Mathematics for Machine Technology

Mathematics for Machine Technology
اسم المؤلف
John C. Peterson , Robert D. Smith
التاريخ
18 فبراير 2021
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Mathematics for Machine Technology
Eighth edition
John C. Peterson
Robert D. Smith
vi PREFACE
1 sECtion 1 Common Fractions and Decimal Fractions
UNIT 1 Introduction to Common Fractions and Mixed Numbers / 1
UNIT 2 Addition of Common Fractions and Mixed Numbers / 8
UNIT 3 Subtraction of Common Fractions and Mixed Numbers / 14
UNIT 4 Multiplication of Common Fractions and Mixed Numbers / 20
UNIT 5 Division of Common Fractions and Mixed Numbers / 27
UNIT 6 Combined Operations of Common Fractions and Mixed Numbers / 33
UNIT 7 Computing with a Calculator: Fractions and Mixed Numbers / 40
UNIT 8 Computing with a Spreadsheet: Fractions and Mixed Numbers / 49
UNIT 9 Introduction to Decimal Fractions / 57
UNIT 10 Rounding Decimal Fractions and Equivalent Decimal and
Common Fractions / 63
UNIT 11 Addition and Subtraction of Decimal Fractions / 69
UNIT 12 Multiplication of Decimal Fractions / 74
UNIT 13 Division of Decimal Fractions / 78
UNIT 14 Powers / 84
UNIT 15 Roots / 91
UNIT 16 Table of Decimal Equivalents and Combined Operations
of Decimal Fractions / 97
UNIT 17 Computing with a Calculator: Decimals / 103
UNIT 18 Computing with a Spreadsheet: Decimals / 112
UNIT 19 Achievement Review—Section One / 121
126 sECtion 2 Ratio, Propor tion, and Percentage
UNIT 20 Ratio and Propor tion / 126
UNIT 21 Direct and Inverse Proportions / 135
UNIT 22 Introduction to Percents / 141
UNIT 23 Basic Calculations of Percentages, Percents, and Rates / 145
UNIT 24 Percent Practical Applications / 150
UNIT 25 Achievement Review—Section Two / 157
160 sECtion 3 Linear Measurement: Customary (English) and Metric
UNIT 26 Customary (English) Units of Measure / 160
UNIT 27 Metric Units of Linear Measure / 169
UNIT 28 Degree of Precision, Greatest Possible Error, Absolute Error,
and Relative Error / 178
UNIT 29 Tolerance, Clearance, and Interference / 183
Contentsiv Contents
UNIT 30 Customary and Metric Steel Rules / 192
UNIT 31 Customary Vernier Calipers and Height Gages / 203
UNIT 32 Metric Vernier Calipers and Height Gages / 212
UNIT 33 Digital Calipers and Height Gages / 215
UNIT 34 Customary Micrometers / 218
UNIT 35 Metric Vernier Micrometers / 226
UNIT 36 Digital Micrometers / 231
UNIT 37 Customary and Metric Gage Blocks / 234
UNIT 38 Achievement Review—Section Three / 238
244 sECtion 4 Fundamentals of Algebra
UNIT 39 Symbolism and Algebraic Expressions / 244
UNIT 40 Signed Numbers / 253
UNIT 41 Algebraic Operations of Addition, Subtraction, and Multiplication / 267
UNIT 42 Algebraic Operations of Division, Powers, and Roots / 274
UNIT 43 Introduction to Equations / 287
UNIT 44 Solution of Equations by the Subtraction, Addition, and Division
Principles of Equality / 295
UNIT 45 Solution of Equations by the Multiplication, Root, and Power
Principles of Equality / 306
UNIT 46 Solution of Equations Consisting of Combined Operations
and Rearrangement of Formulas / 313
UNIT 47 Applications of Formulas to Cutting Speed, Revolutions
per Minute, and Cutting Time / 324
UNIT 48 Applications of Formulas to Spur Gears / 334
UNIT 49 Achievement Review—Section Four / 343
347 sECtion 5 Fundamentals of Plane Geometry
UNIT 50 Lines and Angular Measure / 347
UNIT 51 Protractors—Simple Semicircular and Vernier / 363
UNIT 52 Types of Angles and Angular Geometric Principles / 370
UNIT 53 Introduction to Triangles / 380
UNIT 54 Geometric Principles for Triangles and Other Common Polygons / 387
UNIT 55 Introduction to Circles / 399
UNIT 56 Arcs and Angles of Circles, Tangent Circles / 409
UNIT 57 Fundamental Geometric Constructions / 420
UNIT 58 Achievement Review—Section Five / 430
437 sECtion 6 Geometric Figures: Areas and Volumes
UNIT 59 Areas of Rectangles, Parallelograms, and Trapezoids / 437
UNIT 60 Areas of Triangles / 449
UNIT 61 Areas of Circles, Sectors, and Segments / 454
UNIT 62 Volumes of Prisms and Cylinders / 462
UNIT 63 Volumes of Pyramids and Cones / 473
UNIT 64 Volumes of Spheres and Composite Solid Figures / 481
UNIT 65 Achievement Review—Section Six / 488Contents v
494 sECtion 7 Trigonometry
UNIT 66 Introduction to Trigonometric Functions / 494
UNIT 67 Analysis of Trigonometric Functions / 506
UNIT 68 Basic Calculations of Angles and Sides of Right Triangles / 511
UNIT 69 Simple Practical Machine Applications / 519
UNIT 70 Complex Practical Machine Applications / 529
UNIT 71 The Cartesian Coordinate System / 542
UNIT 72 Oblique Triangles: Law of Sines and Law of Cosines / 546
UNIT 73 Achievement Review—Section Seven / 558
563 sECtion 8 Compound Angles
UNIT 74 Introduction to Compound Angles / 563
UNIT 75 Drilling and Boring Compound-Angular Holes: Computing Angles
of Rotation and Tilt Using Given Lengths / 567
UNIT 76 Drilling and Boring Compound-Angular Holes: Computing Angles
of Rotation and Tilt Using Given Angles / 573
UNIT 77 Machining Compound-Angular Surfaces: Computing Angles
of Rotation and Tilt / 581
UNIT 78 Computing Angles Made by the Intersection
of Two Angular Surfaces / 590
UNIT 79 Computing Compound Angles on Cutting and Forming Tools / 598
UNIT 80 Achievement Review—Section Eight / 608
610 sECtion 9 Computer Numerical Control (CNC)
UNIT 81 Introduction to Computer Numerical Control (CNC) / 610
UNIT 82 Control Systems, Absolute Positioning, Incremental Positioning / 621
UNIT 83 Location of Points: Polar Coordinate System / 631
UNIT 84 Binary Numeration System / 638
UNIT 85 Hexadecimal Numeration System / 645
UNIT 86 BCD (Binary Coded Decimal) Numeration Systems / 650
UNIT 87 An Introduction to G- and M-Codes for CNC Programming / 655
UNIT 88 Achievement Review—Section Nine / 669
674 APPEndixEs
A United States Customary and Metric Units of Measure / 674
B Principles of Plane Geometry / 676
C Formulas for Areas (A) of Plane Figures / 678
D Formulas for Volumes (V) of Solid Figures / 679
E Trigonometry / 680
F Common G- and M-Codes / 682
684 AnswERs to odd-numbEREd APPliCAtions
720 indEx
720
0.0001-inch micrometer, 221–222
0.001-inch micrometer, 219–221
reading and setting, 220–221
0.01-mm micrometer, 226–229
2421 BCD code, 652
5211 BCD code, 652
8421 code/BCD code, 651
A
Abscissa, 542
Absolute error, 180–181
Absolute positioning (absolute coordinates),
622–626
Absolute value, 254–255
Acute angle, 371, 513–515
Acute triangles, 546
Addendum, 335, 336, 339
Addition
algebraic operations, 267–268
of angles, 356
combined operations, 34, 36
of compound numbers, 165, 166–167
of decimal fractions, 69–70
decimals, 113–115
of fractions, 8–11
of linear units, 164
of mixed numbers, 8–11
in order of operations, 33
of signed numbers, 255–256
of whole numbers, 11
Addition principle of equality, 298–299
Adjacent angles, 371–372
Adjacent side, of right triangle, 495
Algebra, 244
Algebraic expressions
defined, 245
evaluation of
application, 246
order of operations for combined
operations, 247–248
overview, 246
overview, 245
symbolism, 244–246
term of, 267
Algebraic operations
addition, 267–268
combined operations, 278
division, 274–275
multiplication, 270–271
overview, 267
powers, 276
removal of parentheses, 277
roots, 277
scientific notation, 278–282
subtraction, 269
Allowance, 185
Alternate interior angles, 372, 373–374
Altitude
of cone, 473, 476
of cylinder, 466
of parallelogram, 441–442
of prism, 465
of pyramid, 473, 476
of triangle, 390
Ambiguous case, 548–549
Angle cuts, 523–524
Angle plate, positioning part on
for drilling, 569
for machining, 583–584
Angles, 349. See also Compound angles;
Protractors
adjacent, 371–372
base, 380
bisecting, 423
central, 400
complementary, 367
computing angles of rotation and tilt using,
573–576
computing true, 565
corresponding, 382
determining functions of, 497–500
in any quadrant, 542–544
determining unknown, of right triangles,
511–515
formed by transversal, 372
formed inside a circle, 409–410
formed on circle, 411
formed outside a circle, 411–413
geometric principles, 372–375
of given functions, 499–500
inscribed, 400
Law of Cosines, 549–552
Law of Sines, 546–549
made by intersection of two angular surfaces,
590–594
naming, 371
overview, 370
supplementary, 367, 374
of triangle, 381
types of, 371–372
Angles of rotation
computing for compound-angular holes
using given angles, 573–576
using given lengths, 567–569
formulas for computing
for machining, 584–586
when drilling, 575
machining compound-angular surfaces,
581–586
Angles of tilt
computing for compound-angular holes
using given angles, 573–576
using given lengths, 567–569
formulas for computing
for drilling, 575–576
for machining, 584–586
machining compound-angular surfaces,
581–586
Angular measure, units of, 349–350
Anvil micrometer, 219
Arc, 400
inverse functions and, 499
Arccosine, 499
Arc length formula, 410–411
Arcsine, 499
Arctangent, 499
Areas
of circles, 454–455
customary units of, 437–438, 440
metric units of, 438–440
overview, 437
of parallelograms, 441–442
of rectangles, 440–441
of sectors, 455
of segments, 456
of trapezoids, 442–443
of triangles, 449–451
Arithmetic numbers, 244
Arithmetic operations
on angular measure in degrees, minutes, and
seconds, 355–358
on compound numbers, 165–167
with customary linear units, 166–167
with linear units, 164
Auxiliary lines, 519
Axioms, 348
Axis (axes)
of cone, 474
of cylinder, 466
x-axis, 542, 612–613
y-axis, 542, 612–613
z-axis, 612–613
B
Balanced scale, 287–288
Bar (—), 1
Bar key, 41
Base
of cone, 475
of cylinder, 467
finding, given percent and percentage, 148
finding in practical applications, 151–152
finding percentage, given rate and, 146–147
finding rate, given percentage and, 147
identifying, in practical applications, 150–151
of percentage, 145
of prism, 464, 467
of pyramid, 475
of trapezoid, 442–443
of triangle, 380
Base angles, 380
Basic dimension, 184
BCD (binary coded decimal) numeration systems
2421 BDC code, 652
5211 BCD code, 652
8421 code/BDC code, 651
converting decimal numbers to 2421 code
numbers, 652
converting decimal numbers to 5211 code
numbers, 652
converting decimal numbers to excess-3 code
numbers, 653
converting from decimal numbers to, 651
converting to decimal numbers, 651–652
excess-3 BCD code, 653
overview, 650–651
Bevel protractors with vernier scale, 366–367
Bevels, 521–522
Bilateral tolerance, 184
expressing unilateral tolerance as, 184–185
Binary digit (bit), 639
Binary numeration system
IndexIndex 721
converting binary coded decimal numbers to,
651–652
converting hexadecimal numbers to, 646–647
converting to binary coded decimal
numbers, 651
converting to hexadecimal numbers, 645–646
expressing as decimal numbers, 640–641
expressing decimal numbers as, 641
overview, 638–639
structure of binary system, 640–641
structure of decimal system, 639–640
Bisect, 390
Bisecting angles, 423
Bit, 639
Boring compound-angular holes
using given angles, 573–576
using given lengths, 567–569
Bow micrometer, 219
Braces, 33
Brackets, 33
Byte, 639
C
Calculator computing
circumference, 401
combined operations, 45–47, 107–111
customary units of linear measure, 163–164
decimal degrees, 350, 351, 354
decimals, 103–104
to express degrees/minutes and seconds as
decimal degrees, 351
fractions, 40–43, 45–47
inverse functions, 499–500
metric units of linear measure, 171–173
mixed numbers, 43–47
powers, 104–105
roots, 105–106
scientific notation, 282
trigonometric functions and angles, 497
Calipers
decimal-inch vernier, 203
overview, 204
reading and setting measurement on, 204–206
dial, 179
digital, 203, 215–217
metric vernier, 212–213
vernier, 179, 203
overview, 204
Cancellation, 21
Caret key, 115, 259, 261
Cartesian coordinate system, 542–544
three-axis, 612–614
two-axis, 611–612
Center
of circle, 399
of sphere, 481
Center point (center of rotation), 631
Center punch, 420
Centimeter (cm), 170
cubic, 463
square, 439
Central angle, 400
Checking equations, 292
Chord, 399
Circles
angles formed inside, 409–410
angles formed on, 411
angles formed outside, 411–413
arc length formula, 410–411
area of, 454–455
circumference formula, 400–401
definitions, 399–400
externally tangent, 413, 414
geometric principles, 401–404
internally tangent, 413
overview, 399
pitch, 335
as ray with fixed point, 349
root, 335
tangents to, 523–524
Circular cone, 474
Circular cylinder, 466
Circular interpolation, 622
Circular motion, 661–663
combined with linear motion, 663–664
Circular pitch, 335, 336, 339
Circumference, 399
formula, 400–401
Clearance, 185, 335, 336, 339
Coefficient, numerical, 267
Combined operations
algebraic operations, 278
calculator computing, 45–47, 107–111
complex fractions, 36
decimal fractions, 98–100, 119–120
fractions and mixed numbers, 34–36
order of operations, 33–36, 98–100,
247–248, 278
signed numbers, 262
solving equations consisting of, 313–315
spreadsheet, 54–56
Common factors, dividing by, 21
Common fractions, 2
expressing as decimal fractions, 64–65
expressing as percents, 142
expressing decimal fractions as, 65
expressing percents as, 143
Complementary angles, 367
functions of, 507–508
Complex fractions, 3, 36
Composite solid figures (composite space figures),
volume of, 482–483
Compound angles
applications, 563
computing on cutting and forming tools,
598–602
computing true lengths and true angles, 565
diagonal of rectangular solid, 564–565
drilling and boring holes
using given angles, 573–576
using given lengths, 567–569
machining surfaces, 581–586
made by intersection of two angular surfaces,
590–594
overview, 563–564
Compound numbers
addition of, 165 , 166–167
defined, 165, 166
division of, 165–166, 167
multiplication of, 165, 167
subtraction of, 165, 166–167
Computer aided drafting/design (CAD), 638
Computer aided manufacturing (CAM), 638
Computer-assisted programming, 611
Computer integrated manufacturing (CIM), 638
Computer numerical control (CNC)
BCD numeration system, 650–653
binary numeration system, 638–641
control systems, 621–627
G-codes for, See G-codes
hexadecimal numeration system, 645–647
M-codes for, See M-codes
overview, 610
polar coordinate system, 631–632
programming, 611
three-axis Cartesian coordinate system,
612–614
two-axis Cartesian coordinate system, 611–612
Cones, 474
frustums of, 475–477
volumes of, 474–475
Congruent triangles, 387
Continuous path control systems, 622
Control systems
absolute positioning, 622–626
incremental positioning, 626–627
overview, 622
programming, 621–622
tool positioning systems, 622
types of, 621
Conversion factor, 161
Conversion factors, 175–177
Coordinates, 611
Coordinate systems. See Cartesian coordinate
system; Polar coordinate system
Corresponding angles, 372, 382
Corresponding sides, 382
Cosecant (csc), 495–496, 497–498
Cosine (cos), 495–496, 497
Cotangent (cot), 495–496, 497–498
Cross-multiplication, 319–320
Cube root, 91, 117
Cubic centimeter, 463
Cubic foot, 462
Cubic inch, 462
Cubic meter, 463
Cubic millimeter, 463
Cubic yard, 462
Cursor key, 354
Curve, involute, 335
Customary units of measure. See also Equivalent
units of measure; Steel rules
of angular measure, 496
area, 437–438, 440
cutting speeds, 324–325
cutting times, 326–327
decimal-inch vernier height gage, 206–207
gage blocks, 234–235
determining gage block combinations, 236
linear measure
arithmetic operations, 166–167
calculator computing, 163–164
expressing larger units as smaller units, 162
expressing smaller units as larger units, 163
measurement definitions, 160–161
overview, 161
steel rules, 192–195
micrometers, 218–222
0.001-inch, 219–221
vernier (0.001-inch), 221–222
revolutions per minute, 325–326
volume, 462–463, 464
Cutting speeds
applications, 324
using customary units of measure, 324–325
using metric units of measure, 327
Cutting speed table, 328–330
Cutting times
using customary units of measure, 324–325
using metric units of measure, 328
Cutting tools, compound angles on, 598–602
Cylinders, 466
volumes of, 466–467
D
Decimal degrees
expressing as degrees, minutes, and seconds,
350–353
expressing degrees, minutes, and seconds as,
353–355
overview, 350
Decimal equivalent table, 97–98
Decimal fractions
addition of, 69–70
alternative way to read, 60
combined operations, 98–100, 119–120
division of, 78–80
expressing as common fractions, 65
expressing as percents, 142722 Index
Decimal fractions (continued)
expressing common fractions as, 64–65
expressing percents as, 142–143
meaning of fractional parts, 58–59
multiplication of, 74–75
overview, 57
reading and writing, 59
rounding, 63–64
simplified method of reading, 60
subtraction of, 69, 70–71
writing from fractions, 60
Decimal-inch rules, 194–195
Decimal-inch spur gears, 336–337
Decimal-inch vernier calipers, 203
overview, 204
reading and setting measurement on, 204–206
Decimal-inch vernier height gage, 206–207
Decimal numbers
converting BCD numbers to, 651–652
converting hexadecimal numbers to, 646–647
converting to 2421 code numbers, 652
converting to 5211 code numbers, 652
converting to BCD numbers, 651
converting to excess-3 code numbers, 653
converting to hexadecimal numbers, 647
expressing as binary numbers, 641
expressing as scientific notation, 278–279
expressing binary numbers as, 640–641
expressing scientific notation as, 279–280
Decimal point key, 103
Decimals
in calculator computing, 103–104
spreadsheet computing, 112–115
Decimeter (dm), 170
Dedendum, 335, 336, 339
Degree of precision
limitation of measuring instruments, 178–179
of numbers, 179–180
overview, 178
Degrees, 350
adding angles expressed in, 356
decimal degrees, 350–355
dividing angles expressed in, 358
expressing as decimal degrees, 353–355
expressing decimal degrees as, 350–353
multiplying angles expressed in, 357–358
relationship to minutes and seconds, 350
subtracting angles expressed in, 356–357
Dekameter (dam), 170
Denominate numbers, 146, 161
Denominator, 2
Depth
whole, 335, 336, 339
working, 335, 336, 339
Diagonal of rectangular solid, 564–565
Dial calipers, 179
Dial indicators (comparison measurement), 179
Diameter
of circle, 399
outside, 335, 336, 339
pitch, 335, 336, 339
of sphere, 481
Diametral pitch system, 336–337
Digital calipers, 203
reading, 215–216
use of, 216–217
Digital micrometers, 179, 231–232
Digits, 639
Dimension
basic, 184
mean, 184
Directed numbers. See Signed numbers
Direct proportions, 135–136
Dividend, 27
Dividers, 420
Division
algebraic operations, 274–275
of angles, 358
by common factors (cancellation), 21
of compound numbers, 165–166, 167
of decimal fractions, 78–80
decimals, 113–115
of fractions, as inverse of multiplying fractions,
27–29
of linear units, 164
of mixed numbers, 28–29
in order of operations, 33
of signed numbers, 258
using scientific notation, 281–282
Division principle of equality, 299–300
Divisor, 27
Dovetails, 523–524
Drilling compound-angular holes
positioning part on angle plate for, 569
using given angles, 573–576
using given lengths, 567–569
Drill presses
cutting speed, 324, 325
cutting times, 327
revolutions per minute, 326
Dual dimensioning, 175
E
English units of measure. See Customary units of
measure
Equality. See also Principles of equality
expression of, 287–288
Equations
addition principle of equality, 298–299
checking, 292
consisting of combined operations, 313–315
defined, 287
division principle of equality, 299–300
expression of equality, 287–288
multiplication principle of equality, 306–307
overview, 287
power principle of equality, 308
proportions, 319–320
rearranging formulas, 317–319
root principle of equality, 307–308
solving
principles of equality, 295
substituting values and solving formulas,
316–317
subtraction principle of equality, 295–296
transposition, 296–297
unknown quantity, 288
writing from word statements, 288–291
Equiangular triangles, 380
Equilateral triangles, 380, 390–391
Equivalent fractions, 3–4, 8
Equivalent units of measure
customary
area, 440
linear, 175–177
overview, 161, 175
volume, 464
metric
area, 440
linear, 175–177
overview, 175
volume, 464
Errors
absolute, 180–181
greatest possible, 180
relative, 180–181
Excess-3 BCD code, 653
Exponents
defined, 84
fractional, 94, 261–262
Externally tangent circles, 413, 414
Extremes, of proportion, 128
F
Factors, 3, 84, 267
literal, 267
Fits of mating parts, 185–186
FOIL method, 271
Foot (ft), 85, 161
cubic, 462
square, 438
Forming tools, compound angles on, 598–602
Formulas, 244. See also Cutting speeds; Cutting
times; Gears; Revolutions per minute
arc length, 410–411
circumference, 400–401
for computing angles of intersecting angular
surfaces, 592–594
for computing angles of rotation and angles
of tilt
when drilling, 575–576
when machining, 584–586
for computing true angles for cutting and
forming tools, 599–600
for computing true angles in front-clearanceangle applications, 600–601
defined, 84, 287
rearranging, 317–319
in spreadsheet, 316–317
substituting values and solving, 316–317
Fourth root, 91, 117
Fractional engineering drawing, 97
Fractional exponents, 94, 261–262
Fractional-inch rules, 193–194
Fraction key, 40
Fractions. See also Decimal fractions
addition of, 8–11
calculator computing, 40–43, 45–47
common, 2
complex, 3, 36
defined, 1, 2–3
division of, 27–29
equivalent, 3–4, 8
expressing as equivalent fractions, 3–4
fractional parts, 1–2
improper, 3
mixed numbers, 3, 4
multiplication of, 20–22
overview, 1
proper, 3
reducing to lowest terms, 3
roots of, 92
spreadsheet, computing with, 50–51, 54–56
subtraction of, 14–16
writing decimal fractions from, 60
Front-clearance-angle applications, computing
true angles in, 600–601
Frustums of pyramids and cones, 475–476
volumes of, 476–477
G
Gage blocks
customary and metric, 234–236
determining gage block combinations, 236
G-codes, 655–656 , 683–684
circular motion, 661–664
linear motion, 657–661, 663–664
Gears
calculations, 337–338
definitions, 335–336
gearing-diametral pitch system, 336–337
gearing-metric module system, 338–339
overview, 334–335
Geometric constructions
applications, 420
bisecting given angle, 423
common marking tools, 420
dividing line segment into equal parts, 425Index 723
line parallel to line, 422–423
overview, 420
perpendicular bisector of line segment, 421
perpendicular to line segment, 421–422
tangents to circle from outside point, 424
Geometric principles
application, 370
for circles, 401–404
arcs, angles, and tangents, 409–414
overview, 401–404
overview, 372–375
for triangles and polygons, 387–393
Geometry. See Plane geometry
Gigabyte, 639
Great circle, 481
Greatest common factor (GCF), 4
Greatest possible error, 180
Grinders
cutting speed, 324, 325
cutting times, 327
revolutions per minute, 326
H
Half-line, 348
Hectometer (hm), 170
Height
of cone, 474, 475
of cylinder, 466, 467
of parallelogram, 441
of prism, 465, 467
of pyramid, 473, 475
Height gages, 203
digital, 215–216
metric vernier, 213
Hemisphere, 482
Hero’s (Heron’s) formula, 450–451
Hexadecimal numeration system
converting binary numbers to, 645–646
converting decimal numbers to, 647
converting to binary numbers, 646
converting to decimal numbers, 646–647
overview, 645
Hexagon, regular, 392
High amplification comparators (mechanical,
optical, pneumatic, electronic), 179
Holes
distance between v-slots and, 522–523
drilling and boring compound-angular
using given angles, 573–576
using given lengths, 567–569
Hypotenuses, 381, 495, 542
I
I-code, 656
Improper fractions
defined, 3
as mixed numbers, 4
mixed numbers as, expressing, 4
Inch (in), 85, 161
cubic, 462
square, 438
Incremental positioning (incremental coordinates), 622, 626–627
Index, 91
Infinity, 507
Inscribed angle, 400
Inside micrometer, 219
Intercepted arc, 411
Interference, 185
Internally tangent circles, 413
Inverse functions, 499–500
Inverse proportions, 136–138
Involute curve, 335
Isosceles triangles, 380, 390–391
applications, 522–523
J
J-code, 656
K
K-code, 656
Kilobyte, 639
Kilometer (km), 170
L
Lateral edges
of prism, 464–465
of pyramid, 473
Lateral faces
of prism, 464
of pyramid, 473
Lateral surface
of cone, 474
of cylinder, 466
Lathe
cutting speed, 325
cutting times, 326–327
revolutions per minute, 326
Law of Cosines, 549–552
Law of Sines, 546–549
Legs, of triangle, 380, 381
Lengths
computing angles of rotation and tilt using
given, 567–569
computing true, 565
Like terms, 267
Limits, 184
Linear interpolation, 622
Linear measurement, 160. See also Customary
units of measure; Metric units of measure
Linear motion, 657–661
combined with circular motion, 663–664
Lines, 348–349
oblique, 349
parallel, 348, 373
to given line, 422–423
perpendicular, 348
Line segment, 348
dividing into equal parts, 425
perpendicular bisector of, 421
perpendicular to, 421–422
Literal factors, 267
Literal numbers, 244
Location of points
polar coordinate system, 631–632
three-axis Cartesian coordinate system, 618
two-axis Cartesian coordinate system, 611–612
Lowest common denominators, 8–11
Lowest terms, fraction in, 3
M
Machine applications of trigonometry
complex practical, 529–535
isosceles triangle, 522–523
method of solution, 519–520, 529–535
overview, 529
sine bars and sine plates, 520–521
tangents to circles, 523–524
tapers and bevels, 521–522
Machining centers, 610, 622
Major arc, 400
Marking tools, 420
Mating parts, fits of, 185–186
M-codes, 656 , 684
Mean dimension, 184
Means, of proportion, 128
Measurements. See also Customary units of measure;
Metric units of measure; Plane geometry
defined, 160
degree of precision, See Degree of precision
greatest possible error, 180
Measuring instruments, limitations of, 178–179
Megabyte, 639
Meter (m), 170
cubic, 463
square, 439
Metric micrometers, 212–213, 227
Metric rules, 195
Metric system, 169
expressing equivalent units within, 170–171
prefixes, 170
Metric units of measure. See also Equivalent units
of measure
angular measure, 496
area, 438–440
cutting speeds, 327
cutting times, 328
gage blocks, 234–235
determining gage block combinations, 236
metric module system, 338–339
micrometer, 226–229
revolutions per minute, 328
steel rules, 195
vernier calipers, 212–213
vernier height gages, 213
vernier micrometer, 227–228
reading, 228–229
volume, 463–464
Metric vernier calipers, 212–213
Micrometer depth gage, 219
Micrometers, 179
customary, 218–222
0.001-inch, 219–221
vernier (0.001-inch), 221–222
digital, 231–232
metric, 212–213, 226–229
reading, 227
Microsoft Excel, xi, 50
Mile, 161
Millimeter (mm), 170
area, 439
cubic, 463
Milling machines, 621–622
cutting speed, 324, 325
cutting times, 327
revolutions per minute, 326
Minor arc, 400
Minutes
adding angles expressed as, 356
dividing angles expressed in, 358
expressing as decimal degrees, 353–355
expressing decimal degrees as, 350–353
multiplying angles expressed in, 357–358
relation to degrees and seconds, 350
subtracting angles expressed in, 356–357
Miscellaneous (auxiliary) functions, 656
Mixed numbers, 3
addition of, 8–11
calculator computing, 43–47
division of, 28–29
expressing as improper fractions, 4
expressing as percents, 142
expressing improper fractions as, 4
multiplication of, 22
spreadsheet, computing with, 51–53, 54–56
subtraction of, 15–16
Module, of gear, 338
Multiplication
algebraic operations, 270–271
of angles, 357–358
combined operations, 35–36
of compound numbers, 165, 167
of decimal fractions, 74–75
decimals, 113–115
FOIL method, 271724 Index
Multiplication (continued)
of fractions, 20–22
of linear units, 164
of mixed numbers, 22
in order of operations, 33
of signed numbers, 257–258
using scientific notation, 281–282
Multiplication principle of equality, 306–307
Multiplication sign, 245
Multiplicative inverse, 27
Multiplier, 161
N
Naming angles, 371
Negative key, 260
Negative numbers, 254
addition of, 255–256
Nibble, 639, 645
Nonterminating (repeating) decimal, 64
Number of teeth, 337
Numbers. See also Compound numbers; Decimal
numbers; Mixed numbers; Negative
numbers; Positive numbers; Signed
numbers
arithmetic, 244
degree of precision of, 179–180
denominate, 146, 161
literal, 244
Number scale, 254
Numeration systems
binary, 638–641
binary coded decimal, 650–653
hexadecimal, 645–647
Numerator, 2
Numerical coefficient, 267
Numerical control (NC) machines, 610
O
Oblique lines, 349
Oblique prism, 465
Oblique rectangular prism, 465
Oblique triangles
Law of Cosines, 549–552
Law of Sines, 546–549
overview, 546
practical machine shop problems, 556–557
Oblique triangular prism, 465
Obtuse angle, 371
Opposite side, of right triangle, 495
Order of operations
algebraic expressions, 247–248
for combined operations, 33–36, 98–100,
247–248, 278
Order of terms, in ratios, 127
Ordinate, 542
Origin, Cartesian coordinate system, 542
Outside diameter, 335, 336, 339
P
Parallel lines, 348, 373
Parallelogram, 392, 441
area of, 441–442
Parentheses
to indicate multiplication, 245
order of operations and, 33
removal of, 277
use with powers, 85–86
PEMDAS (parentheses, exponents, multiplication,
division, addition, subtraction), 33, 99,
247, 278
Percent
applications, 150–153
defined, 141–142
expressing as common fractions, 143
expressing as decimal fractions, 142–143
expressing common fractions as, 142
expressing decimal fractions as, 142
expressing mixed numbers as, 142
finding, given base and percentage, 147
finding base, given percentage and, 148
finding in practical applications, 151
Percentages
complex practical applications, 152–153
defined, 145
finding, given base and rate, 146–147
finding base, given percent and, 148
finding in practical applications, 151
finding percent, given base and, 147
identifying, in practical applications, 150–151
simple problems, 145–146
Percent symbol (%), 141
Perpendicular, to line segment, 421–422
Perpendicular bisector of line segment, 402, 421
Perpendicular lines, 348
Pi (π), 86, 247, 454–455
Pi (π) key, 336
Pinion, 335
Pitch, 336
circular, 336, 339
diametral, 335
Pitch circles, 335
Pitch diameter, 335, 336, 339
Place values
of binary numbers, 640
of decimal numbers, 639
Plane, 347
Plane geometry. See also Areas; Circles; Polygons;
Protractors; Triangles; Volume
angles, 349
adding, 356
dividing, 358
multiplying, 357–358
subtracting, 356–357
arithmetic operations on angular measure, 355
axioms, 348
decimal degrees, 350–355
lines, 348–349
overview, 347
points, 348–349
postulates, 348
units of angular measure, 349–350
Pocket scriber, 420
Point of tangency, 400
Points, 348–349
on three-axis Cartesian coordinate system,
612–614
Point-to-point control systems, 622
Polar coordinate system, 631–632
Polygons
defined, 388
other than triangles, 391–393
Polyhedron, 464
Positioning
part on angle plate for drilling, 569
part on angle plate for machining, 583–584
systems, tool, 622
Positive numbers, 253–254
addition of, 255–256
powers of, 259–260
roots of, 260–261
sign of, 253
square root of, 105
subtraction of, 256–257
Postulates, 348
Power principle of equality, 308
Powers
algebraic operations, 276
calculator computing, 104–105
defined, 84
description of, 84–85
of signed numbers, 259–260
spreadsheet computing, 115–116
use of parentheses, 85–86
Powers of 10, 57
and writing decimal fractions from fractions, 60
Precision gage blocks (comparison measurement), 179
Prefixes, metric system, 170
Preparatory commands, 656. See also G-codes
Principal cube root, 260
Principal square root, 260
Principles of equality
addition, 298–299
division, 299–300
multiplication, 306–307
overview, 295
power, 308
root, 307–308
subtraction, 295–296
transposition, 296–297
Prisms, 464–465
volumes of, 465–466
Programming, CNC
control systems, 621–622
overview, 611
Proper fraction, 3
Proportions, 319–320
defined, 128
description of, 128–129
direct, 135–136
expressing, 128
inverse, 136–138
Protractors
bevel, with vernier scale, 366–367
complementary and supplementary
angles, 367
overview, 363
simple semicircular, 363–366
Punch, center, 420
Pyramids, 473–474
frustums of, 475–477
volumes of, 474–475
Pythagorean Theorem, 390, 403, 450
Q
Quadrants, 542
determining functions of angles in any,
542–544
Quadrilaterals, 391
Quotient, 27
R
Radicals, 91
expressing numbers with fractional exponents
as, 261–262
Radical sign, 91, 92
expressions enclosed within, 92–93
Radicand, 91
Radius
of circle, 400
of sphere, 481
Raised dot, 245
Rate (percent), 145
finding, given base and percentage, 147
finding base, given percentage and, 148
finding in practical applications, 151
finding percentage, given base and, 146–147
identifying, in practical applications, 150–151
Ratio method, 495–496
Ratios
defined, 126
description of, 126–127
expressing, 127
in lowest terms, 128
order of terms, 127
of right triangle sides, 495Index 725
Ray, 348
in circle, 349
Reciprocal, 27
Rectangles, 392, 440
area of, 440–441
Rectangular coordinate system. See Cartesian
coordinate system
Rectangular solid, diagonal of, 564–565
Reducing fractions, 3
Reference angle, 495, 542
Reflex angle, 371
Regular hexagon, 392
Regular hexagonal pyramid, 474
height and base of, 475
volume of frustums of, 475–477
volumes of, 474–475
Regular polygon, 391
Regular pyramid, 474
Regular quadrangular pyramid, 474
Regular triangular pyramid, 474
Relative error, 180–181
Repeating (nonterminating) decimal, 64
Revolutions per minute
cutting speed table and, 328
using customary units of measure, 325–326
using metric units of measure, 328
Right angle, 371
Right circular cone, 474
height and base of, 475
volume of frustums of, 475–477
volumes of, 474–475
Right circular cylinder, 466
Right hexagonal prism, 465
Right prism, 465
Right rectangular prism, 465
Right triangles, 381, 390–391
identifying sides by name, 495
ratio of sides, 495
unknown sides and angles of, 511–515
Root circle, 335
Root diameter, 335, 336
Root principle of equality, 307–308
Roots
algebraic operations, 277
calculator computing, 105–106
defined, 91
description of, 91–92
expressions enclosed within radical symbol,
92–93
fractional exponents, 94
of fractions, 92
that are not whole numbers, 93–94
whole number, 93
Rounding, decimal fractions, 63–64
r value, 631
S
Scalene triangle, 380
Scale readings, complements and supplements
of, 367
Scientific notation, 278–282
division using, 281–282
expressing as decimal numbers, 279–280
expressing decimal numbers in, 278–279
multiplication using, 281–282
Screw thread micrometer, 219
Scroll key, 354
Secant (sec), 495–496, 497–498
Secants, 400, 412–413
Seconds
adding angles expressed as, 356
dividing angles expressed in, 358
expressing as decimal degrees, 353–355
expressing decimal degrees as, 350–353
multiplying angles expressed in, 357–358
relation to degrees and minutes, 350
subtracting angles expressed in, 356–357
Sectors, 400, 455
area of, 455
Segment of circle, 400, 456
area of, 456
Semicircle, 400
Sides
corresponding, 382
determining unknown, of right triangles, 511–515
identifying by name, of right triangles, 495
using Law of Cosines, 549–552
using Law of Sines, 546–549
Signed numbers
absolute value, 254–255
addition of, 255–256
combined operations, 262
division of, 258
meaning of, 253–254
multiplication of, 257–258
number scale, 254
numbers with fractional exponents as radicals,
261–262
operations using, 254–255
overview, 253
powers of, 259–260
roots of, 260–261
subtraction of, 256–257
Signs of operation, 253
Similar figures, 388
Similar polygons, 388
Similar triangles, 388–390
Simple semicircular protractors, 363–366
Sine (sin), 495–496, 497
Sine bar, 520–521
Sine plate, 520–521
Slash ( ⁄ ), 1
Small circle, 481
Spheres, 481–482
volume of, 482
Spreadsheet
combined operations, 54–56
decimal computing, 112–115
to express degrees/minutes and seconds as
decimal degrees, 352–353
formulas in, 316–317
fractions, entering/computing in, 50–51, 54–56
mixed numbers, computing in, 51–53, 54–56
numerical values in, 316–317
overview, 50
powers, computation of, 115–116
scientific notation in, 280
square root computation using, 117
and trigonometric functions, 498–499
Spur gears
definitions, 335–336
gear calculations, 337–338
gearing-diametral pitch system, 336–337
gearing-metric module system, 338–339
overview, 334–335
SQRT command, 117
Square, 391
Square centimeter, 439
Square foot, 438
Square inches, 438
Square key, 104
Square meter, 439
Square millimeter, 439
Square roots, 91
spreadsheet computing, 117
Square yard, 438
Steel rules, 179
correct procedure in use of, 192–193
customary, 192–195
decimal-inch, 194–195
fractional-inch, 193–194
metric, 195
Straight angle, 371
Subtraction
algebraic operations, 269
of angles, 356–357
combined operations, 34, 36
of compound numbers, 165, 166–167
of decimal fractions, 69, 70–71
decimals, 113–115
of fractions, 14–16
of linear units, 164
of mixed numbers, 15–16
in order of operations, 33
of signed numbers, 256–257
Subtraction principle of equality, 295–296
Supplementary angles, 367, 374
Surface measure. See Areas
Surfaces
angles made by intersection of two angular,
590–594
machining compound-angular, 581–586
Surface speed. See Cutting speeds
Symbolism, 244–246
Symbols
arc, 400
customary units of linear measure, 161
degrees, 349
feet, 85
inches, 85
indicating fractions, 1
multiplication, 245
second, 350
T
Table of decimal equivalents, 97–98
Tangent (tan), 495–496, 497
Tangent circles, 412–413
externally, 413, 414
internally, 413
Tangent point, 400
Tangents to circles
angles formed on circle, 411
angles formed outside circle, 411–413
constructing from outside point, 424
defined, 400
geometric principles, 401–404
trigonometry applications, 523–524
Tapers, 521–522
Teeth, number of, 337
Terabyte, 639
Terminating decimal, 64
Terms
of algebraic expression, 267
of fraction, 2
like, 267
of ratio, 127
unlike, 267
Theta (θ) values, 631
Threat wire checking dimensions, 523–524
Three-axis Cartesian coordinate system, 612–614
Tolerance
bilateral, 184–185
defined, 184
fits of mating parts, 185–186
overview, 184
unilateral, 184–185
Tool positioning (coordinate) systems, 622
Tooth thickness, 335, 336, 339
Trammels, 420
Transistors, 638–639
Transposition, 296–297
Transversal, angles formed by, 372
Trapezoid, 442
area of, 442–443
Triangles. See also specific types of triangles by
name
angles of, 381726 Index
Triangles (continued)
area of, 449–451
corresponding parts of, 382
geometric principles for, 387–393
congruent triangles, 387
equilateral triangles, 390–391
isosceles triangles, 390–391
right triangles, 390–391
similar triangles, 388–390
overview, 380
types of, 380–381
Trigonometric functions. See also Oblique
triangles
Cartesian coordinate system, 542–544
of complementary angles, 507–508
customary units of angular measure, 496
determining, 497–500
metric units of angular measure, 496
overview, 494
ratio method, 495–496
right triangles
identifying sides by name, 495
ratio of sides, 495
unknown sides and angles of, 511–515
spreadsheet and, 498–499
variation of, 506–507
Trigonometric keys, calculator, 497, 498
Trigonometry, 494. See also Compound angles;
Machine applications of trigonometry;
Trigonometric functions
Turning centers, 610
Two-axis Cartesian coordinate system, 611–612
U
Unilateral tolerance, 184
expressing as bilateral tolerance, 184–185
Units of angular measure, 349–350
Unity fraction method, 161
Unity (unit circle) method, 495
Universal power key, 104, 259, 261
Unknown quantity, 288
Unlike terms, 267
V
Values, substituting, 316–317
Variation of functions, 506–507
V-blocks, 523–524
Vernier calipers, 179, 203
metric, 212–213
overview, 204
reading and setting measurement on, 204–206
Vernier height gages
decimal-inch, 206–207
metric, 213
Vernier micrometers
customary, 221–222
reading and setting, 221–222
metric, 226–229
overview, 227–228
reading, 228–229
Vernier protractors, 366–367
Vertex
of angle, 349
of cone, 474
of pyramid, 473
Volume
composite solid figures, 482–483
cones, 474–475, 476–477
customary units of measure, 462–463, 464
cylinders, 466–467
metric units of measure, 463–464
overview, 462, 473
prisms, 465–466
pyramids, 474–475, 476–477
spheres, 482
V-slots, distance between holes and, 522–523
W
Whole depth, 335, 336, 339
Whole numbers
addition of, 11
roots that are not, 93–94
Word lengths, 639
Word statements, writing equations from,
288–291
Working depth, 335, 336, 339
Wringing blocks, 234
X x
-axis, 542, 612–613
Y
Yard, 161
cubic, 462
square, 438
y-axis, 542, 612–613
Z z
-axis, 612–613
Zero
addition of, 70
division of, 258
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