bi is a + bi, so the complex conjugate of 2 – i is 2 + i.

8

4.3.4: Dividing Complex Numbers

Page 9: Introduction

Guided Practice: Example 2, continued3. Rationalize the fraction by multiplying

both the numerator and denominator by the complex conjugate of the denominator.

9

4.3.4: Dividing Complex Numbers

Page 10: Introduction

Guided Practice: Example 2, continued4. If possible, simplify the fraction. The

answer can be left as a fraction, or simplified by dividing both terms in the numerator by the quantity in the denominator.

10

4.3.4: Dividing Complex Numbers

✔

Page 11: Introduction

Guided Practice: Example 2, continued

11

4.3.4: Dividing Complex Numbers

http://www.walch.com/ei/00229
Page 12: Introduction

Guided PracticeExample 3Find the result of (4 – 4i) ÷ (3 – 4i

3).

12

4.3.4: Dividing Complex Numbers

Page 13: Introduction

Guided Practice: Example 3, continued1. Simplify any powers of i.

i 3 = –i

13

4.3.4: Dividing Complex Numbers

Page 14: Introduction

Guided Practice: Example 3, continued2. Simplify any expressions containing a

power of i. 3 – 4i

3 = 3 – 4(–i) = 3 + 4i

14

4.3.4: Dividing Complex Numbers

Page 15: Introduction

Guided Practice: Example 3, continued3. Rewrite the expression as a fraction, using

the simplified expression. Both numbers should be in the form a + bi.

15

4.3.4: Dividing Complex Numbers

Page 16: Introduction

Guided Practice: Example 3, continued4. Find the complex conjugate of the

denominator. The complex conjugate of a + bi is a – bi, so the complex conjugate of 3 + 4i is 3 – 4i.

16

4.3.4: Dividing Complex Numbers

Page 17: Introduction

Guided Practice: Example 3, continued5. Rationalize the fraction by multiplying

both the numerator and denominator by the complex conjugate of the denominator.

17

4.3.4: Dividing Complex Numbers

Page 18: Introduction

Guided Practice: Example 3, continued6. If possible, simplify the fraction. The

answer can be left as a fraction, or simplified by dividing both terms in the numerator by the quantity in the denominator.

18

4.3.4: Dividing Complex Numbers

✔

Page 19: Introduction

Guided Practice: Example 3, continued

19

4.3.4: Dividing Complex Numbers

http://www.walch.com/ei/00230

Introduction€¦ · Introduction ThisdocumentprovidesinformationabouttheIOSXEsoftwarereleasefortheCiscoNCS4201andCisco …

Introduction Section I: Introduction

voip-iptel-introduction-SRK1.ppt voip-iptel-introduction-SRK1.pptvoip-iptel-introduction-SRK1.pptvoip-iptel-introduction-SRK1.pptvoip-iptel-introduction-SRK1.pptvoip-iptel-introduction-SRK1.pptvoip-iptel-introduction-SRK1.pptvoip-iptel-introduction-SRK1.pptvoip-iptel-introduction-SRK1.pptvoip-iptel-introduction-SRK1.pptvoip-iptel-introduction-SRK1.pptvoip-iptel-introduction-SRK1.pptvoip-iptel-introduction-SRK1.pptvoip-iptel-introduction-SRK1.pptvoip-iptel-introduction-SRK1.pptvoip-iptel-introduction-SRK1.pptvoip-iptel-introduction-SRK1.pptvoip-iptel-introduction-SRK1.pptvoip-iptel-introduction-SRK1.pptvoip-iptel-introduction-SRK1.pptvoip-iptel-introduction-SRK1.pptvoip-iptel-introduction-SRK1.pptvoip-iptel-introduction-SRK1.pptvoip-iptel-introduction-SRK1.pptvoip-iptel-introduction-SRK1.pptvoip-iptel-introduction-SRK1.pptvoip-iptel-introduction-SRK1.pptvoip-iptel-introduction-SRK1.pptvoip-iptel-introduction-SRK1.pptvoip-iptel-introduction-SRK1.pptvoip-iptel-introduction-SRK1.pptvoip-ip

Module: Introduction...Module: Introduction Page: Introduction CC0.1 Introduction Please give a general description and introduction to your organization. Providing electrical, hydraulic

Topics Introduction 1 / 3 Introduction 2 / 3 Introduction 3 / 3

VisionaryScancombinesarichsetofqualitycontrolandproductivi ... · Introduction Introduction VisionaryScancombinesarichsetofqualitycontrolandproductivityfeatureswithasimpleinterfacedesignedspecificallyforscanningandcodingdepositionexhib-

Introduction to introduction to introduction to … Optimization

General introduction 1.1 Introduction

Introduction Introduction to Brittle Deformation

Introduction to Transducers, Introduction to Transducers ... on... · Introduction to Transducers, Introduction to Transducers, Sensors and ActuatorsSensors and Actuators Introduction

An Introduction to NSAn Introduction to NS2An Introduction to NS2 · 2016. 8. 26. · An Introduction to NSAn Introduction to NS2An Introduction to NS2 Textbook: T. Issariyakul and

Industry introduction Industry introduction Company introduction Company introduction Production introduction Display time time content

Introduction | Introduction | Millennium Serials · 2018. 4. 17. · MARC Display | Print | Introduction | Millennium Serials INTRODUCTION Introduction This tutorial introduces the

R as a Web Service Introduction Introduction Introduction

Introduction to Elasticsearch and Logstash. - KIT · PDF fileOutline 1 Introduction 2 Logstash Introduction 3 Elasticsearch Introduction 4 Conclusions Introduction Logstash ElasticsearchConclusions

An Introduction to NSAn Introduction to NS2An Introduction

INTRODUCTIONS (LEADS) Academic introduction Creative introduction Action introduction Dialogue introduction Overarching societal statements Personal introduction

 · Introduction . 6 . Contents . Introduction

PRODUCT INTRODUCTION Company Introduction

URBAN STREAM REHABILITATION. INTRODUCTION, OBJECTIVES & IMPACTS INTRODUCTION, OBJECTIVES & IMPACTS INTRODUCTION, OBJECTIVES & IMPACTS INTRODUCTION, OBJECTIVES

Introducción / Introduction / Introduction

 · Contents INTRODUCTION Introduction

Introduction Introduction (ctnd.)

1. Introduction 10 - therackonline.com€¦ · 1. Introduction 10 - therackonline.com ... Introduction)

Introduction to Logic Introduction

introduction | EinlEitung | introduction | introducción · introduction | EinlEitung | introduction | introducción En Description Operation Maintenance Safety instructions dE Benennung

Introduction and Welcome€¦ · Introduction and Welcome Introduction 4

Introduction Chapitre 1 · Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

The Sacrament of Baptism Introduction Video Introduction Video The Sacrament of Baptism Introduction Video Introduction Video

Introduction and Thesis Introduction - …...1 Introduction and Thesis Introduction: The introduction relates the topic to the main point of the essay. The introduction is composed

Introduction 1. Introduction

Introduction to Transducers, Introduction to Transducers ...engtech.weebly.com/.../more_on_transducers_sensors_actuators.pdf · Introduction to Transducers, Introduction to Transducers,

1. Introduction 1. Introduction 1. Introduction 1 ... · Introduction 1. Introduction 1. Introduction 1. Introduction 6061B0242401. The Phone Screen From the Phone screen, you can

1. INTRODUCTION • INTRODUCTION • EINFÜHRUNG • INLEIDING • INTRODUCCIÓN … · 2016-12-16 · 1. INTRODUCTION • INTRODUCTION • EINFÜHRUNG • INLEIDING • INTRODUCCIÓN

CrowPiwithMineCraftPiEdition · 2 CrowPiwithMineCraftPiEdition-Introduction-MinecraftPiEdition-Introduction-Whatyouwillneed-Introduction-RunningMinecraft-Introduction