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Technical ShopMathematics

Third Edition

THOMAS ACHATZ, P.E.

WITH

JOHN G. ANDERSON

Contributing Author

KATHLEEN MCKENZIE

Contributing Editor

2005

INDUSTRIAL PRESS INC.

NEW YORK

Library of Congress Cataloging-in-Publication Data

Achatz, Thomas.Technical shop mathematics / Thomas Achatz and John G. Anderson – 3rd ed. p. cm.Previous eds. by John G. Anderson

ISBN 0-8311-3086-5TJ1165.A56 2004510’.246--dc22

2004056762

COPYRIGHT © 1974, © 1988, © 2006 by Industrial Press Inc., New York, NY.

All rights reserved. This book or parts thereof may not be reproduced, stored in a retrievalsystem, or transmitted in any form without permission of the publishers.

Printed and bound in the United States of America by Edwards Bros. Company, Philadelphia, Pa.

10 9 8 7 6 5 4 3 2 1

TECHNICAL SHOP MATHEMATICSThird Edition

INDUSTRIAL PRESS, INC.200 MADISON AVENUE

NEW YORK, NEW YORK 10016-4078

FIRST PRINTING, NOVEMBER 2005

To John, Ruth, Diane, and Elizabeth

PREFACE

Technical Shop Mathematics, 3rd edition, is a major revision with many new topics added, old topics updated,illustrations improved, and a larger, cleaner format. The use of two colors offers easier reading and better de-lineation of key points, while margin notes include historical information, caveats, and other useful references.Building on the strengths of the original editions, the 3rd edition delivers an expanded number of real-worldexercises in a consistent manner using straightforward language throughout.

This versatile edition may be used as a classroom textbook, a self-study refresher or a convenient on-the-jobreference. Community colleges, high school vocational programs, and trade schools will appreciate the system-atic organization of topics that are well suited for a thorough two-semester course or an accelerated one-semes-ter course. For those who are pursuing higher education, this edition serves as an excellent review of funda-mental mathematical skills or as a primer for advanced algebra, trigonometry, or calculus. Industry profession-als such as machinists, HVAC technicians, mechanics, electricians, surveyors, and others interested in thepractical application of mathematics will find that the individual topics are comprehensive and clearly identi-fied, thereby allowing for easy navigation and quick reference.

The fundamental areas of arithmetic, algebra, geometry, and trigonometry are further divided into chaptersconcentrating on particular topics. This format allows for either a cumulative, sequential approach to learningnew subjects, or for use as a reference on specific points of interest. The review of arithmetic includes signednumber operations, place values, Roman numerals, fractions, percents, rounding, and measurement systems.The algebra topics build a strong foundation by extending arithmetic to include exponents, logarithms, ratioand proportion, Cartesian coordinates, graphing linear functions, solving equations and word problems, manip-ulating literal variables, working with radicals, factoring, and finding quadratic roots. Covered within the ge-ometry topics are practical applications of Euclid’s axioms, postulates, and theorems while proofs are present-ed as a motivation for solving problems from a series of reasoned steps. The final chapters provide a structuredapproach to right angle and oblique trigonometry. Graphing trigonometric functions is emphasized to buildmathematical intuition.

The math skills required to solve technical problems are the foundation of critical thinking. Conceptual under-standing, practical application, and the ability to adapt and extend underlying principles are far more valuablein the work environment than mere memorization. My hope is that everyone who uses this book, regardless ofprior mathematical skills or experiences, gains an increased ability to solve practical mathematics problems,develops an appreciation for the study of technical mathematics, and finds improved career prospects.

ACKNOWLEDGMENTS

This edition of Technical Shop Mathematics has been a long time in the making. Many people, far too numer-ous to mention, have contributed one way or another, directly or indirectly, to this project. They have my grat-itude even if their contributions are not explicitly recognized here. Among them are professors from my yearsat the University of Michigan and Rensselaer Polytechnic Institute, as well as former mathematics and engi-neering students who have inspired my goals of clear presentation and meaningful application of concepts.

One of the earliest participants in the effort to produce this book was Kathleen McKenzie from Radical X Edit-ing Services who provided constructive criticism, content suggestions, and final proofreading. Countless ver-sions of the manuscript in various formats were rendered by Elena Godina who tirelessly revised chapter lay-outs for optimal visual appearance. Robert Weinstein provided meticulous and expert copy editing. Many ofthe illustrations from the second edition were recast electronically by Michigan Technological University engi-neering student James Kramer. My colleagues, Dennis Bila and James Egan from Washtenaw CommunityCollege, and Debi Cohoon from General Motors University, provided encouragement to complete the projectand offered many opportunities to develop my presentation style and teaching skills. The attractive cover wasdesigned and produced by William Newhouse, Split3Studio.com. Lisa Patishnock, Mary Walker, and MaryBest furnished technical guidance, typing assistance, and cross-referencing services. I am particularly gratefulfor the timely support from Charlie Achatz, who carefully read the manuscript multiple times, corrected errors,and re-worked all of the exercises. Lastly, special recognition is due to John G. Anderson whose legacy contin-ues through many of the exercises preserved from earlier editions.

The staff at Industrial Press, especially John Carleo, have provided support and encouraging feedback through-out this project. Christopher McCauley, Riccardo Heald, and Janet Romano made the painstaking details ofgetting from manuscript to finished product achievable and enjoyable.

Notwithstanding the able and dedicated efforts of so many, errors and omissions may nonetheless be present.Please provide suggestions for improvement by visiting the home page for this book at www.industrial-press.com and clicking on the link to “Email the Author.” Your feedback would be greatly appreciated.

Thomas Achatz, PE

TABLE OF CONTENTS

1 THE LANGUAGE OF MATHEMATICS

Symbols — The Alphabet of Mathematics 1

Properties of Real Numbers 5

Real Number Set and Subsets 6

The Multiplication Table 9

Operations in Arithmetic 10

2 SIGNED NUMBER OPERATIONS

Addition and Subtraction on the Number Line 17

Absolute Value 20

Combining More than Two Numbers through Addition and Subtraction 22

Multiplication and Division 24

Combining All Signed Number Operations 26

3 COMMON FRACTIONS

Common Fractions as Division 27

Converting Improper Fractions and Mixed Numbers 29

Raising a Common Fraction to Higher Terms 31

Reducing a Common Fraction to Lowest Terms 32

Addition and Subtraction of Common Fractions 37

Addition and Subtraction of Mixed Numbers 43

Multiplication and Division of Common Fractions 48

Multiplication and Division of Mixed Numbers 52

Complex Fractions 55

TABLE OF CONTENTSxii

4 DECIMAL FRACTIONS

Meaning of a Decimal Fraction 57

Converting Common Fractions to Decimal Fractions 59

Converting Decimal Fractions to Common Fractions 61

Addition and Subtraction of Decimal Fractions 63

Multiplication of Decimal Fractions 65

Division of Decimal Fractions 67

Place Value and Rounding 69

Measurement Arithmetic 73

Decimal Tolerances 76

5 OPERATIONS WITH PERCENTS

Working with Percents 83

Solving Percent Problems 90

Simple Interest 95

List Price and Discounts 96

6 EXPONENTS: POWERS AND ROOTS

Powers of Positive and Negative Bases 99

Exponent Rules Part 1 101

Exponent Rules Part 2 104

Scientific Notation 108

Logarithms 111

7 MEASUREMENT

Systems of Measurement 115

Measures of Length, Area, and Volume 118

Angle Measure 126

Weight and Mass Measure 131

Measures of Temperature and Heat 132

Measures of Pressure 135

Strain 140

8 ALGEBRAIC EXPRESSIONS

Working with Algebraic Expressions 141

Operations on Expressions — Exponents 146

Operations on Expressions — Radicals 149

Operations on Expressions — Rationalizing the Denominator 151

Operations on Expressions — Combining Like Terms 152

9 SOLVING EQUATIONS AND INEQUALITIES IN X

Solving Linear Equations in One Variable 155

Solving Inequalities in x 163

10 GRAPHING LINEAR EQUATIONS

The Cartesian Plane 169

Graphing Points of a Line 172

The Slope of a Line 175

Applying Linear Equation Forms to Graphs 180

xiiiTABLE OF CONTENTS

11 TRANSFORMING AND SOLVING SHOP FORMULAS

Literal Equations 187

Applications of Literal Equations in Shop Mathematics 190

12 RATIO AND PROPORTION

Statements of Comparison 213

Mixture Proportions 223

Tapers and Other Tooling CalculationsRequiring Proportions 225

Variation 233

13 OPERATIONS ON POLYNOMIALS

Expanding Algebraic Expressions 243

Factoring Polynomials 248

Binomial Factors of a Trinomial 251

Special Products 256

Algebraic Fractions 259

14 SOLVING QUADRATIC EQUATIONS

Solving Quadratic Equations of Form x2 = Constant 271

The Quadratic Formula 276

15 LINES, ANGLES, POLYGONS, AND SOLIDS

Points, Lines, and Planes 279

Polygons 286

Polyhedrons and Other Solid Figures 294

16 PERIMETER, AREA, AND VOLUME

Perimeter 297

Area of a Polygon 301

Surface Area and Volume of a Solid 312

17 AXIOMS, POSTULATES, AND THEOREMS

Axioms and Postulates 321

Theorems About Lines and Angles in a Plane 325

18 TRIANGLES

Special Lines in Triangles 341

Similar Triangles 344

Pythagorean Theorem 353

Congruent Triangles 360

The Projection Formula 365

Hero’s Formula 369

19 THE CIRCLE

Definitions 375

Theorems Involving Circles 381

20 TRIGONOMETRY FUNDAMENTALS

Some Key Definitions Used in Trigonometry 401

Solving Sides of Triangles Using Trigonometric Functions 410

Special Triangles and the Unit Circle 427

Graphing the Trigonometric Functions 433

21 OBLIQUE ANGLE TRIGONOMETRY

Solving Oblique Triangles Using Right Triangles 439

Special Laws of Trigonometry 445

TABLE OF CONTENTSxiv

22 SHOP TRIGONOMETRY

Sine Bars and Sine Plates 463

Hole Circle Spacing 468

Coordinate Distances 470

Solving Practical Shop Problems 476

Trigonometric Shop Formulas 485

A APPENDIX

Greek Letters and Standard Abbreviations 489

Factors and Prefixes for Decimal Multiples of SI Units 489

Linear Measure Conversion Factors 490

Square Measure Conversion Factors 491

Cubic Measure Conversion Factors 492

Circular and Angular Measure Conversion Factors 493

Mass and Weight Conversion Factors 494

Pressure and Stress Conversion Factors 495

Energy Conversion Factors 495

Power Conversion Factors 496

Heat Conversion Factors 496

Temperature Conversion Formulas 496

Gage Block Sets — Inch Sizes 497

Gage Block Sets — Metric Sizes 498

B ANSWERS TO SELECTED EXERCISES

Chapter 1 Exercises 499






















INDEX 557

1
THE LANGUAGE OFMATHEMATICS
Mathematics is a universal language that has evolved over thousands of years. Itdraws on contributions from every civilization and corner of the world—fromthe ancient worlds of the Middle East, Greece, and Rome, to India, China, Rus-sia, Africa, and pre-Columbian Mayan culture.

Mathematics is used all over the world to solve problems in economics, engi-neering, manufacturing, construction, electronics, social science and myriadother disciplines. Through mathematics, people can communicate abstract ideaswith each other even though they may speak different languages and may comefrom different cultures.

The language of mathematics consists of many dialects, or subdisciplines.These include arithmetic, algebra, geometry, trigonometry, and statistics, toname a few. This book concentrates on the rudimentary skills needed to studymathematics and solve practical problems encountered in technical fields.

As with any language, mathematics has established rules and terminology.These are written with symbols—a sort of mathematical alphabet that is used toconstruct complicated expressions and convey abstract concepts in a compact,unambiguous form. Unlike the English alphabet, which has twenty-six symbols,the mathematical language has numerous symbols and is not recited in any par-ticular order.

Many common mathematical symbols are listed in Table 1.1. Become familiarwith these symbols and refer to them throughout this book.

Greek letters are often used to represent angles.

Greek letters, such as π (pi) in the familiar circle formulas, are sometimes usedto represent operations, constants, or variables. Part of the Greek alphabet that iscommonly used in mathematics is given in Table 1.1.

1.1 Symbols — The Alphabet of Mathematics

THE LANGUAGE OF MATHEMATICS2 CHAPTER 1

TABLE 1.1: Common Mathematical Symbols

Symbol Symbol+ Plus (sign of addition), or positive ∪ Union of sets– Minus (sign of subtraction), or negative ⊂, ⊆ Subset of

Plus or minus (minus or plus) ∅ Empty set

×, ⋅ Multiplication ∩ Intersection of sets

÷, / Division α Alpha: Is to (ratio or proportion) λ Lamda (wavelength)= Is equal to µ Mu (coefficient of friction)≠ Is not equal to π Pi (3.1416…)≡ Is identical to Σ Sigma (sign of summation)

Approximately equals β Beta

≅ Is congruent to Triangle∼ Is similar to sin Sine> Is greater than cos Cosine< Is less than tan Tangent≥ Is greater than or equal to cot Cotangent≤ Is less than or equal to sec Secant

Varies directly as csc Cosecant∞ Infinity sin–1 Inverse sine

∴ Therefore an a sub n

Square root a′ a prime

Cube root a′′ a double prime

nth root a1 a sub one

i (or j) Imaginary number ∠ Angle

a2 a squared (second power of a) || Is parallel to

a3 a cubed (third power of a) ⊥ Is perpendicular to

an nth power of a ° Degree (circular or temperature)

1—n Reciprocal value of n ′ Minutes or feet

|n| Absolute value of n ′′ Seconds or inches ∫ Integral (in calculus) AB Segment AB

( ), { }, [ ] Parentheses / Braces / Brackets Ray AB

! Factorial (5! = 5 × 4 × 3 × 2 × 1) Line AB

log Logarithm (base 10)ln Natural logarithm (base e)

± ( )∓

≈�

∝

3

n

1–( )

AB��

AB��

SYMBOLS — THE ALPHABET OF MATHEMATICS 3

Roman Numerals

Some historians say that the Roman civilization fell because it lacked mathemati-cal science.

There are many symbols used to represent actions and operations; even numer-als themselves are symbols. Roman numerals, for example, are seen on the cor-nerstones of some public buildings, on clocks and watches, in outlines, in tablesof contents, and many other places.

The Roman numeral system is not suited to the work of complex mathematics.Nonetheless, knowledge of the basic Roman numeral system is useful. The ba-sic elements of the Roman numeral system are provided in the sidebar. Noticethat these elements are not digits in the sense of our familiar Arabic number sys-tem, and they do not have place value.

In the Roman numeral system, individual elements are combined to build num-bers in such a way that when added together, they result in a value. One general-ly writes the element of highest value first and decreases the value of elementsfrom left to right. For example, 29 is XXVIIII. Of course this can result in verylong strings of numbers. To get around this problem and write numbers morecompactly, a subtractive rule was devised. The subtractive rule states that whenan element of smaller value appears before one of larger value, the individualvalues are subtracted. For example, 9 can be written as VIIII using the additionrule or as IX using the subtractive rule. Both rules are used in some cases, as inMDCCIX whose Arabic equivalent is 1709.

Other examples of Roman numerals in comparison to their Arabic equivalentsare provided in Table 1.2.

TABLE 1.2: Comparison of Arabic and Roman Numerals

Arabic Numerals Roman Numerals7 VII14 XIV19 XIX38 XXXVIII44 XLIV80 LXXX99 XCIX

150 CL627 DCXXVII1234 MCCXXXIV2000 MM

Common Roman Numerals I = 1II = 2III =3IV = 4V =5VI = 6VII = 7VIII = 8IX = 9X = 10L = 50C =100D = 500M =1000

Years are sometimes written in Roman numerals. For exam-ple, the year 2000 is written as MM.


Arabic Numerals

Our number system came to us many centuries ago. We call its symbols Arabicnumerals, but they really first came from ancient India where the Hindu peopleoriginated them. Arab merchants and traders of the Middle or Dark Ages adopt-ed the Hindu number system to help them in commerce. While people of West-ern Europe were still struggling with the Latin language and the Roman numer-als in their schools and universities, science was waiting for a breakthrough incommunications, particularly in mathematics; Arabic numerals saved the day.

dig·it (dîj′ît)A human finger or toe.

Arabic numerals have a great advantage over Roman numerals because they arebuilt on the base 10 number system. In this system, the magnitude of a numberis based on the place values of its digits, so named because fingers are so oftenused for counting.

The Arabic system has ten distinct digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. Alone,the value of each digit is a quantity that can be counted. For any number in theArabic system, a digit has place value equal to a power of 10 as determined bythe digit’s location in the number.

Consider this simple illustration of how powers of 10 are generated:

The small number above and to the right of each 10 is called an exponent. It in-dicates the number of factors of 10 are multiplied to produce a correspondingpower of 10. Table 1.3 shows these powers of 10 in a grid over the number52,307.

The 5 has a value of 5 × 10,000, or 50,000. The 2 has a value of 2 × 1000 or2000. The 3 is really 3 × 100 or 300. The 0 is 0 × 10, and the 7 is 7 × 1. Writingthese out in this way is called expanded notation of a number.

To read the number 4271 we say “four thousand two hundred seventy-one.” Wewould not say “four two seven one” as a rule unless we were reading certainkinds of numbers, such as a telephone number.

We read large numbers by reading one group at a time: thousands, millions, bil-lions, and so on. Hence, for the number 7,952,024 we say “seven million nine-

10 0 = 1 ones or units

10 1 = 10 tens

10 2 = 10 × 10 = 100 hundreds

10 3 = 10 × 10 × 10 = 1000 thousands

10 4 = 10 × 10 × 10 × 10 = 10,000 ten thousands

10 5 = 10 × 10 × 10 × 10 × 10 = 100,000 hundred thousands

10 6 = 10 × 10 × 10 × 10 × 10 × 10 = 1,000,000 millions

TABLE 1.3: Powers of 10

Ten thousands Thousands Hundreds Tens Ones or Units10 4 = 10,000 103 = 1000 102 = 100 101 = 10 100 = 1

5 2 3 0 7

PROPERTIES OF REAL NUMBERS 5

hundred fifty-two thousand twenty-four.” Notice that for the thousands group,we say “nine hundred fifty-two,” then the word “thousand.” We do not say“seven million nine hundred fifty-two thousand and twenty-four.” The word“and” is reserved for the decimal point. We review how to read decimal num-bers in Chapter 4.

E X E R C I S E S

1.1 Referring to Table 1.1 write the indicated symbol.

1.2 Change the indicated Arabic numeral to a Roman numeral.

1.3 Change the indicated Roman numeral to an Arabic numeral.

1.4 Write and say in words the indicated number.

Having established a number system, we note that the structure of mathematicsis built upon properties, definitions, and operations of real numbers. We beginby stating several properties, or axioms, of the set of real numbers as given inTable 1.4.

For the stated properties, the letters a, b, and c represent real numbers.

a) Is greater than b) Is less than or equal toc) Square root d) Is parallel toe) Alpha f) Betag) Is perpendicular to h) Angle

a) 3 b) 18c) 160 d) 133e) 524 f) 1001g) 2005 h) 4262

a) IV b) XXIVc) CCLI d) CDXIVe) CMVI f) CCCXXXIIIg) DCCVI h) MMXVII

a) 67,000 b) 3,880,131c) 205,009 d) 27,393e) 7,123,226 f) 18,323,516g) 104,362,456 h) 932,418,207

1.2 Properties of Real Numbers

An axiom is a statement whose truth is accepted with-out proof.


E X E R C I S E S

1.5 Name the property illustrated in the example.

Real numbers form a set. Basic definitions and properties of sets are necessaryfor a clear discussion of the real numbers and the various subsets of real num-bers.

Definitions Relating to Sets

TABLE 1.4: Properties of Real Numbers

a = a Reflexive property

a = b ⇒ b = a Symmetric property

a = b, b = c ⇒ a = c Transitive property

a + b = b + a Commutative property of addition

ab = ba Commutative property of multiplication

(a + b) + c = a + (b + c) Associative property of addition

(ab)c = a(bc) Associative property of multiplication

a(b + c) = ab + ac

(a + b)c = ac + bcDistributive property of multiplication over addition

a + 0 = 0 + a = a Additive identity

a + (–a) = 0 Additive inverse

a ⋅ 1 = 1 ⋅ a = a Multiplicative identity

Multiplicative inverse

a ⋅ 0 = 0 ⋅ a = 0 Zero property of multiplication

a) x + 1 = x + 1 b) x = 7 ⇒ 7 = xc) a = 2, 2 = x ⇒ a = x d) 1448 × 1 = 1448e) 14 = 14 f) 52 + 0 = 52g) 4 + 11 = 11 + 4 h) (1)(77) = 77

i) (6)(7 ⋅ 9) = (6 ⋅ 7)(9) j) x(2 + 4) = x × 2 + x × 4 = 2x + 4x

k) (15)(27) = (27)(15) l) (3 + 12)5 = 3 × 5 + 12 × 5

The symbol ⇒ means “implies.”

a1a---⋅ 1=

1.3 Real Number Set and Subsets

SetA collection of objects, called elements, often related in some obvious way. Usually symbolized by an uppercase letter, for example, set S. The elements are often represented by lowercase letters, such as a, b.

REAL NUMBER SET AND SUBSETS 7

Real Number Subsets

The set of real numbers is the universal set in the discussion that follows. It con-tains various subsets, all defined below. These include integers, which is the setwe will use to illustrate the rules of signed number operations.

A repeating pattern is indi-cated by three dots, called an ellipsis, or by a bar over the repeating digits.

The set of real numbers, represented by the symbol �, is depicted by the realnumber line. This unbroken line symbolizes the union of the real number sub-sets, the rational and irrational numbers. These two subsets of real numbers

A set within a set. Every set is a subset of itself and the set containing no ele-ments is also a subset of any set. Symbolized by A ⊂ S or A ⊆ S and read, “A is a subset of S.” This means A is contained in S. Another way of saying this: “All elements in A are also in S.”

Subset of a set

The set of real numberscontains various subsets.

A set whose elements are all the members of both sets A and B. Symbolized by A ∪ B and read “A union B.” Sets are joined in a set union, but no elements are listed more than once.

Union of sets A and B

A set whose elements are in both A and B. Symbolized by A ∩ B. The sets overlap, forming a new set containing those elements the intersected sets have in common.

Intersection of setsA and B

A relative term meaning the set of all elements from which subsets under con-sideration are drawn. The set of real numbers is, generally, the universal set ina discussion of real numbers.

Universal set

An integer greater than 1 is called a prime number if its only positive divisors are 1 and itself.

The set having no elements. Certain set intersections produce empty sets (i.e., when the sets intersected have no elements in common, an empty set results). The empty set is symbolized as ∅ or { }.

Empty or null set

EXAMPLE 1.1: Union and Intersection of Sets

If P = {prime numbers} and O = {odd numbers} then P ∪ O = {odd numbersand 2}. This is the case because the set of prime numbers is included in the setof odd numbers, with the exception of the number 2, an element not found inthe odd numbers.

If A = {Democrats in Baltimore} and B = {Republicans in Baltimore}then A ∩ B = ∅.

{ } { }{ } { }

If 1,2,3,4,5 and 2,3,5,7,11

then 1,2,3,4,5,7,11 and 2,3,5 .

A B

A B A B

= =

∪ = ∩ =

{ } { }{ } { }

If 2, 4, 6 and 1,3,5

then 1, 2,3, 4,5,6 and or .

A B

A B A B

= =

∪ = ∩ = ∅


have no elements in common. Thus, a real number is either rational or irratio-nal, but it cannot be both, as the following definitions show:

An interesting fact to note is that the square root of any number that is not a perfect square is irrational.

The scheme of real number subset inclusion is illustrated in the following dia-gram; subsets nest in other subsets contained in the universal set, the real num-bers.

Real numbers, � � = The set of rational numbers and irrational numbers.

The real numbers make up the real number line.

Natural numbers, � Natural numbers are the counting numbers.

� = {1, 2, 3, …}

08− + 8

Whole numbers, � Whole numbers are the natural numbers and zero.

� = {0, 1, 2, 3, …}

Natural numbers ⊂ Whole numbers

� ⊂ �

0 21 3 4 5 6 7 + 8

Integers, � Integers are the whole numbers and their negative counterparts.

� = {…, –3, –2, –1, 0, 1, 2, 3, …}

Whole numbers ⊂ Integers

� ⊂ �

08− + 821 3 4 5 6 7−2 −1−3−4−5−6−7

Rational numbers, � � = the set of numbers whose decimal form repeats or terminates. Alternately,the set of numbers that can be represented in fraction form. Note that this in-cludes integers, as they can be written with a denominator of one. Writing an

integer this way illustrates its rational form .

Examples of rational numbers:

Irrational numbers, �′ � ′ = The set of numbers whose decimal form does not repeat or terminate.

The sets of rational and irrational numbers have no element in common.

Examples of irrational numbers:

n1---

6 7, ,0.333...,5, 8.341

13 1−

3.141592..., 2.7182..., 2, 11eπ = = −

THE MULTIPLICATION TABLE 9

E X E R C I S E S

1.6 Let A = {Dog, Cat, Bird, Bear} and B = {Dolphin, Goat, Fish}. Find the indicated sets.

1.7 Let A = {0, 3, 12, 131} and B = {3, 10, 23, 99}. Find the indicated sets.

1.8 True or false: a) All rational numbers are integers.b) All integers are natural numbers.c) All whole numbers are integers.d) All irrational numbers are real numbers.e) The real number line is completely composed of integers and rational num-

bers.f) All integers are real numbers, but not all real numbers are integers.

The best way to start learning technical shop math is to memorize and masterthe multiplication table given in Table 1.5.

To find the product of any two numbers, themselves called factors of the prod-uct, determine where the column for the first number intersects the row for thesecond number. Once the multiplication table is committed to memory, any oth-er product of two numbers can be easily calculated. The table can also be usedin reverse to learn division.

Naturalnumbers

⊂ Wholenumbers

⊂ Integers ⊂ Rationalnumbers

⊂ Realnumbers

� ⊂ � ⊂ � ⊂ � ⊂ �

Rational ∪ Irrational = Real numbers � ∪ �′ = �

Rational ∩ Irrational = ∅ � ∩ �′ = ∅

a) A ∪ B b) A ∩ B

a) A ∪ B b) A ∩ B

1.4 The Multiplication Table

TABLE 1.5: Multiplication Table

2 3 4 5 6 7 8 92 4 6 8 10 12 14 16 183 6 9 12 15 18 21 24 274 8 12 16 20 24 28 32 365 10 15 20 25 30 35 40 456 12 18 24 30 36 42 48 547 14 21 28 35 42 49 56 638 16 24 32 40 48 56 64 729 18 27 36 45 54 63 72 81

Make a set of flash cards to help memorize the products in the multiplication table. Use 3 by 5 index cards with the multi-plication on one side of the card and the answer on the reverse side. Shuffle the cards and try to answer each multi-plication. Check the answer by looking at the back of the card. Practice until the multiplica-tion table becomes second nature.


E X E R C I S E S

1.9 Use the multiplication table to find the indicated products.

Addition and subtraction are operations in the additive process. Multiplicationand division are closely related to addition and subtraction, and are therefore al-so part of the additive process.

Addition and Subtraction

When numbers are added, the sequence of addition may be taken in any order.For example, 3 + 2 + 7 + 8 = 20. Rearranging the sequence does not change theresult: 2 + 3 + 8 + 7 = 20 and 8 + 3 + 7 + 2 = 20.

EXAMPLE 1.2: Using the Multiplication Table

Using the multiplication table, multiply 7 by 8.

Solution:

STEP 1: Locate 7 along the top row.

STEP 2: Locate 8 down the left side column.

STEP 3: Read answer, 56, at intersection of the 7 column and the 8 row.

EXAMPLE 1.3: Using the Multiplication Table for Division

Divide 24 by 6.

Solution:

STEP 1: Locate 6 along the top row.

STEP 2: Locate 24 down the column identified in Step 1.

STEP 3: Read answer, 4, in the left column along the row identified in Step 2.

a) 3 × 4 b) 7 × 6 c) 6 × 7 d) 9 × 2e) 5 ×3 f) 2 × 8 g) 4 × 3 h) 8 × 4

1.5 Operations in Arithmetic

EXAMPLE 1.4: Addition and Subtraction Operations

Solve using steps: 90 – 15 + 10 – 5

Solution:

STEP 1: 90 – 15 = 75

STEP 2: 75 + 10 = 85

STEP 3: 85 – 5 = 80

OPERATIONS IN ARITHMETIC 11

Series Multiplication and Division

A series of multiplications may be performed in any sequence.

For instance: 2 × 5 × 7 × 3 = 210; also, 7 × 2 × 3 × 5 = 210.

However, a series of divisions must be done in the sequence given:

For instance: 90 ÷ 15 ÷ 3 = ?

by steps: 90 ÷ 15 = 6

6 ÷ 3 = 2.

If the sequence is not followed, an error will be made; for instance:

15 ÷ 3 = 5

90 ÷ 5 = 18,

which is not the correct answer to the original problem.

E X E R C I S E S

1.10 Perform the indicated operations.

1.11 Perform the indicated operations.

Short and Long Division

Two methods of division are used in arithmetic. The first, called “short divi-sion,” is used when the divisor has only one digit. The second, called “long divi-sion,” is used when the divisor has two or more digits. Examples 1.5 and 1.6 il-lustrate these methods.

a) 12 + 18 – 10 – 4 b) 6 – 1 + 8 – 3 c) 13 – 4 + 6 – 3 d) 7 – 2 – 3 + 17 e) 9 + 3 – 4 + 6 f) 7 + 3 + 10 – 12 g) 19 – 10 – 6 – 1 h) 21 + 4 + 11 – 30

a) 9 × 6 × 2 b) 4 × 2 × 6 × 8 × 4 c) 45 ÷ 3 ÷ 5 d) 11 × 24 × 4 e) 128 ÷ 4 ÷ 4 ÷ 2 f) 98 ÷ 7 ÷ 7 g) 54 ÷ 3 ÷ 6 h) 8 × 6 × 3 × 7i) 6 × 5 × 10 × 2 j) 75 ÷ 5 ÷ 5


EXAMPLE 1.5: Short Division

Divide 636 by 6.

STEP 1: Determine whether the divisor 6 will divide the first digit of the dividend 636. It will since it is not greater than this digit. The result of division is 1, which is placed under the 6.

STEP 2: Determine whether the divisor 6 will divide the second digit of the dividend. Since 6 will not divide 3, a zero is placed under the 3.

STEP 3: The 3 is now taken with the third digit 6 to become 36. The divisor 6 divides 36 and the quotient 6 is placed under the 6. The answer is 106.

EXAMPLE 1.6: Long Division

Divide 6048 by 56.

Solution:

STEP 1: Set up the problem by placing a division symbol over the dividend 6048 with the divisor 56 to the left.

STEP 2: Start from the left of the dividend and find the smallest string of digits that the divisor will divide. In this case, the number is 60. Place the quotient 1 above the division symbol directly over the 0 of 60.

STEP 3: Multiply the divisor 56 by the quotient 1 and place the answer under the dividend found in Step 2.

STEP 4: Subtract the product, 56, from the first two digits of the number above it, 60.

6 636|1

6 636|10

6 636|106

|56 6048

1|56 6048

1|56 604856

1|56 604856

4


STEP 5: Bring down the next digit in the dividend to form a partial remainder, 44.

STEP 6: Divide the divisor 56 into the partial remainder 44 and place the quotient above the division symbol. Since 56 cannot divide 44, a zero is placed to the right of the 1.

STEP 7: Bring down the next digit in the dividend to form a new partial remainder, 448.

STEP 8: Divide 56 into the partial remainder 448 and place the quotient above the division symbol.

STEP 9: Multiply the divisor 56 by the quotient 8 and place the product below the partial remainder.

STEP 10: Subtract the product, 448, from the previous partial remainder, 448.

EXAMPLE 1.6: Long Division (Continued)

1|56 604856

44

10|56 60485644

10|56 604856448

108|56 604856

448

108|56 604856

448448

108|56 604856448448

0


In many problems the answer may have a remainder; that is, the divisor is not afactor of the dividend. The remainder is handled as shown in Example 1.7.

E X E R C I S E S

1.12 Perform the indicated division:

Order of Mixed Operations

In most technical mathematics problems several different operations need to beperformed. To arrive at the correct answer, the operations must be performed inthe proper sequence. The rules for the proper sequence are known as the orderof operations.

As a simple example of applying the correct order of operations to a problem,consider this: 7 + 2 × 4 – 6 + 15 ÷ 3 – 2.

The correct way to group the operations is 7 + (2 × 4) – 6 + (15 ÷ 3) – 2, whereinthe portions in ( ) are done first, working from left to right.

EXAMPLE 1.7: Division with a Remainder

Divide 4789 by 25.

The answer is:

Since we have used all of the digits given in the problem and are not left with azero at the bottom, the leftover 14 is the remainder. The final result is written

as 191 .

a) 390 ÷ 13 b) 9134 ÷ 17 c) 35,000 ÷ 128d) 1000 ÷ 33 e) 50,412 ÷ 24 f) 1357 ÷ 19

A factor of a number divides the number without a remainder.

191|25 478925228225

392514

1425------

7 (2 4) 6 (15 3) 2 7 8 6 (15 3) 2

7 8 6 5 2

15 6 5 2

9 5 2

14 2

12

multiplication

division

addition

subtraction

addition

subtraction

+ × − + ÷ − = + − + ÷ −= + − + −= − + −= + −= −=


Operations are performed in a specific sequence: Multiplication and division aredone first, in the order they appear, left to right. Then additiona and subtractionare done in the order they appear, left to right.

This is called the MDAS rule for Multiplication, Division, Addition, Subtrac-tion.

We often combine numbers with several operations somewhat automatically.Usually parentheses are included, but if no order is intended other than MDAS,the parentheses may be left out. That is why knowing the order of operations isso important.

When an operational order is intended other than the order provided by theMDAS rule, grouping symbols are necessary. Grouping symbols include paren-theses, brackets, and braces. Accordingly, we must add “P—Parentheses” to thememory device to get PMDAS. Operations in parentheses are always taken careof first.

For example, if the previous problem were written with parentheses inserted asshown, the answer would have been different:

(7 + 2) × 4 – 6 + 15 ÷ (3 – 2) = 9 × 4 – 6 + 15 ÷ 1 = 36 – 6 + 15 = 30 + 15 = 45

Grouping symbols may be “nested,” that is, one set may appear within anotherset. Often when this happens, other symbols—namely, brackets [ ] and braces{ }—further define the order. For example,

{7 – [3 × (4 – 2)] ÷ 2} + 1 = {7– [3 × 2] ÷ 2} + 1

= {7 – 6 ÷ 2} + 1

= {7 – 3} +1

= 4 + 1

= 5

A mnemonic device for remembering the order of operations is the phrase, “Please Excuse My Dear Aunt Sally.”

Finally, we insert “E—Exponents” in the phrase, so that if any term inside oroutside the parentheses is raised to an exponent, the exponent is taken before theother operations. Exponents are discussed in Chapter 6. The phrase is nowPEMDAS, which stands for: Parentheses, Exponents, Multiplication, Division,Addition, and Subtraction.

E X E R C I S E S

1.13 Perform according to the order of operations:

a) 24 ÷ 3 + 4 × 5 – 6 b) 60 – 3 × 8 + 6 × 5 – 14 ÷ 2 + 33 ÷ 11c) 148 – 34 × 2 – 37 d) 25 ÷ 5 + 3 × 6e) 24 ÷ 8 + 6 × 4 – 10 ÷ 2 f) 5 × 7 + 6 – 4 × 7g) 32 ÷ 4 + 8 × 3 – 5 h) 25 ÷ (2 + 3) × 5i) [(7 × 2) + 3] × 2 j) 6 + [(9 × 2) +1] × 3

INDEX

AAA theorem for similarity 345Abscissa 169Absolute error 74Absolute pressure 137–138Absolute temperature 133Absolute value 20–22Accuracy 75Acre 120Acute angles 282Acute triangles

oblique 365–366Addition 10, 17–20, 22–24

absolute value 21–22algebraic fractions 263–266common fractions 37–42decimal fractions 63–65measurement 73mixed numbers 43–47

Additive identity 6Additive inverse 6, 20, 24, 157–159Additive process 10Adjacent angles 282Air pressure 135–136Algebra 1, 141, 155Algebraic expressions 141–154

expanding 243–248Algebraic fractions 259–269Algorithms 141Altitude 343–344, 347Amplitude 434Angle bisector 341–343Angles 281–284

central 376complementary 326congruent 287inscribed 377measure 126–131supplementary 326–327

Angular measure 493Antilog 111–112Arabic numerals 3–5

Arcs 376, 381Area 120–121

polygons 301–312triangle 369–373

Arithmetic 1operations 10–15

ASA theorem for congruence 362Associative property

addition 6multiplication 6

Axioms 5, 321–322

BBase 90–94, 99–100Base 10 4Binary operation 22, 25Binomials 142, 245–248, 262

factoring 251–255Bisector 341–343, 385Board feet 122–123Board measure 122–126Boyle’s law 237–238British thermal unit 134–135Btu 134–135

CCalorie 134Cancellation 48–49, 55–56Cartesian plane 169–172Celsius 132Central angle 376, 387–390Charles’ law 234Chord 376, 381, 391Circles 299–300, 375–400

area 306circumscribed 342inscribed 342–343

Circular measure 493Circumference 299–300, 376Circumscribed circle 342

Closed intervals 165Coefficient 142Combined variation 239–241Common denominator 37Common factor 32Common fractions 27–56

converting to decimal 59–63converting to percent 83–86equivalent 58raising to higher terms 31–32reducing 32–36

Common logarithm 111Common multiple 37Commutative property

addition 6multiplication 6, 141

Commutative property of multiplation 251

Complementary angles 283, 326Complex fractions 55–56Compound proportion 217–218Compound ratio 217–218Compound shapes

area 306–308volume 317–318

Compressible fluids 136Concave polygons 286Concentric circles 378Cones 296

volume 316–317Congruence 287–288, 325Congruent angles 287, 326Congruent circles 385Congruent triangles 360–364Conical tapers 226–229Constant 141Constant coefficient 253Conversion factors 490–496Converting

common fractions 59–63decimal fractions 59–63decimal to common fractions 161–162improper fractions 29–30

558 INDEX

measurement units 117–118mixed numbers 29–30percent 83–90temperature 133–134to scientific notation 109–110weights and mass 131–132

Convex polygons 286Coordinate distances

trigonometry 470–476Coordinate plane 169Corresponding angles 284, 344–345Corresponding sides 344–345Cosecant 402–407Cosine 402–407, 434

law 447–450Cotangent 402–407

law 450–452Coulomb’s law 239Cube 294

surface area 313–314Cubic measure 492–493Cutting speed 198Cylinders 296

volume 316

DDecimal 57–59Decimal fractions 57–81

addition 63–65converting to common fraction 59–63division 67–69equivalent 58multiplication 65–66subtraction 63–65

Decimal places 66Decimal point 57Decimal tolerances 76–81Degree of a polynomial 142Degrees 126Degrees decimal 127Denominator 27

common 37rationalizing 151–152

Depth of cut 197–198Diagonal 286, 299Diagonals 290Diameter 299, 376, 381Diamond 290Difference of squares 246, 256–257Direct current electrical formulas 190–191

Direct variation 233–236Discount 96–98Distance formula 187–188Distributive property 6Division 24–25

algebraic fractions 261–263common fractions 48–51decimal fractions 67–69measurement 74mixed numbers 52–54

short and long 11–15with scientific notation 110

Dodecahedron 294

EElectrical formulas 190–191, 204–205Elements 6Ellipse 379–380Empty set 7Energy 495English system of measurement 115Equality 322Equations 143

solving 155–163Equiangular polygon 287, 290Equilateral triangles 288, 342–343Equilaterial polygon 287Equivalent fractions 31, 89Equivalent percents 89Error

measurement 74–75Euclid’s postulates 322–324Expanded notation 4–5Expanding

algebraic expressions 243–248binomials 245–246polynomials 243

Exponents 4, 99–113operations 102–103operations on expressions 146–149rules 101–108

Expression 142Expressions

combining like terms 152–154evaluating 143–145exponents 146–149radicals 149–151rationalizing the denominator 151–

152Exterior angles 284Exterior point

circle 375

FFactoring 273–274

imaginary numbers 257–258polynomials 248–250

Factorizationprime 33–34

Factor-label method 117–118, 217Factors 9, 32Fahrenheit 132First degree polynomial 142Flat tapers 226–229Focus 379–380FOIL method 245–246, 251–253, 256–257

Force 131Formulas 144–145, 187–211

trigonometry 485–488

Fraction bar 27Fractions

algebraic 259common 27–56decimal 57–81equivalent 31improper 28proper 28unity 28

Frustumvolume 316–317

GGage blocks 79–80, 497–498Gauge (see Gage)Gauge pressure 137–138GCF (see Greatest common factor)Gear ratios 216Geometry 1, 333–335Gradians 126Graphs

inequalities 163–164linear equations 169–185slope-intercept form 180–185

Gravity 131Greatest common factor 33–36, 38, 249Greek letters 1–2, 282, 489Grouping symbols 15

HHeat 132–135, 496Hectare 120Hero’s formula 369–373Hexagons 286, 292, 345, 379

area 305Higher roots 105Higher terms 31–32Hole circle 468–469Hooke’s law 234Horizontal lines 179–180Horsepower 190, 193–197, 205Hydraulic cylinders 202–204Hydraulic formulas 200–202, 210–211Hydraulic hoist 201–202Hypotenuse 289, 343–344, 347

IIcosahedron 294Imaginary numbers 257–258Improper fractions 28

converting 29–30Indexing 199Indirect variation (see Inverse variation)Inequalities

graphing 164solving 155, 163–168

Inequality 143Inscribed angle 377

559

Inscribed circle 342–343, 386Integers 7–8

operations 17Interest

simple 95–96Interest rate 95Interior angles 284, 287, 327–333Interior point

circle 375International system of units (see SI

measurement)Intersection of sets

7Interval notation 163, 165–168Inverse

operations 20Inverse logarithm 112Inverse natural logarithm 112–113Inverse operations 156Inverse trigonometric functions 414–415Inverse variation 233, 236–238Irrational nubmers 7–8Isosceles

triangle 343Isosceles triangles 288

JJig-boring 470–476Joint variation 239–241Joule 134, 190

KKelvin 133

LLaw of cosines 447–450Law of sines 445–447LCD (see Least common denominator)Least common denominator 37–40, 43–45, 263

Least common multiple 263Length 118–119Like terms 152–154, 158–159Limits

dimensions 76–78Line segments 281

congruent 287Linear equations

graphing 169–185one variable 155–163

Linear measure 118–119, 490Lines 279–281

graphing points 172–174parallel 327perpendicular 327point-slope form 176–177slope 175–180slope-intercept form 177–178

List price 96–98Literal equations 187–211Log (see Logarithms)Logarithms 111–113Long division 11–13Lowest terms 32–36, 48

MMajor axis 380Mass 131–132, 494Mean proportional 219, 347Measurement 73–76, 115–140

converting units 118–123systems 115–118

Metric system 116–118Minor axis 381Minuend 44Mixed number percent 86–88Mixed numbers 29

addition 43–47converting 29–30division 52–54multiplication 52–54subtraction 43–47

Mixed operations 14–15Mixture proportions 223–225Monomials 142, 243–244Motors 196–197, 206Multiples 31Multiplication 11, 24–25

algebraic fractions 261–263common fractions 48–51decimal fractions 65–66measurement 74mixed numbers 52–54monomials 243–244polynomials 243–244, 266table 9with scientific notation 110zero property 6

Multiplicative identity 6, 31Multiplicative inverse 6, 157

NNatural logarithms 112Natural numbers 8Negative base 100Negative error 74Negative exponent 102Negative numbers 17–20Negative slope 176Newton’s law of gravitational force 239–240

Newton-meter 190Null set 7Number line 17–20

7–8Number system

base 10 4Numerals 3–5

Numerator 27

OOblique triangles 289, 344, 439–462

acute 365–366obtuse 366–368

Obtuse angles 283, 289Obtuse triangles 289

oblique 366–368Octagons 293, 345Octahedron 294Ohm’s Law 144, 190–191Open intervals 165Operations

inverse 20, 156order 14–15percent 83–98polynomials 243signed numbers 17–26

Order of operations 14–15Ordered pair 169Origin 169

PParallel lines 280, 327, 345Parallelograms 290

area 303Partial root 149Pascal’s law 200–201PEMDAS 15, 26, 143–144Pentagons 286, 292Percentage formula 188Percents 83–98

converting to common fraction 85–88converting to decimal fraction 85mixed numbers 86–88solving problems 90–94

Perimeter 297–301square 298

Period 434Perpendicular lines 280, 327Phase shift 434Pi 1, 299–300Place value 4, 69–73Plane geometry 279–296Planes 284Plotting points 170–172Points 169, 279

graphing 172–174plotting 170–172

Point-slope form 176–177Polygons 286–294, 329–333

area 301–312perimeter 297–298

Polyhedrons 294–295Polynomial 142Polynomials 243

expanding 247–248factoring 248–250multiplication 266

560 INDEX

operations 243Positive base 100Positive error 74Positive numbers 17–20Positive slope 176Postulates 321–325Pound force 135Power 4, 193, 496

of 10 99–101of a power 103power of 10 4

Power formula 190–191Power of 10 61Powers of x 146–148Precision 75Precision gage blocks 79–80Pressure 135–139, 200, 495Price 96–98Prime factorization 33–36Prime number 7, 33Principal 95Principle root 104Prisms 295

surface area 312–314volume 314–315

Product 9Projection 348Projection formula 365–368Proper fractions 28Proportion 213–241

compound 217–218mixture 223–225simple 214–215tapers 225–233

Proportionality constant 233, 238Pumps 196–197, 206Pyramids 295

volume 315Pythagorean theorem 353–360

QQuadrants 170, 429–432Quadratic equations 268–278Quadratic formula 276–278Quadrilaterals 286, 289Quotient 12

RRadians 126–131Radical equations 162–163Radicals 104

operations on expressions 149–151simplifying 106–107, 150

Radicand 104Radius 299, 376, 383Raising

common fractions 31–32Rankine 133Rate 90–95Ratio 213–241

compound 217–218gears 216simple 213

Rational expression 143Rational expressions 259Rational numbers 7–8, 161Rays 281Real numbers 5–9

number line 7–8properties 5–6, 141, 321

Reciprocal 49, 56, 158–159Rectangles 290

area 302–303Rectilinear sides 301Reducing

common fractions 32–36mixed numbers 36

Reducing mixed numbers 36Reflexive property 6Reflexivity 321Regular polygon 287Regular polyhedrons 294Relative pressure 137–138Relative temperature 132Remainder 14, 29Rhombi 290Right angles 282, 289Right triangle 347–348, 353–360, 401, 410–437, 439–443

Right triangles 289Rise 175Rod 297Roman numerals 3Roots 99–100, 271

higher 105of negative numbers 105–106square 104–105

Rounding 69–73Run 175

SSAS theorem for congruence 361SAS theorem for similarity 345Scalene triangles 288, 343–344Scientific notation 108–111Secant 376, 402–407Second degree polynomial 142–143Sector 376Segment

circle 376Semicircles 376Set notation 165–168Sets 6–9

defined 6real number 6–7

Short division 11–12SI measurement 116–118SI units 489Signed numbers

operations 17–26Significant digits 70–72

Similar triangles 291, 344–353Simple interest 95–96Simple proportion 214–215Simplifying

algebraic fractions 259–261radicals 106–107

Sine 402–407, 434Sine bars 463–488Sine plates 463–468Sines

law 445–447Skew lines 280Slope 175–180Slope-intercept form 177–178

graphing 180–185Solids

surface area 312–314Solution set 163Special exponents 101–102Special products 256–258Spheres 296

volume 316Spindle speed 198–199Square measure 491Square root 149Square roots 104–105Squares 291

area 301–302perimeter 298–299

SSS theorem for congruence 362Standard atmosphere 136Statistics 1Straigth angles 282Strain 140, 234–235Stress 234–235, 495Subsets 6–9

real number 7–9Subtraction 10, 17–20, 22–24

absolute value 21–22algebraic fractions 263common fractions 37–42decimal fractions 63–65measurement 73mixed numbers 43–47

Subtrahend 44Sum of squares 257–258Supplementary angles 283, 326–327Surface area

solids 312–314Surface area formula 188–189Symbols 1–5Symmetric property 6Symmetry 321Systems of measurement 115–118

TTangent 376, 383–385, 402–407, 434Tangent circles 378Tapers 225–233Temperature 132–135, 496Term 142

561

Tessellations 292Tetrahedron 294Theorems 325–335, 361–362

circle 381–400Thread translation 191–192Tolerances 76–81Torque 196–197Transitive property 6Transitivity 321–322Translation 191–192Transversal 327Transversal line 283–284Trapezoids 291

area 303–304Triangles 286–289, 327–330, 341–373

area 304–305congruent 304, 360–364equilateral 342–343isosceles 343oblique 344right 347–348scalene 343–344special 427–429

Triangulation 341Trigonometry 1

cofunction identities 407–408complementary identities 407functions 402–407graphing 433–437oblique angle 439–462

reciprocal identities 408right triangle 401–437shop 463–488solving sides 410–427

Trinomials 142, 247–248factoring 251–255

UUndefined slope 176, 179Union of sets

7Unit circle 429–432Unity 28Universal set 7

VVaccums 138–139Variable 141Variables 155–163

applying exponent rules 146–148Variation 233–241Velocity 195Vertex 287Vertical angles 283–284, 325Vertical lines 179–180Volume 121

solids 314–318Volume mixture 223–224

WWater flow rate 202Watt 190Weight 131–132, 494Weight mixture 224–225Whole numbers 8Worm gear ratios 192–193

XX-axis 169, 179X-intercept 179X-values 173–174

YY-axis 169, 180Y-intercept 178Y-values 173–174

ZZero degree polynomial 142–143Zero property of multiplication 6Zero slope 176

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