Basic mathematics with calculus 10e by washington

Contents 1.3 Measurement, Calculation, and 1.7 Addition and Subtraction of Algebraic 1.8 Multiplication of Algebraic Expressions 36 Equations, Review Exercises, and Practice Test 51 Equa

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Basic Technical Mathematics with Calculus

SI Version

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Basic Technical Mathematics with Calculus, Tenth Edition, by Allyn J Washington

Introduction to Technical Mathematics, Fifth Edition, by Allyn J Washington, Mario F Triola, and Ellena Reda

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TENTH EDITION

Basic Technical Mathematics

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Library and Archives Canada Cataloguing in Publication

Washington, Allyn J., author

Basic technical mathematics with calculus : SI version / Allyn

J Washington, Michelle Boué Tenth edition

Includes indexes

ISBN 978-0-13-276283-0 (bound)

1 Mathematics Textbooks. I Boué, Michelle, author II Title

QA37.3.W37 2014 510 C2014-900075-8

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ISBN 978-0-13-276283-0

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In memory of my loving wife, Millie ~Allyn J Washington

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Contents

1.3 Measurement, Calculation, and

1.7 Addition and Subtraction of Algebraic

1.8 Multiplication of Algebraic Expressions 36

Equations, Review Exercises, and Practice Test 51

3.6 Graphs of Functions Defined by

5.3 Solving Systems of Two Linear Equations

5.6 Solving Systems of Three Linear Equations

5.7 Solving Systems of Three Linear Equations

6.6 Multiplication and Division of Fractions 202

7.1 Quadratic Equations; Solution by

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8 Trigonometric Functions of

10 Graphs of The Trigonometric

10.1 Graphs of y = a sin x and y = a cos x 297

10.2 Graphs of y = a sin bx and y = a cos bx 300

10.3 Graphs of y = a sin (bx + c) and y = a cos (bx + c) 303

10.4 Graphs of y = tan x, y = cot x, y = sec x, y = csc x 307

10.5 Applications of the Trigonometric Graphs 310

11.1 Simplifying Expressions with

11.5 Multiplication and Division of Radicals 335

12.6 Products, Quotients, Powers, and Roots

13 Exponential and Logarithmic

14 Additional Types of Equations

14.1 Graphical Solution of Systems of Equations 40014.2 Algebraic Solution of Systems of Equations 403

15.1 The Remainder and Factor Theorems;

16 Matrices; Systems of Linear

16.1 Matrices: Definitions and Basic Operations 436

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17.4 Inequalities Involving Absolute Values 482

17.5 Graphical Solution of Inequalities with

19 Sequences and The Binomial

20 Additional Topics in Trigonometry 531

24.2 Newton’s Method for Solving Equations 714

24.7 Applied Maximum and Minimum Problems 73724.8 Differentials and Linear Approximations 743

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27.5 Derivative of the Logarithmic Function 830

27.6 Derivative of the Exponential Function 834

28.8 Integration by Trigonometric

28.9 Integration by Partial Fractions:

28.10 Integration by Partial Fractions:

29 Partial Derivatives and Double

29.2 Curves and Surfaces in Three Dimensions 899

30.4 Computations by Use of Series Expansions 928

31.4 The Linear Differential Equation

31.5 Numerical Solutions of First-Order

31.8 Auxiliary Equation with Repeated

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Preface

Basic Technical Mathematics with Calculus, SI Version, tenth edition, is intended marily for students in technical and pre-engineering technology programs or other pro-grams for which coverage of basic mathematics is required Chapters 1 through 20 provide the necessary background for further study, with an integrated treatment of algebra and trigonometry Chapter 21 covers the basic topics of analytic geometry, and Chapter 22 gives an introduction to statistics Fundamental topics of calculus are cov-ered in Chapters 23 through 31 In the examples and exercises, numerous applications from many fields of technology are included, primarily to indicate where and how mathematical techniques are used However, it is not necessary that the student have a specific knowledge of the technical area from which any given problem is taken.Most students using this text will have a background that includes some algebra and geometry However, the material is presented in adequate detail for those who may need more study in these areas The material presented here is sufficient for three to four semesters

pri-One of the principal reasons for the arrangement of topics in this text is to present material in an order that allows a student to take courses concurrently in allied technical areas, such as physics and electricity These allied courses normally require a student to know certain mathematical topics by certain definite times; yet the traditional order of topics in mathematics courses makes it difficult to attain this coverage without loss of continuity However, the material in this book can be rearranged to fit any appropriate sequence of topics Another feature of this text is that certain topics traditionally included for mathematical completeness have been covered only briefly or have been omitted The approach used in this text is not unduly rigorous mathematically, although all appropriate terms and concepts are introduced as needed and given an intuitive or algebraic foundation The aim is to help the student develop an understanding of math-ematical methods without simply providing a collection of formulas The text material has been developed with the recognition that it is essential for the student to have a sound background in algebra and trigonometry in order to understand and succeed in any subsequent work in mathematics

Scope of the Book

New Features In this tenth edition of Basic Technical Mathematics with Calculus, SI Version, we

have retained all the basic features of successful previous editions and have also duced a number of improvements, described here

intro-NEW AND REVISED COVERAGE

The topics of units and measurement covered in an appendix in the ninth edition have been expanded and integrated into Chapter 1, together with new discussions on round-ing and on engineering notation Interval notation is introduced in Chapter 3 and is used in several sections throughout the text Chapter 31 includes a new subsection on solving nonhomogeneous differential equations using Fourier series

Chapter 22 has been revised and expanded; a new section on summarizing data covers measures of central tendency, measures of spread, and new material on Chebychev’s theorem; the section on normal distributions now includes a subsection on sampling distributions In addition, the chapter now includes a completely new section on confi-dence intervals

EXPANDED PEDAGOGY

r sizes valuable warnings against common mistakes or areas where students frequently IBWFEJGGJDVMUZ5IFTFCPYFTSFQMBDFUIFOPUFTGMBHHFECZBi$BVUJPOuJOEJDBUPSJOthe previous edition

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/&8i$PNNPO&SSPSuCPYFTBQQFBSUISPVHIPVUUIFUFYU"GSFTIEFTJHOFNQIB-ful boxes highlight the underlying rationale of using specific mathematical functions and encourage students to think strategically about how and why specific mathemati-cal concepts are needed and applied They also focus attention on material that is of particular importance in understanding the topic under discussion These boxes replace UIFOPUFTGMBHHFECZBi/PUFTuJOEJDBUPSJOUIFQSFWJPVTFEJUJPO

r /&8i1SPDFEVSFuCPYFTJODMVEFTUFQCZTUFQJOTUSVDUJPOTPOIPXUPQFSGPSNTFMFDUcalculations

FEWER CALCULATOR SCREENS

Many figures involving screens from a graphic calculator have been either removed from the text or replaced by regular graphs The calculator displays that remain are, for the most part, related to topics that require the use of technology (such as the graphical solution of systems of equations) or topics where technology can greatly simplify a process (such as obtaining the inverse of a large matrix) The appendix on graphing cal-culators from the previous edition dedicated to the graphing calculator will be available

in Chapter 34 of the Study Plan in both MyMathLab and MathXL versions of this course Students will also have easy access to it through the eText in MyMathLab

FUNCTIONAL USE OF COLOUR

The new full-colour design of this edition uses colour effectively for didactical poses Many figures and graphs have been enhanced with colour Moreover, colour is used to identify and focus attention on the text’s new pedagogical features Colour is also used to highlight the question numbers of writing exercises so that students and instructors can identify them easily

pur-NOTATION

Symbols used in accordance with professional Canadian standards are applied ently throughout the text

consist-INCREASED BREADTH OF APPLICATIONS

New examples and exercises have been added in order to increase the range of tions covered by the text New material can be found involving statics, fluid mechanics, optics, acoustics, cryptography, forestry, reliability, and quality control, to name but a few

applica-INTERNATIONAL AND CANADIAN CONTENT

New Canadian content appears either in the form of examples within the text (some of which are linked to chapter openers, so they are accompanied by a full colour image),

or as exercises at the end of a section or chapter All material of global interest has been retained or updated, and some new exercises were also added

LEARNING OUTCOMES

A list of Learning Outcomes appears on the introductory page of each chapter, ing the list of key topics for each section in the previous edition This new learning tool reflects the current emphasis on learning outcomes and gives the student and instructor

replac-a quick wreplac-ay of checking threplac-at they hreplac-ave covered key contents of the chreplac-apter

Continuing Features EXAMPLE DESCRIPTIONS

A brief descriptive title is given with each example number This gives an easy ence for the example, which is particularly helpful when a student is reviewing the contents of the section

refer-PRACTICE EXERCISES

Throughout the text, there are practice exercises in the margin Most sections have at least

one (and up to as many as four) of these basic exercises They are included so that a student

is more actively involved in the learning process and can check his or her understanding of

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the material to that point in the section They can also be used for classroom exercises The answers to these exercises are given at the end of the exercise set for the section.

SPECIAL EXPLANATORY COMMENTS

Throughout the book, special explanatory comments in colour have been used in the examples to emphasize and clarify certain important points Arrows are often used to indicate clearly the part of the example to which reference is made

IMPORTANT FORMULAS

Throughout the book, important formulas are set off and displayed so that they can be easily referenced

SUBHEADS AND KEY TERMS

Many sections include subheads to indicate where the discussion of a new topic starts within the section Key terms are noted in the margin for emphasis and easy reference

EXERCISES DIRECTLY REFERENCED TO TEXT EXAMPLES

The first few exercises in most of the text sections are referenced directly to a specific example of the section These exercises are worded so that it is necessary for the stu-dent to refer to the example in order to complete the required solution In this way, the student should be able to review and understand the text material better before attempt-ing to solve the exercises that follow

WRITING EXERCISES

One specific writing exercise is included at the end of each chapter These exercises give the students practice in explaining their solutions Also, there are more than 400 additional exercises throughout the book (at least 8 in each chapter) that require at least

a sentence or two of explanation as part of the answer The question numbers of writing FYFSDJTFTBSFIJHIMJHIUFEJODPMPVS"TQFDJBMi*OEFYPG8SJUJOH&YFSDJTFTuJTJODMVEFE

at the back of the book

WORD PROBLEMS

There are more than 120 examples throughout the text that show the complete solutions

of word problems There are also more than 850 exercises in which word problems are

to be solved

CHAPTER EQUATIONS, REVIEW EXERCISES, AND PRACTICE TESTS

At the end of each chapter, all important equations are listed together for easy reference Each chapter is also followed by a set of review exercises that covers all the material in the chapter Following the chapter equations and review exercises is a chapter practice test that students can use to check their understanding of the material Solutions to all practice test problems are given in the back of the book

APPLICATIONS

Examples and exercises illustrate the application of mathematics in all fields of nology Many relate to modern technology such as computer design, electronics, solar

tech-"QQMJDBUJPOTuJTJODMVEFEOFBSUIFFOEPGUIFCPPL

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There are more than 1400 worked examples in this text Of these, more than 300 trate technical applications.

ANSWERS TO EXERCISES

The answers to all odd-numbered exercises (except the end-of-chapter writing exercises) are given at the back of the book

FLEXIBILITY OF MATERIAL COVERAGE

The order of material coverage can be changed in many places, and certain sections may be omitted without loss of continuity of coverage Users of earlier editions have indicated the successful use of numerous variations in coverage Any changes will de-pend on the type of course and completeness required

Extensively updated by text author Michelle Boué, the Students Solutions Manual

con-tains revised solutions for every other odd-numbered exercise These step-by-step tions have been expanded for even greater accuracy, clarity, and consistency to improve

solu-student problem-solving skills The Students Solutions Manual is included in MyMathLab

and is also available as a printed supplement via the Pearson Custom Library (Please contact your local Pearson representative to learn more about this option.)

SUPPLEMENTS FOR THE INSTRUCTOR

Instructor’s resources include the following supplements

Instructor’s Solutions Manual

The Instructor’s Solution Manual contains detailed solutions to every section exercise,

including review exercises These in-depth, step-by-step solutions have been thoroughly revised by text author Michelle Boué for greater clarity and consistency; note that this ex-

pansion has been carried through to the Student Solutions Manual as well The

Instruc-tors Solutions Manual can be downloaded from Pearson’s online catalogue at www

.pearsoned.ca The Instructor’s Solution Manual contains solutions for all section exercises.

Animated PowerPoint Presentations

More than 150 animated slides are available for download from a protected location on Pearson Education’s online catalogue, at www.pearsoned.ca

Each slide offers a step-by-step mini lesson on an individual section, or key concept, formula, or equation from the first 28 chapters of the book For instance, 15 steps for using the “General Power Formula for Integration” are beautifully illustrated in the ani-mated slide for Chapter 28 There are two sets of slides for “Operations with Complex Numbers” for section 2 of Chapter 12; the 9 steps to perform addition are shown on one slide, and the 13 steps to perform subtraction appear on the second slide

These animated slides offer bite-sized chunks of key information for students to view and process prior to going to the homework questions for practice Please note that not every section in every chapter is accompanied by an animated slide as some topics lend themselves to this approach more than others These PowerPoint slide are also integrated in the Pearson eText within MyMathLab

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re-TestGen with Algorithmically Generated Questions

Instructors can easily create tests from textbook section objectives Algorithmically generated questions allow unlimited versions Instructors can edit problems or create their own by using the built-in question editor to generate graphs; import graphics; and insert math notation, variable numbers, or text Tests can be printed or administered online via the Web or other network

MyMathLab ® Online Course MyMathLab delivers proven results in helping individual students succeed:

r Z.BUI-BCIBTBDPOTJTUFOUMZQPTJUJWFJNQBDUPOUIFRVBMJUZPGMFBSOJOHJOIJHIFSeducation math instruction MyMathLab can be successfully implemented in any environment—lab-based, hybrid, fully online, traditional—and demonstrates the quantifiable difference that integrated usage has on student retention, subsequent success, and overall achievement

r sults on tests, quizzes, and homework and in the study plan You can use the grade-book to quickly intervene if your students have trouble or to provide positive feed-back on a job well done The data within MyMathLab is easily exported to a variety

.Z.BUI-BCTDPNQSFIFOTJWFPOMJOFHSBEFCPPLBVUPNBUJDBMMZUSBDLTTUVEFOUTSF-of spreadsheet programs, such as Micros.Z.BUI-BCTDPNQSFIFOTJWFPOMJOFHSBEFCPPLBVUPNBUJDBMMZUSBDLTTUVEFOUTSF-oft Excel You can determine which points

of data you want to export and then analyze the results to determine success

MyMathLab provides engaging experiences that personalize, stimulate, and measure

learning for each student:

r Exercises: The homework and practice exercises in MyMathLab are correlated to

the exercises in the textbook, and they regenerate algorithmically to give students unlimited opportunity for practice and mastery The software offers immediate, helpful feedback when students enter incorrect answers

r Multimedia learning aids: Exercises include guided solutions, sample problems,

animations, videos, and eText clips for extra help at point-of-use

r Expert tutoring: Although many students describe the whole of MyMathLab as

to live tutoring from Pearson, from qualified mathematics and statistics instructors who provide tutoring sessions for students via MyMathLab

And MyMathLab comes from a trusted partner with educational expertise and an eye

on the future:

r ity content Our eTexts are accurate, and our assessment tools work Whether you are just getting started with MyMathLab or have a question along the way, we’re here to help you learn about our technologies and how to incorporate them into your course

,OPXJOHUIBUZPVBSFVTJOHB1FBSTPOQSPEVDUNFBOTLOPXJOHUIBUZPVBSFVTJOHRVBM-To learn more about how MyMathLab combines proven learning applications with

pow-erful assessment, visit www.mymathlab.com or contact your Pearson representative MathXL ® Online Course

MathXL® is the homework and assessment engine that runs MyMathLab (MyMathLab

is MathXL plus a learning management system.) With MathXL, instructors can:exercises correlated at the objective level to the textbook

r $SFBUFBOEBTTJHOUIFJSPXOPOMJOFFYFSDJTFTBOEJNQPSU5FTU(FOUFTUTGPSBEEFEflexibility

r BJOUBJOSFDPSETPGBMMTUVEFOUXPSLUSBDLFEJO.BUI9-TPOMJOFHSBEFCPPL

With MathXL, students can:

r ized homework assignments based on their test results

5BLFDIBQUFSUFTUTJO.BUI9-BOESFDFJWFQFSTPOBMJ[FETUVEZQMBOTBOEPSQFSTPOBM-r 6TFUIFTUVEZQMBOBOEPSUIFIPNFXPSLUPMJOLEJSFDUMZUPUVUPSJBMFYFSDJTFTGPSUIFobjectives they need to study

r "DDFTTTVQQMFNFOUBMBOJNBUJPOTBOEWJEFPDMJQTEJSFDUMZGSPNTFMFDUFEFYFSDJTFT

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www.mathxl.com, or contact your Pearson representative.

Pearson Custom Library

For enrollments of at least 25 students, you can create your own textbook by choosing the chapters that best suit your own course needs To begin building your custom text, visit www.pearsoncustomlibrary.com You may also work with a dedicated Pearson Custom editor to create your ideal text—publishing your own original content or mixing and matching Pearson content Contact your local Pearson representative to get started

CourseSmart for Instructors

CourseSmart goes beyond traditional expectations—providing instant, online access to textbooks and course materials You can save time and hassle with a digital eTextbook that allows you to search for the most relevant content at the very moment you need it Whether it’s evaluating textbooks or creating lecture notes to help students with diffi-cult concepts, CourseSmart can make life a little easier See how by visiting www coursesmart.com/instructors

The team at Pearson Canada—Gary Bennett, Laura Armstrong, Cathleen Sullivan, Mary Wat, Michelle Bish—made this new edition possible

Also of great assistance during the production of this edition were Kimberley Blakey; Heidi Allgair; Kitty Wilson, copyeditor; Denne Wesolowski, proofreader; and Robert Brooker, tech checker

The authors gratefully acknowledge the contributions of the following reviewers, whose detailed comments and many suggestions were of great assistance in preparing this tenth edition:

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in the 1900s and 2000s, mathematics has been vital to the development of electronics and space travel.

Basic Algebraic

Operations

LEARNING OUTCOMES After completion of this chapter, the student should

be able to:

t Identify real, imaginary, rational, and irrational numbers

t Perform mathematical operations on integers, decimals, fractions, and radicals

t Use the fundamental laws

of algebra in numeric and algebraic equations

t Employ mathematical order

of operations

t Understand technical measurement and approximation, as well as the use of significant digits and rounding

t Use scientific and engineering notations

t Convert units of measurement

t Rearrange and solve basic algebraic expressions

t Interpret word problems using algebraic symbols

Interest in things such as the land on which they lived, the structures they built, and the

motion of the planets led people in early civilizations to keep records and to create

meth-ods of counting and measuring In turn, some of the early ideas of arithmetic, geometry,

and trigonometry were developed From such beginnings, mathematics has played a key role

in the great advances in science and technology.

Often, mathematical methods were developed from studies made in sciences, such as

astron-omy and physics, to better describe, measure, and understand the subject being studied Some

of these methods resulted from the needs in a particular area of application.

Many people were interested in the mathematics itself and added to what was then known

Although this additional mathematical knowledge may not have been related to applications

at the time it was developed, it often later became useful in applied areas.

In the chapter introductions that follow, examples of the interaction of technology and

math-ematics are given From these examples and the text material, it is hoped you will better

understand the important role that mathematics has had and still has in technology In this

text, there are applications from technologies including (but not limited to) aeronautical,

busi-ness, communications, electricity, electronics, engineering, environmental, heat and air

con-ditioning, mechanical, medical, meteorology, petroleum, product design, solar, and space To

solve the applied problems in this text will require a knowledge of the mathematics presented

but will not require prior knowledge of the field of application.

We begin by reviewing the concepts that deal with numbers and symbols This will enable us

to develop topics in algebra, an understanding of which is essential for progress in other areas

such as geometry, trigonometry, and calculus.

1

The Great Pyramid at Giza in Egypt

was built about 4500 years ago.

In the 1500s, 1600s, and 1700s, discoveries

in astronomy and the need for more accurate maps and instruments in navigation were very important in leading scientists and mathematicians to develop useful new ideas and methods in mathematics.

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In technology and science, as well as in everyday life, we use the very familiar ing numbers 1, 2, 3, and so on They are also called natural numbers or positive integers The negative integers -1, -2, -3, and so on are also very necessary and

count-useful in mathematics and its applications The integers include the positive integers

and the negative integers and zero, which is neither positive nor negative This means

the integers are the numbersc, -3, -2, -1, 0, 1, 2, 3, and so on

To specify parts of a quantity, rational numbers are used A rational number is any

number that can be represented by the division of one integer by another nonzero ger Another type of number, an irrational number, cannot be written as the division

inte-of one integer by another

E X A M P L E 1 Identifying rational numbers and irrational numbers

The numbers 5 and -19 are integers They are also rational numbers since they can be written as 51 and - 191 , respectively Normally, we do not write the 1’s in the denominators

The numbers 58 and - 113 are rational numbers because the numerator and the nator of each are integers

denomi-The numbers 12 and p are irrational numbers It is not possible to find two gers, one divided by the other, to represent either of these numbers It can be shown that square roots (and other roots) that cannot be expressed exactly in decimal form are irrational Also, 227 is sometimes used as an approximation for p, but it is not equal

inte-exactly to p We must remember that 227 is rational and p is irrational

The decimal number 1.5 is rational since it can be written as 32 Any such

terminat-ing decimal is rational The number 0.6666c, where the 6’s continue on indefinitely,

is rational since we may write it as 23 In fact, any repeating decimal (in decimal form, a

specific sequence of digits is repeated indefinitely) is rational The decimal number 0.673 273 273 2cis a repeating decimal where the sequence of digits 732 is repeated indefinitely 10.673 273 273 2 c = 1121

The integers, the rational numbers, and the irrational numbers, including all such numbers that are positive, negative, or zero, make up the real number system (see

Fig 1.1) There are times we will encounter an imaginary number, the name given to

the square root of a negative number Imaginary numbers are not real numbers and will

be discussed in Chapter 12 However, unless specifically noted, we will use real

num-bers Until Chapter 12, it will be necessary to only recognize imaginary numbers when

they occur

Also in Chapter 12, we will consider complex numbers, which include both the real

numbers and imaginary numbers See Exercise 37 of this section

E X A M P L E 2 Identifying real numbers and imaginary numbers

The number 7 is an integer It is also rational since 7 = 71, and it is a real number since the real numbers include all the rational numbers

The number 3p is irrational, and it is real since the real numbers include all the tional numbers

irra-The numbers 1-10 and - 1-7 are imaginary numbers

The number - 37 is rational and real The number - 17 is irrational and real

The number p

6 is irrational and real The number 1 - 32 is imaginary ■

A fraction may contain any number or symbol representing a number in its

numer-ator or in its denominnumer-ator The fraction indicates the division of the numerator by the denominator, as we previously indicated in writing rational numbers Therefore, a frac-tion may be a number that is rational, irrational, or imaginary A fraction can represent

■ Irrational numbers were discussed by the

Greek mathematician Pythagoras in about

540 B C E

A notation that is often used for

repeating decimals is to place a bar

over the digits that repeat Using this

notation we can write

Fig 1.1

■ Real numbers and imaginary numbers are

both included in the complex number system

See Exercise 37.

■ Fractions were used by early Egyptians and

Babylonians They were used for calculations

that involved parts of measurements, property,

and possessions.

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a part of a whole, and sometimes it can represent the number of equal-sized parts that a whole is divided into For example, in Fig 1.2, a whole circle has been divided into eight equal pieces The shaded portion represents five of those eight pieces, or 5/8 of the whole circle.

E X A M P L E 3 Fractions

The numbers 2

7 and - 32 are fractions, and they are rational

The numbers 129 and p6 are fractions, but they are not rational numbers It is not sible to express either as one integer divided by another integer

pos-The number 1 - 56 is a fraction, and it is an imaginary number ■Real numbers may be represented by points on a line We draw a horizontal line and

designate some point on it by O, which we call the origin (see Fig 1.3) The integer

zero is located at this point Equal intervals are marked to the right of the origin, and the positive integers are placed at these positions The other positive rational numbers are located between the integers The points that cannot be defined as rational numbers represent irrational numbers We cannot tell whether a given point represents a rational number or an irrational number unless it is specifically marked to indicate its value

2 3 4 5 6

The negative numbers are located on the number line by starting at the origin and

marking off equal intervals to the left, which is the negative direction As shown in

Fig 1.3, the positive numbers are to the right of the origin and the negative numbers

are to the left of the origin Representing numbers in this way is especially useful for graphical methods

We next define another important concept of a number The absolute value of a

number is the numerical value (magnitude) of the number without regard to its sign The absolute value of a positive number is the number itself, and the absolute value of

a negative number is just the number, without the negative sign On the number line,

we may interpret the absolute value of a number as the distance (which is always tive) between the origin and the number Absolute value is denoted by writing the num-ber between vertical lines, as shown in the following example

E X A M P L E 4 Absolute value

The absolute value of 6 is 6, and the absolute value of -7 is 7 We write these as

060 = 6 and 0-70 = 7 See Fig 1.4

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number is said to be greater than the second If the first number is to the left of the

sec-ond, it is less than the second number The symbol 7 designates “is greater than,” and the symbol 6 designates “is less than.” These are called signs of inequality See Fig 1.5.

E X A M P L E 5 Signs of inequality

by English mathematicians in the late 1500s.

Practice Exercises

Place the correct sign of inequality ( 6 or 7)

between the given numbers.

2 is to the right of −4

2 > −4 3 < 6

0 > −4

5 < 9 −3 > −7 −1 < 0

Pointed toward smaller number

Every number, except zero, has a reciprocal The reciprocal of a number is 1

divided by the number

E X A M P L E 6 Reciprocal

The reciprocal of 7 is 17 The reciprocal of 23 is

12 3

= 1 * 32 = 32 invert denominator and multiply (from arithmetic)The reciprocal of 0.5 is 0.51 = 2 The reciprocal of -p is -1

p Note that the negative sign is retained in the reciprocal of a negative number

We showed the multiplication of 1 and 32 as 1 * 3

2 We could also show it as 1# 3

2

or 113

22 We will often find the form with parentheses is preferable ■

In applications, numbers that represent a measurement and are written with units of

measurement are called denominate numbers The next example illustrates the use of

units and the symbols that represent them

To show that the speed of a rocket is 1500 metres per second, we write the speed as

1500 m>s (Note the use of s for second We use s rather than sec.)

To show that the area of a computer chip is 0.75 square centimetres, we write the area as 0.75 cm2 (We will not use sq cm.)

To show that the volume of water in a glass tube is 25 cubic centimetres, we write

It is usually more convenient to state definitions and operations on numbers in a

general form To do this, we represent the numbers by letters, called literal numbers

For example, if we want to say “If a first number is to the right of a second number on the number line, then the first number is greater than the second number,” we can write

“If a is to the right of b on the number line, then a 7 b.” Another example of using a literal number is “The reciprocal of n is 1 >n.”

Certain literal numbers may take on any allowable value, whereas other literal

num-bers represent the same value throughout the discussion Those literal numnum-bers that

may vary in a given problem are called variables, and those literal numbers that are

held fixed are called constants.

■ For reference, see Section 1.3 for units of

measurement and the symbols used for them.

Literal Numbers

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E X A M P L E 8 Variables and constants

(a) The resistance of an electric resistor is R The current I in the resistor equals the

voltage V divided by R, written as I = V>R For this resistor, I and V may take on various values, and R is fixed This means I and V are variables and R is a constant For a different resistor, the value of R may differ.

(b) The fixed cost for a calculator manufacturer to operate a certain plant is b dollars

per day, and it costs a dollars to produce each calculator The total daily cost C to produce n calculators is

C = an + b Here, C and n are variables, and a and b are constants, and the product of a and n is shown as an For another plant, the values of a and b would probably differ.

If specific numerical values of a and b are known, say a = $7 per calculator and

b = $3000, then C = 7a + 3000 Thus, constants may be numerical or literal ■

EXERCISES 1.1

In Exercises 1–4, make the given changes in the indicated examples of

this section, and then answer the given questions.

1 In the first line of Example 1, change the 5 to -3 and the -19 to

14 What other changes must then be made in the first paragraph?

2 In Example 4, change the 6 to -6 What other changes must then

be made in the first paragraph?

3 In the left figure of Example 5, change the 2 to -6 What other

changes must then be made?

4 In Example 6, change the 2 to 3 What other changes must then be

made?

In Exercises 5 and 6, designate each of the given numbers as being an

integer, rational, irrational, real, or imaginary (More than one

designation may be correct.)

In Exercises 7 and 8, find the absolute value of each number.

In Exercises 9–16, insert the correct sign of inequality ( 7 or 6)

between the given numbers.

writ-25 Find a rational number between 0.13 and 0.14 that can be written

with a numerator of 3 and an integer in the denominator.

29 If a and b are positive integers and b 7 a, what type of number is

represented by the following?

(a) b - a (b) a - b (c) b b - a

+ a

30 If a and b represent positive integers, what kind of number is

rep-resented by (a) a + b, (b) a>b, and (c) a * b?

31 For any positive or negative integer: (a) Is its absolute value

always an integer? (b) Is its reciprocal always a rational number?

32 For any positive or negative rational number: (a) Is its absolute

value always a rational number? (b) Is its reciprocal always a rational number?

33 Describe the location of a number x on the number line when

(a) x 7 0 and (b) x 6 -4.

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(a) 0x0 6 1 and (b) 0x0 7 2.

35 For a number x7 1, describe the location on the number line of

the reciprocal of x.

36 For a number x6 0, describe the location on the number line of

the number with a value of 0x0.

37 A complex number is defined as a + bj, where a and b are real

numbers and j = 1-1 For what values of a and b is the

com-plex number a + bj a real number? (All real numbers and all

imaginary numbers are also complex numbers.)

38 A sensitive gauge measures the total weight w of a container and

the water that forms in it as vapor condenses It is found that

w = c 10.1t + 1, where c is the weight of the container and t is

the time of condensation Identify the variables and constants.

39 In an electric circuit, the reciprocal of the total capacitance of two

capacitors in series is the sum of the reciprocals of the

capaci-tances Find the total capacitance of two capacitances of 0.0040 F

and 0.0010 F connected in series.

40 Alternating-current (ac) voltages change rapidly between positive

and negative values If a voltage of 100 V changes to -200 V,

which is greater in absolute value?

Express the number N of bits in n kilobytes in an equation (A bit

is a single digit, and bits are grouped in bytes in order to represent

special characters Generally, there are 8 bits per byte If

neces-sary, see Fig 1.10 for the meaning of kilo.)

42 The computer design of the base of a truss is x m long Later it is

redesigned and shortened by y cm Give an equation for the length L, in centimetres, of the base in the second design.

43 In a laboratory report, a student wrote “-20°C 7 -30°C.” Is this statement correct? Explain.

44 After 5 s, the pressure on a valve is less than 600 kPa Using t to

represent time and p to represent pressure, this statement can be written “for t 7 5 s, p 6 600 kPa.” In this way, write the statement “when the current I in a circuit is less than 4 A, the voltage

exam-called the commutative law for addition It states that the sum of two numbers is the

same, regardless of the order in which they are added We make no attempt to prove

this law in general, but accept that it is true

In the same way, we have the associative law for addition, which states that the sum

of three or more numbers is the same, regardless of the way in which they are grouped for addition For example, 3 + 15 + 62 = 13 + 52 + 6

The laws just stated for addition are also true for multiplication Therefore, the

prod-uct of two numbers is the same, regardless of the order in which they are multiplied, and the product of three or more numbers is the same, regardless of the way in which

they are grouped for multiplication For example, 2 * 5 = 5 * 2, and

5 * 14 * 22 = 15 * 42 * 2

Another very important law is the distributive law It states that the product of one

number and the sum of two or more other numbers is equal to the sum of the products

of the first number and each of the other numbers of the sum For example,

514 + 22 = 5 * 4 + 5 * 2

In this case, it can be seen that the total is 30 on each side

In practice, these fundamental laws of algebra are used naturally without thinking

about them, except perhaps for the distributive law

Not all operations are commutative and associative For example, division is not commutative, since the order of division of two numbers does matter For instance, 6

5 ≠ 5

6 (≠ is read “does not equal”) (Also, see Exercise 50.)Using literal numbers, the fundamental laws of algebra are as follows:

'VOEBNFOUBM-BXTPG"MHFCSB t

Operations on Positive and Negative

/VNCFST t 0SEFSPG0QFSBUJPOT t

Operations with Zero

The Commutative and Associative Laws

The Distributive Law

■ Note carefully the difference:

associative law: 5 * 14 * 22

distributive law: 5 * 14 + 22

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Associative law of multiplication: a(bc) = (ab)c

Each of these laws is an example of an identity, in that the expression to the left of the = sign equals the expression to the right for any value of each of a, b, and c.

OPERATIONS ON POSITIVE AND NEGATIVE NUMBERS

When using the basic operations (addition, subtraction, multiplication, division) on positive and negative numbers, we determine the result to be either positive or negative according to the following rules

Addition of two numbers of the same sign Add their absolute values and assign

the sum their common sign.

E X A M P L E 1 Adding numbers of the same sign

(b) -2 + 1 -62 = - 12 + 62 = -8 the sum of two negative numbers is negative The negative number -6 is placed in parentheses since it is also preceded by a plus sign showing addition It is not necessary to place the -2 in parentheses ■

Addition of two numbers of different signs Subtract the number of smaller

abso-lute value from the number of larger absoabso-lute value and assign to the result the sign

of the number of larger absolute value. Alternatively, one can visualize addition ing the number line concept discussed in Section 1.1 Start with the number line loca-tion of the first number in the addition problem Then, if you add a positive number,

us-move right along the number line to the total If you add a negative number, us-move left

along the number line until you arrive at the solution

E X A M P L E 2 Adding numbers of different signs

Subtraction of one number from another Change the sign of the number being

subtracted and change the subtraction to addition Perform the addition.

E X A M P L E 3 Subtracting positive and negative numbers

(a) 2 - 6 = 2 + 1 -62 = - 16 - 22 = -4 Note that after changing the subtraction to addition, and changing the sign of 6 to make it -6, we have precisely the same illustration as Example 2(a)

(b) -2 - 6 = -2 + 1 -62 = - 12 + 62 = -8 Note that after changing the subtraction to addition, and changing the sign of 6 to make it -6, we have precisely the same illustration as Example 1(b)

(c) -a - 1 -a2 = -a + a = 0

This shows that subtracting a number from itself results in zero, even if the number

is negative Therefore, subtracting a negative number is equivalent to adding a

positive number of the same absolute value ■

Multiplication and division of two numbers The product (or quotient) of two

num-bers of the same sign is positive The product (or quotient) of two numnum-bers of ent signs is negative.

differ-■ From Section 1.1, we recall that a positive

number is preceded by no sign Therefore, in

using these rules, we show the “sign” of a

positive number by simply writing the number

itself.

Subtraction of a Negative Number

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(a) 31122 = 3 * 12 = 36 123 = 4 result is positive if both numbers are positive

result is positive if both numbers are negative

(c) 31 -122 = - 13 * 122 = -36 -123 = -123 = -4 result is negative if one number is positive and

the other is negative

When mathematical operation symbols separate a series of numbers in an expression, it

is important to follow an unambiguous order for completing those operations.

Order of Operations

1 Perform operations within specific groupings first—that is, inside parentheses

( ), brackets [ ], or absolute values % %

2 Exponents and roots/radicals are evaluated next

These will be discussed in Section 1.4 and Section 1.6, respectively.

3 Perform multiplications and divisions (from left to right).

4 Perform additions and subtractions (from left to right).

E X A M P L E 5 Order of operations

(a) 20 , 12 + 32 is evaluated by first adding 2 + 3 and then dividing The grouping

of 2 + 3 is clearly shown by the parentheses Therefore,

20 , 12 + 32 = 20 , 5 = 4

(b) 20 , 2 + 3 is evaluated by first dividing 20 by 2 and then adding No specific grouping is shown, and therefore the division is done before the addition This means 20 , 2 + 3 = 10 + 3 = 13

(c) 16 - 2 * 3 is evaluated by first multiplying 2 by 3 and then subtracting We do not first subtract 2 from 16 Therefore, 16 - 2 * 3 = 16 - 6 = 10

(d) 16 , 2 * 4 is evaluated by first dividing 16 by 2 and then multiplying From left

to right, the division occurs first Therefore, 16 , 2 * 4 = 8 * 4 = 32

(e) % 3 - 5 % - % -3 - 6 % is evaluated by first performing the subtractions within the absolute value vertical bars, then evaluating the absolute values, and then subtracting This means that % 3 - 5 % - % -3 - 6 % = % -2% - % -9 % = 2 - 9 = -7 ■

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E X A M P L E 6 Evaluating numerical expressions

In illustration (b), we see that the division and multiplication were done before the addition and subtraction In (c) and (d), we see that the groupings were evaluated first Then we did the divisions, and finally the subtraction and addition ■

E X A M P L E 7 Evaluating in an application

A 1500-kg van going at 40 km>h ran head-on into a 1000-kg car going at 20 km>h An insurance investigator determined the velocity of the vehicles immediately after the collision from the following calculation See Fig 1.6

The numerator and the denominator must be evaluated before the division is formed The multiplications in the numerator are performed first, followed by the addi-

OPERATIONS WITH ZERO

Since operations with zero tend to cause some difficulty, we will show them here

If a is a real number, the operations of addition, subtraction, multiplication, and

division with zero are as follows:

a + 0 = a

a : 0 = 0

0 ÷ a = 0a = 0 1if a 3 02 (≠ means “is not equal to”)

E X A M P L E 8 Operations with zero

(a) 5 + 0 = 5 (b) -6 - 0 = -6 (c) 0 - 4 = -4 (d) 0

6 = 0 (e)

0-3 = 0 (f)

Division by zero is undefined because

no real value can be associated with

that division.

If c = 40, then c* 0 = 4, which is

not true, since c * 0 = 0 for any

value of c.

There is a special case of division

by zero termed indeterminate

because no specific value can be

determined from the division, but

many real values are indeed possible.

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see bottom of page 92

Division by zero is an undefined operation in mathematics Even when trying to solve

equations, every time you perform a division, you must specify that you are not

commit-ting a division by zero error.

For example, to solve x#x = 3#x one might be tempted to divide both sides of the

equation by x This is fine as long as x ≠ 0.

x#x

x =

3#x x

x = 3 Notice, however, that a solution to the equation has been missed x = 0 is also a valid

solution (0#0 = 3#0), yet it was missed because when x = 0, an invalid division by x

took place.

C O M M O N E R R O R

EXERCISES 1.2

this section and then solve the resulting problems.

1 In Example 5(c), change 3 to 1 -32 and then evaluate.

2 In Example 6(b), change 18 to -18 and then evaluate.

3 In Example 6(d), interchange the 2 and 8 in the first denominator

and then evaluate.

4 In the rightmost illustration in Example 9, interchange the 6 and

the 0 above the 6 Is any other change needed?

In Exercises 5–36, evaluate each of the given expressions by performing

the indicated operations.

49 (a) What is the sign of the product of an even number of negative

numbers? (b) What is the sign of the product of an odd number of negative numbers?

50 Is subtraction commutative? Explain.

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51 Explain why the following definition of the absolute value of a

real number x is either correct or incorrect (the symbol Ú means

“is equal to or greater than”: If x Ú 0, then 0x0 = x; if x6 0,

then 0x0 = -x).

52 Explain what the error is if the expression 24 - 6 , 2#3 is

evaluated as 27 What is the correct value?

53 Describe the values of x and y for which (a) -xy = 1 and

(b) x - y

x - y = 1.

54 Describe the values of x and y for which (a) 0x + y0 = 0x0 + 0y0

and (b) 0x - y0 = 0x0 + 0y0.

55 Some solar energy systems are used to supplement the utility

company power supplied to a home such that the meter runs

backward if the solar energy being generated is greater than the

energy being used With such a system, if the solar power

aver-ages 1.5 kW for a 3.0-h period and only 2.1 kW#h is used during

this period, what will be the change in the meter reading for this

period?

56 A baseball player’s batting average (total number of hits divided

by total number of at-bats) is expressed in decimal form from

0.000 (no hits for all at-bats) to 1.000 (one hit for each at-bat) A

player’s batting average is often shown as 0.000 before the first

at-bat of the season Is this a correct batting average? Explain.

57 The daily high temperatures (in °C) in the Falkland Islands in the

southern Atlantic Ocean during the first week in July were

recorded as 7, 3, -2, -3, -1, 4, and 6 What was the average

daily temperature for the week? (Divide the algebraic sum of the

readings by the number of readings.)

58 A flare is shot up from the top of a tower Distances above the

flare gun are positive and those below it are negative After 5 s

the vertical distance (in m) of the flare from the flare gun is found

by evaluating 1202 152 + 1 -52 1252 Find this distance.

59 Find the sum of the voltages of the batteries shown in Fig 1.7

Note the directions in which they are connected.

6 V −2 V 8 V −5 V 3 V

+ − + −

+ − − + − +

Fig 1.7

60 The electric current was measured in a given ac circuit at equal

intervals as 0.7 mA, -0.2 mA, -0.9 mA, and -0.6 mA What was the change in the current between (a) the first two readings, (b) the middle two readings, and (c) the last two readings?

61 One oil-well drilling rig drills 100 m deep the first day and 200 m

deeper the second day A second rig drills 200 m deep the first day and 100 m deeper the second day In showing that the total depth drilled by each rig was the same, state what fundamental law of algebra is illustrated.

62 A water tank leaks 12 L each hour for 7 h, and a second tank leaks

7 L each hour for 12 h In showing that the total amount leaked is the same for the two tanks, what fundamental law of algebra is illustrated?

63 Each of four persons spends 8 min browsing one website and 6

min browsing a second website Set up the expression for the total time these persons spent browsing these websites What fundamental law of algebra is illustrated?

64 A jet travels 600 km>h relative to the air The wind is blowing at

50 km >h If the jet travels with the wind for 3 h, set up the sion for the distance travelled What fundamental law of algebra

expres-is illustrated?

Answers to Practice Exercises

1 9 2 2 3 -4 4 8

$"-$6-"5034

You will be doing many of your calculations on a calculator, and a graphing calculator

can be used for these calculations and many other operations In this text, we will

restrict our coverage of calculator use to graphing calculators because a scientific

cal-culator cannot perform many of the required operations we will cover

A discussion regarding the use of a graphing calculator can be found at the text’s companion web site Since there are many models of graphing calculators, the notation and screen appearance for many operations will differ from one model to another Therefore, although we include some calculator screens throughout the book, not every calculator discussion will be accompanied by a sample screen

You should practice using your calculator and review its manual to be sure how it

is used Following is an example of a basic calculation done on a graphing calculator

E X A M P L E 1 Calculating on a graphing calculator

In order to calculate the value of 38.3 - 12.91 -3.582, the numbers are entered as lows The calculator will perform the multiplication first, following the order of opera-tions shown in Section 1.2 The sign of -3.58 is entered using the (-) key, before 3.58 is entered The display on the calculator screen is shown in Fig 1.8

■ The calculator screens shown with text

material are for a TI-83 or TI-84 They are

intended only as an illustration of a calculator

screen for the particular operation Screens for

other models may differ.

Fig 1.8

Trang 29

Note in the display that the negative sign of -3.58 is smaller and a little higher to distinguish it from the minus sign for subtraction Also note the * shown for multiplica-tion; the asterisk is the standard computer symbol for multiplication ■

Looking back into Section 1.2, we see that the minus sign is used in two different

ways: (1) to indicate subtraction and (2) to designate a negative number This is clearly shown on a graphing calculator because there is a key for each purpose The - key

is used for subtraction, and the (-) key is used before a number to make it negative

We will first use a graphing calculator for the purpose of graphing in Section 3.5 Before then, we will show some calculational uses of a graphing calculator

6/*540'.&"463&.&/5

Most scientific and technical calculations involve numbers that represent a

measure-ment or count of a specific physical quantity A measuremeasure-ment represents an estimate

of the value of the physical quantity that exists in reality, and is usually accompanied

by an uncertainty or error in that measured value To report a measurement in a meaningful way, the units of measurement, which indicate a specific size or magni-

tude of a physical measurement, have to be expressed For example, if the length of an object is measured to be 12.5, it is critical to know if that is measured in centimetres, metres, feet, or some other unit of length

The definition and practical use of units of measurement has spawned many ent systems of counting and units throughout human history Many of the ancient sys-tems invented were largely based on dimensions of the human body Consequently, measurements varied from place to place, and communication of the measured values was inconsistent since each unit did not have a universally recognized size The metric system, first adopted in France in the late 1700s, incorporated the feature of standardi-zation of units, wherein everyone using the system agreed to a specific size for each unit The SI metric system of units (International System of Units) has been agreed upon by international committees of scientists and engineers and was established in

differ-1960 Most scientific endeavours worldwide employ the SI system of units It is tant for scientists, engineers, and technologists to be able to communicate measure-ments to each other easily and without confusion

impor-The SI system consists of seven base units (from which all other units are structed), supplementary units (used for measuring plane and solid angles), and derived units (which are formed by multiplication and division of the seven base units)

con-Each unit measures a specific physical quantity, has a standard symbol, and has a single spelling when written out in full (Exception: The United States has different spellings for deca, metre, and litre, writing them as deka, meter, and liter.)

Fig 1.9 summarizes some SI physical quantities and common variable symbols, their unit names and SI unit symbols, and any re-expression of a derived unit in terms

of more fundamental base units

Among the units for time, for which the standard unit is the second, other units like minute (min), hour (h), day (d), and year (y or yr) are also acceptable For angles, divi-sions such as the degree, minute of arc, and second of arc are also permitted

The kilogram is the SI unit for mass (not weight) It is different because it also tains an SI prefix kilo, which denotes a power of 103 Please note that weight and mass are different: Mass is the amount of material in an object (in kg), and weight is the gravitational force (in N) exerted on that mass Weight changes with the local strength

con-of the gravity field, whereas mass remains constant

Originally the metre was defined as one ten-millionth of the length along the globe from the North Pole to the equator Today it is defined as the distance travelled by light in

a vacuum in 1>299 792 458 s Similarly, the second was once defined as the fraction

1>86 400 of the mean solar day It is now defined as the time required for 9 192 631 770 cycles of the radiation corresponding to the transition between the two lowest energy states of the cesium-133 atom

■ Some calculator keys on different models

are labelled differently For example, on some

models, the EXE key is equivalent to the

ENTER key.

■ Calculator keystrokes will generally not be

shown, except as they appear in the display

screens They may vary from one model to

another.

Trang 30

When writing units, there are several conventions that one must follow:

Exception: Degrees Celsius

Isaac Newton, Pa for Blaise Pascal) Exception: The litre symbol is L, which is not named for a person It used to be l or l but it was easily confused with the digit 1 (one) so it was altered The l symbol still has some international accept-ance Both °C and L were added to the SI system due to their practical importance

r &OTVSFUIBUB# symbol appears between units that are multiplied (e.g., kg#m2>s2

not kgm2>s2) This will prevent confusion between units and SI prefixes, some of which use the same symbol (e.g., mm is millimetres, but m#m is metres squared)

Trang 31

are italicized (e.g., V is the quantity of electrical potential, and V is the unit volts).

r tional practices of different interpretations of commas (e.g., 10 585 is accepta-ble, while 10,585 means 10.585 in some countries)

4QBDFTNBZCFVTFEUPTFQBSBUFUIPVTBOETUPBWPJEDPOGVTJPOXJUINBOZJOUFSOB-SI PREFIXES

In science, it is common to deal with measurements that consist of very large numbers,

or very small numbers In order to avoid the problem of having to write many zeros in

a decimal (whether trailing or leading zeros), one can utilize some common unit fixes allowing for a quick way to write a specific multiple of 10 applied to the unit

pre-These prefixes have specific names and symbols, just like units, but are written

preced-ing the unit, as a normal prefix There can never be more than one prefix for a single unit Scientific and engineering notations, which are used to report very large or very small measurements using these prefixes, will be discussed in Section 1.5

(b) We also use the definitions of the SI prefixes to give the name and meaning of the

units corresponding to the following symbols:

Trang 32

This process is more fully discussed in Sections 1.7 to 1.12, but it is important to discuss the principle here, since measurements and units have a fundamental role in most subse-quent applied problems.

To convert a set of units, you multiply the measurement by a fraction equal to one, where the fraction represents the equivalency ratio between the two units You put the units you want to eliminate on the opposite side of the fraction of the converting ratio from where they are in the original measurement when you multiply By multiplying by a fraction equal to one, the measurement is not changing To convert multiple units at the same time, just use more than one conversion fraction multiplication This is illustrated in Example 3

been determined by some measurement Certain other numbers are exact numbers,

having been determined by a definition or counting process.

E X A M P L E 4 Approximate numbers and exact numbers

If a voltage on a voltmeter is read as 116 V, the 116 is approximate Another voltmeter

might show the voltage as 115.7 V However, the voltage cannot be determined exactly.

If a computer prints out the number of names on a list of 97, this 97 is exact We know it is not 96 or 98 Since 97 was found from precise counting, it is exact

Significant digits are digits in a measurement or result that you can confidently

esti-mate That is to say, those digits that are not swamped by the error or uncertainty in the

measurement are significant The accuracy of a measurement refers to the number of

significant digits it has

The measurements 5.00 m and 5.000 m may not seem to be very different, but to a

scientist, an engineer, or a technologist, they are not the same thing The first

measure-ment has been measured to the nearest centimetre and the second measuremeasure-ment to the

nearest millimetre The precision of a measurement is defined as the last decimal place

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measured (significant) digit in the measurement For instance, 5.00 m has precision 0.01 m = 1 cm, and 5.000 m has precision 0.001 m = 1 mm Therefore, the second

measurement is more precise (it has a smaller precision) The concept of precision is

important when finding the proper significant digits in a calculated result

To find the number of significant digits in a single measurement, you start counting

at the first nonzero digit, and finish counting once the precision of the measurement is reached Some rules to remember are:

r "MMOPO[FSPEJHJUTare significant r ;FSPTCFUXFFOOPO[FSPEJHJUTare significant r ;FSPTUPUIFMFGUPGUIFGJSTUOPO[FSPEJHJUBSFnot significant r 5SBJMJOH[FSPTBGUFSBEFDJNBMare significant

E X A M P L E 5 Accuracy and precision

(a) Suppose that an electric current is measured to be 0.31 A on one ammeter and

0.312 A on another ammeter The measurement 0.312 A is measured to the nearest thousandth ampere, so it is more precise than 0.31 A, which is measured to the nearest hundredth ampere 0.312 A is also more accurate, since it contains three significant digits, whereas 0.31 A contains only two

(b) If a concrete driveway is measured to be 135 m long and 0.1 m thick, the measurement

0.1 m (measured to the nearest tenth metre) is more precise than the measurement 135 m (measured to the nearest metre) On the other hand, 135 m is more accurate, since it con-tains three significant digits, whereas 0.1 m contains only one ■

E X A M P L E 6 Significant digits

All numbers in this example are assumed to be approximate

34.7 has three significant digits

0.039 has two significant digits The zeros properly locate the decimal point.706.1 has four significant digits The zero is not used for the location of the decimal point It shows the number of tens in 706.1

5.90 has three significant digits

1400 has two significant digits, unless information is known about the number that makes either or both zeros significant (A temperature shown as 1400°C has two sig-nificant digits If a price list gives all costs in dollars, a price shown as $1400 has four significant digits.) Without such information, we assume that the zeros are placehold-ers for proper location of the decimal point

Other approximate numbers with the number of significant digits are 0.0005 (one),

960 000 (two), 0.0709 (three), 1.070 (four), and 700.00 (five) ■

■ To show that zeros at the end of a whole

number are significant, a notation that can be

used is to place a bar over the last significant

zero Using this notation, 78 000 is shown to

have four significant digits.

Do not write trailing zeros if they are not significant The measurement 15 m is different from 15.0 m because the precision is different.

C O M M O N E R R O R

Trang 34

From Example 6, we see that all nonzero digits are significant Also, zeros not used

as placeholders (for location of the decimal point) are significant.The last significant digit of an approximate number is not exact It has usually been

determined by estimating or rounding off However, it is not off by more than one-half

of a unit in its place value

E X A M P L E 7 .FBOJOHPGUIFMBTUEJHJUPGBOBQQSPYJNBUFOVNCFS

When we write the voltage in Example 4 as 115.7 V, we are saying that the voltage is more than 115.65 V and less than 115.75 V Any such value, rounded off to tenths, would be expressed as 115.7 V

In changing the fraction 2

3 to the approximate decimal value 0.667, we are saying

The method of unbiased rounding (also known as round half to even) for rounding off any measurement to a specific precision, or a number to a specified number of sig-

nificant digits, consists of three simple rules Locate the last significant digit (the digit

to be rounded) Then:

round up (increase the rounded digit by one, discard the rest);

card the rest);

nearest even (make the rounded digit the nearest even number and discard

the rest)

This last rule ensures proper statistical treatment of all the measurements falling

in this category, as half will round up, and half will round down This technique will not statistically bias your measurements to be consistently larger upon rounding

We will use unbiased rounding throughout the text However, there are many ent rules that can be followed when rounding For example, in the common method of

differ-round half up, if the first discarded digit is 5, then the number is always differ-rounded up It

can be seen that the two methods are identical except for their treatment of those bers where the digit following the rounding digit is a five and has no nonzero digits after it

E X A M P L E 8 Rounding off

70 360 rounded off to three significant digits is 70 400 Here, 3 is the third significant digit, and the next digit is 6 Since 6 7 5, we add 1 to 3 and the result, 4, becomes the third significant digit of the approximation The 6 is then replaced with a zero in order

to keep the decimal point in the proper position

70 430 rounded off to three significant digits, or to the nearest hundred, is 70 400 Here the 3 is replaced with a zero

187.35 rounded off to four significant digits, or to tenths, is 187.4, because 4 is the nearest even to 3.5

187.349 rounded off to four significant digits is 187.3 We do not round up the 4 and

then round up the 3

35.003 rounded off to four significant digits is 35.00 We do not discard the

zeros since they are significant and are not used only to properly place the decimal point

187.45 rounded off to four significant digits is 187.4 since 4 is the nearest even

■ On graphing calculators, it is possible to set

the number of decimal places (to the right of

the decimal point) to which results will be

rounded off Note that calculators round

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When performing operations on approximate numbers or measurements, we must not express the result to an accuracy or precision that is not valid Measurement uncertainty restricts how many significant digits can exist in a calculated result.

off to tenths, the precision of the least precise length, and it is written as 17.2 m ■

E X A M P L E 1 0 Application of accuracy

We find the area of the rectangular piece of land in Fig 1.11 by multiplying the length, 207.54 m, by the width, 81.4 m Using a calculator, we find that 1207.542181.42 = 16 893.756 This apparently means the area is 16 893.756 m2

However, the area should not be expressed with this accuracy Since the length and width are both approximate, we have

1207.535 m2 181.35 m2 = 16 882.972 25 m2 least possible area1207.545 m2 181.45 m2 = 16 904.540 25 m2 greatest possible areaThese values agree when rounded off to three significant digits (16 900 m2) but do not agree when rounded off to a greater accuracy Thus, we conclude that the result is accu-

rate only to three significant digits, the accuracy of the least accurate measurement, and

Following are the rules used in expressing the result when we perform basic tions on approximate numbers They are based on reasoning similar to that shown in Examples 9 and 10

opera- smallest values largest values

81.4 m

0.05 m 207.54 m

Operations with Approximate Numbers

1 When approximate numbers are added or subtracted, the result is expressed

with the precision of the least precise number.

2 When approximate numbers are multiplied or divided, the result is expressed

with the accuracy of the least accurate number.

3 When the root of an approximate number is found, the result is expressed

with the accuracy of the number.

4 When approximate numbers and exact numbers are involved, the accuracy of

the result is limited only by the approximate numbers.

Always express the result of a calculation with the proper accuracy or precision When using a calculator, if additional digits are displayed, round off the final result

(do not round off in any of the intermediate steps).

L E A R N I N G T I P

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E X A M P L E 1 1 Adding approximate numbers

Find the sum of the approximate numbers 73.2, 8.0627, and 93.57

Showing the addition in the standard way and using a calculator, we have

73.2 least precise number (expressed to tenths) 8.0627

93.57 174.8327 final display must be rounded to tenths

E X A M P L E 1 2 Combined operations

In finding the product of the approximate numbers 2.4832 and 30.5 on a calculator, the final display shows 75.7376 However, since 30.5 has only three significant digits, the product is 75.7

In Example 1, we calculated that 38.3 - 12.91 -3.582 = 84.482 We know that 38.3 - 12.91 -3.582 = 38.3 + 46.182 = 84.482 If these numbers are approximate,

we must round off the result to tenths, which means the sum is 84.5 We see that when there is a combination of operations, we must examine the individual steps of the calcula-tion and determine how many significant digits can carry through to the final result ■

E X A M P L E 1 3 Operations with exact numbers and approximate numbers

Using the exact number 600 and the approximate number 2.7, we express the result to tenths if the numbers are added or subtracted If they are multiplied or divided, we express the result to two significant digits Since 600 is exact, the accuracy of the result depends only on the approximate number 2.7

600 + 2.7 = 602.7 600 - 2.7 = 597.3

You should make a rough estimate of the result when using a calculator An

estima-tion may prevent accepting an incorrect result after using an incorrect calculator sequence, particularly if the calculator result is far from the estimated value

E X A M P L E 1 4 Estimating results

In Example 1, we found that

38.3 - 12.91 -3.582 = 84.482 using exact numbersWhen using the calculator, if we forgot to make 3.58 negative, the display would be -7.882, or if we incorrectly entered 38.3 as 83.3, the display would be 129.482.However, if we estimate the result as

40 - 101 -42 = 80

we know that a result of -7.882 or 129.482 cannot be correct

When estimating, we can often use one-significant-digit approximations If the culator result is far from the estimate, we should do the calculation again ■

cal-■ When rounding off a number, it may seem

difficult to discard the extra digits However, if

you keep those digits, you show a number with

too great an accuracy, and it is incorrect to

do so.

Practice Exercises

Evaluate using a calculator.

3 40.5 + -60.0413275 (Numbers are approximate.)

A note regarding the equal sign (=)

is in order We will use it for its

defined meaning of “equals exactly”

and when the result is an

approxi-mate number that has been properly

rounded off Although 127.8 ≈ 5.27,

where ≈ means “equals

approxi-mately,” we write 127.8 = 5.27, since

5.27 has been properly rounded off.

EXERCISES 1.3

this section, and then solve the given problems.

1 In Example 6, change 0.039 (the second number discussed) to

0.390 Is there any change in the conclusion?

2 In the next-to-last paragraph of Example 8, change 35.003 to

35.303 and then find the result.

3 In the first paragraph of Example 12, change 2.4832 to 2.483 and

then find the result.

4 In Example 14, change 12.9 to 21.9 and then find the estimated

value.

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In Exercises 5–8, give the symbol and the meaning for the given unit.

5 megahertz 6 kilowatt 7 millimetre 8 picosecond

In Exercises 9–12, give the name and the meaning for the units whose

symbols are given.

21 45.0 m>s to centimetres per second.

22 1.32 km>h to metres per second.

23 9.80 m>s 2 to centimetres per minute squared.

24 5.10 g>cm 3 to kilograms per cubic metre.

25 25 h to milliseconds.

26 5.25 mV to watts per ampere.

27 15.0 mF to millicoulombs per volt.

28 Determine how many metres light travels in one year.

29 Determine the speed (in km>h) of the earth moving around the

sun Assume it is a circular path of radius 150 000 000 km.

30 At sea level, atmospheric pressure is about 101 300 Pa How many

33 The velocity of some seismic waves is 6800 m>s What is this

velocity in kilometres per hour?

34 The memory of a 1985 computer was 64 kB (B is the symbol for

byte), and the memory of a 2012 computer is 1.50 TB How many

times greater is the memory of the 2012 computer?

35 The recorded surface area of a DVD is 112 cm2 What is this area

in square metres?

36 A solar panel can generate 0.024 MW#h each day Convert this

to joules.

37 The density of water is 1000 kg>m 3 Change this to grams per litre.

38 Water flows from a kitchen faucet at the rate of 8500 mL>min

What is this rate in cubic metres per second?

39 The speed of sound is about 332 m>s Change this speed to

kilo-metres per hour.

40 Fifteen grams of a medication are to be dissolved in 0.060 L of

water Express this concentration in milligrams per decilitre.

41 The earth’s surface receives energy from the sun at the rate of

1.35 kW >m 2 Reduce this to joules per second per square

centimetre.

42 The moon travels about 2 400 000 km in about 28 d in one

rota-tion about the earth Express its velocity in metres per second.

43 A typical electric current density in a wire is 1.2 * 10 6 A >m 2 Express this in milliamperes per square centimetre.

44 A certain car travels 24 km on 2.0 L of gas Express the fuel

con-sumption in litres per 100 kilometres.

In Exercises 45–48, determine whether the given numbers are approximate or exact.

45 A car with 8 cylinders travels at 55 km>h.

46 A computer chip 0.002 mm thick is priced at $7.50.

47 In 24 h there are 1440 min.

48 A calculator has 50 keys, and its battery lasted for 50 h of use.

In Exercises 49–54, determine the number of significant digits in each

of the given approximate numbers.

In Exercises 77–80, perform the indicated operations The first number is approximate, and the second number is exact.

In Exercises 81–84, answer the given questions.

81 The manual for a heart monitor lists the frequency of the

ultra-sound wave as 2.75 MHz What are the least possible and the greatest possible frequencies?

82 A car manufacturer states that the engine displacement for a

cer-tain model is 2400 cm3 What should be the least possible and greatest possible displacements?

83 A flash of lightning struck a tower 5.23 km from a person The

thun-der was heard 15 s later The person calculated the speed of sound and reported it as 348.7 m >s What is wrong with this conclusion?

84 A technician records 4.4 s as the time for a robot arm to swing

from the extreme left to the extreme right, 2.72 s as the time for the return swing, and 1.68 s as the difference in these times What

is wrong with this conclusion?

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1.4 Exponents 21

In Exercises 85–100, perform the calculations on a calculator.

85 Evaluate: (a) 2.2 + 3.8 * 4.5 (b) 12.2 + 3.82 * 4.5

86 Evaluate: (a) 6.03 , 2.25 + 1.77 (b) 6.03 , 12.25 + 1.772

87 Evaluate: (a) 2 + 0 (b) 2 - 0 (c) 0 - 2 (d) 2 * 0 (e) 2 , 0

Compare with operations with zero in Section 1.2.

88 Evaluate: (a) 2 , 0.0001 and 2 , 0 (b) 0.0001 , 0.0001 and

0 , 0 (c) Explain why the displays differ.

89 Enter a positive integer x (five or six digits is suggested) and then

rearrange the same digits to form another integer y Evaluate

1x - y2 , 9 What type of number is the result?

90 Enter the digits in the order 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, using

between them any of the operations 1 +, - , *, , 2 that will

lead to a result of 100.

91 Show that p is not equal exactly to (a) 3.1416, or (b) 22>7.

92 At some point in the decimal equivalent of a rational number,

some sequence of digits will start repeating endlessly An

irra-tional number never has an endlessly repeating sequence of

dig-its Find the decimal equivalents of (a) 8 >33 and (b) p Note the

repetition for 8 >33 and that no such repetition occurs for p.

93 Following Exercise 92, show that the decimal equivalents of the

following fractions indicate they are rational: (a) 1 >3 (b) 5>11 (c)

2 >5 What is the repeating part of the decimal in (c)?

94 Following Exercise 92, show that the decimal equivalent of the

fraction 124 >990 indicates that it is rational Why is the last digit

different?

95 In 3 successive days, a home solar system produced 32.4 MJ,

26.704 MJ, and 36.23 MJ of energy What was the total energy produced in these 3 days?

96 Two jets flew at 938 km>h and 1450 km>h, respectively How much faster was the second jet?

97 If 1 K of computer memory has 1024 bytes, how many bytes are

there in 256 K of memory? (All numbers are exact.)

98 Find the voltage in a certain electric circuit by multiplying the

sum of the resistances 15.2 Ω, 5.64 Ω, and 101.23 Ω by the rent 3.55 A.

99 The percent of alcohol in a certain car engine coolant is found by

performing the calculation 100140.63 + 52.962

105.30 + 52.96 Find this percent of alcohol The number 100 is exact.

100 The tension (in N) in a pulley cable lifting a certain crate was

found by calculating the value of 50.4519.802

1 + 100.9 , 23, where the

1 is exact Calculate the tension.

Answers to Practice Exercises

1 2020 2 0.300 3 -14.0

In mathematics and its applications, we often have a number multiplied by itself

sev-eral times To show this type of product, we use the notation a n , where a is the number and n is the number of times it appears In the expression a n , the number a is called the

base, and n is called the exponent; in words, a n is read as “the nth power of a.”

E X A M P L E 1 .FBOJOHPGFYQPOFOUT

(a) 4 * 4 * 4 * 4 * 4 = 45 the fifth power of 4

(b) 1 -22 1 -22 1 -22 1 -22 = 1 -224 the fourth power of -2

(d) a15 b a15 b a15 b = a15 b3 the third power of 1 , called “ 1 cubed” ■

We now state the basic operations with exponents using positive integers as

expo-nents Therefore, with m and n as positive integers, we have the following operations:

■ Two forms are shown for Eq (1.4) in order

that the resulting exponent is a positive integer

We consider negative and zero exponents after

the next three examples.

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E X A M P L E 2 Illustrating Eqs (1.3) and (1.4)

Using Eq (1.3): Using the meaning of exponents:

(3 factors of a)(5 factors of a)

E X A M P L E 3 Illustrating Eqs (1.5) and (1.6)

multiply exponents

1a523 = a5132 = a15 1a523 = 1a52 1a52 1a52 = a5 +5+5 = a15

Using first form Eq (1.6): Using the meaning of exponents:

Using second form Eq (1.6): Using the meaning of exponents:

■ In a 3 , which equals a * a * a, each a is

called a factor A more general definition of

factor is given in Section 1.7.

■ Here we are using the fact that a (not zero)

divided by itself equals 1, or a>a = 1.

When an expression involves a

prod-uct or a quotient of different bases,

only exponents of the same base

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1132 142 142EI =

PL3

48EI

L is the length of the beam, and P is the force applied to it E and I are constants related

to the beam In simplifying this expression, we combined exponents of L and divided

ZERO AND NEGATIVE EXPONENTS

If we let n = m in Eq (1.4), we would have a m >a m = a m -m = a0 Also, a m >a m = 1, since any nonzero quantity divided by itself equals 1 Therefore, for Eq (1.4) to hold,

when m = n, we have

Eq (1.7) states that any nonzero expression raised to the zero power is 1 Zero

expo-nents can be used with any of the operations for expoexpo-nents

E X A M P L E 6 Zero as an exponent

(a) 50 = 1 (b) 1 -320 = 1 (c) - 1 -320 = -1 (d) 12x20 = 1

(e) 1ax + b20 = 1 (f) 1a2b0c22 = a4c2 (g) 2t0 = 2112 = 2

We note in illustration (g) that only t is raised to the zero power If the quantity 2t were

If we apply the first form of Eq (1.4) to the case where n 7 m, the resulting

expo-nent is negative This leads to the definition of a negative expoexpo-nent

E X A M P L E 7 Basis for negative exponents

Applying both forms of Eq (1.4) to a2>a7, we have

Following the reasoning of Example 7, if we define

Although positive exponents are

generally preferred in a final result,

there are some cases in which zero or

negative exponents are to be used

Also, negative exponents are very

useful in some operations that we

will use later.

then all of the laws of exponents will hold for negative integers

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