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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
Chapter 2 Exercises 501
Chapter 3 Exercises 502
Chapter 4 Exercises 506
Chapter 5 Exercises 509
Chapter 6 Exercises 510
Chapter 7 Exercises 512
Chapter 8 Exercises 515
Chapter 9 Exercises 517
Chapter 10 Exercises 519
Chapter 11 Exercises 524
Chapter 12 Exercises 525
Chapter 13 Exercises 527
Chapter 14 Exercises 531
Chapter 15 Exercises 533
Chapter 16 Exercises 534
Chapter 17 Exercises 537
Chapter 18 Exercises 538
Chapter 19 Exercises 539
Chapter 20 Exercises 540
Chapter 21 Exercises 551
Chapter 22 Exercises 552
INDEX 557
1
THE LANGUAGE OFMATHEMATICSMathematics 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.
THE LANGUAGE OF MATHEMATICS4 CHAPTER 1
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.
THE LANGUAGE OF MATHEMATICS6 CHAPTER 1
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
= =
∪ = ∩ = ∅
THE LANGUAGE OF MATHEMATICS8 CHAPTER 1
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.
THE LANGUAGE OF MATHEMATICS10 CHAPTER 1
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
THE LANGUAGE OF MATHEMATICS12 CHAPTER 1
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
OPERATIONS IN ARITHMETIC 13
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
THE LANGUAGE OF MATHEMATICS14 CHAPTER 1
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 IN ARITHMETIC 15
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