Mathematics for Machine Technology

Mathematics for Machine Technology
اسم المؤلف
Robert D. Smith
التاريخ
9 يوليو 2018
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Mathematics for Machine Technology
Fourth Edition
Robert D. Smith
CONTENTS
Preface ix
SECTION 1 Common Fractions, Decimal Fractions, and Percentage 1
UNIT 1
UNIT 2
UNIT 3
UNIT 4
UNIT 5
UNIT 6
Introduction to Common Fractions and Mixed Numbers
Addition of Common Fractions and Mixed Numbers
Subtraction of Common Fractions and Mixed Numbers
Multiplication of Common Fractions and Mixed Numbers
Division of Common Fractions and Mixed Numbers
Combined Operations of Common Fractions and
Mixed Numbers
Computing with a Calculator: Fractions and Mixed Numbers
Introduction to Decimal Fractions
Rounding Decimal Fractions and Equivalent Decimal and
Common Fractions
Addition and Subtraction of Decimal Fractions
Multiplication of Decimal Fractions
Division of Decimal Fractions
Powers
Roots
Table of Decimal Equivalents and Combined Operations of
Decimal Fractions
Computing with a Calculator: Decimals
Introduction to Percents
Basic Calculations of Percentages, Percents, and Rates
Percent Practical Applications
Achievement Review—Section One
16
11
15
19
22
UNIT 7
UNIT 8
UNIT 9
27
32
36
UNIT 10
UNIT 11
UNIT 12
UNIT 13
UNIT 14
UNIT 15
41
45
47
51
57
62
UNIT 16
UNIT 17
UNIT 18
UNIT 19
UNIT 20
68
74
77
82
88
SECTION 2 Linear Measurement: English and Metric 95
UNIT 21 English and Metric Units of Measure. .
UNIT 22 Degree of Precision, Greatest Possible Error, Absolute Error,
and Relative Error
UNIT 23 Tolerance, Clearance, and Interference
UNIT 24 English and Metric Steel Rules
UNIT 25 English Vernier Calipers and Height Gages
UNIT 26 Metric Vernier Calipers and Height Gages
UNIT 27 English Micrometers
VVi MATHEMATICS FOR MACHINE TECHNOLOGY
UNIT 28 Metric Micrometers 141
UNIT 29 English and Metric Gage Blocks
UNIT 30 Achievement Review— Section Two
SECTION 3 Fundamentals of Algebra 155
UNIT 31 Symbolism 155
UNIT 32 Signed Numbers
UNIT 33 Algebraic Operations of Addition, Subtraction,
and Multiplication
UNIT 34 Algebraic Operations of Division, Powers, and Roots
UNIT 35 Introduction to Equations
UNIT 36 Solution of Equations by the Subtraction, Addition,
and Division Principles of Equality
UNIT 37 Solution of Equations by the Multiplication, Root,
and Power Principles of Equality
UNIT 38 Solution of Equations Consisting of Combined Operations and
Rearrangement of Formulas
UNIT 39 Ratio and Proportion
UNIT 40 Direct and Inverse Proportions
UNIT 41 Applications of Formulas to Cutting Speed, Revolutions
Per Minute, and Cutting Time
UNIT 42 Applications of Formulas to Spur Gears
UNIT 43 Achievement Review—Section Three
SECTION 4 Fundamentals of Plane Geometry 258
UNIT 44 Introduction to Geometric Figures
UNIT 45 Protractors—Simple and Vernier
UNIT 46 Angles
UNIT 47 Introduction to Triangles
UNIT 48 Geometric Principles for Triangles and Other Common
Polygons
UNIT 49 Introduction to Circles
UNIT 50 Arcs and Angles of Circles
UNIT 51 Fundamental Geometric Constructions
UNIT 52 Achievement Review—Section Four
SECTION 5 Trigonometry 333
UNIT 53 Introduction to Trigonometric Functions
UNIT 54 Analysis of Trigonometric Functions
UNIT 55 Basic Calculations of Angles and Sides of Right Triangles
UNIT 56 Simple Practical Machine Applications
354CONTENTS Vii
UNIT 57 Complex Practical Machine Applications
UNIT 58 The Cartesian Coordinate System
UNIT 59 Oblique Triangles: Law of Sines and Law of Cosines
UNIT 60 Achievement Review
—Section Five
SECTION 6 Compound Angles 390
UNIT 61 Introduction to Compound Angles
UNIT 62 Drilling and Boring Compound-Angular Holes: Computing
Angles of Rotation and Tilt Using Given Lengths
UNIT 63 Drilling and Boring Compound-Angular Holes: Computing
Angles of Rotation and Tilt Using Given Angles
UNIT 64 Machining Compound-Angular Surfaces: Computing Angles
of Rotation and Tilt
UNIT 65 Computing Angles Made by the Intersection of Two
Angular Surfaces
UNIT 66 Computing Compound Angles on Cutting and Forming Tools . . . . 418
UNIT 67 Achievement Review
—Section Six
SECTION 7 Computer Numerical Control (CNC) 428
UNIT 68 Introduction to Computer Numerical Control (CNC)
UNIT 69 Control Systems, Absolute Positioning, Incremental
Positioning
UNIT 70 Binary Numeration System
UNIT 71 Achievement Review
—Section Seven
Appendix
Answers to Odd-Numbered Applications .
Index 477INDEX
signed numbers in, 162-171
signs of operation in, 155, 163-171
for spur gears, 246-251
substituting values in, 218-219
subtraction in, 177-178, 200-201
symbolism in, 155-158
terms vs. factors in , 176
transposition in, 201-202
unknown quantity in, 193
word statements and, 193-196
Algebraic expressions, 155-158, 179
Allowance, 109
steel rules and , 115
vernier height gage and, 122
Alternate interior angles, 278, 279* 280
Altitude, 292-293, 356
Ambiguous cases, 376-377
American National Standards Institute
(ANSI), 95
Angle cuts, 357-358
Angles, 259-280
acute, 277, 347, 349-351, 372
adjacent, 278
alternate interior, 278. 279, 280
base, 284-285, 292, 356
bisecting, 322-323
calculator and, 262-266, 335-338,
347-351
coordinates and, 371-373. See also
Cartesian coordinate system
central, 301, 303. 309-311
of circles, 309-310
complementary, 274, 279, 344
compound, 390-422
corresponding. See Corresponding
angles
English measurement of, 260, 261,
335, 347
formed on circles, 311
formed outside circles, 312
functions and, 335-338, 342-385,
See also Cartesian coordinate
system
geometric principles and, 278-280
increasing, 342-344
inscribed, 301, 302, 310
mean, 337-338
metric measurement of, 260,
261-263, 335, 347
naming, 277
obtuse, 277
of polygons, 294
protractors and, 271-274
reference, 334, 372
reflex, 277
right, 277
of right triangles, 347-351 . See
also Right triangle
of rotation , 393—415
sine bar/plate and, 355
straight, 277
supplementary, 274, 279
taper, 355-356
of of triangles tilt, 393—, 415 285-286
trigonometry and, 333-385. See
also Right triangle;
Trigonometry
true, 391-392, 418-419
types of, 277 278
units of measure of, 260-261
vertex of, 271, 292, 372
ANSI. See American National
Standards Institute (ANSI)
Anvil micrometers, 135
Arc, 301, 303, 309-313
intercepted, 309-312
trigonometry and, 337-338,
363-367
Arccos, 337-338
Arcsin, 337-338
Arctan, 337-338
Area measurements, 95, 157-158. See
also Powers; Roots
Arithmetic
algebra and, 155, 179, 182-184
in angular measurement, 263-266
Arithmetic numbers, 155
Auxiliary lines. 254-257, 361-367 ,
398-401 , 405 422
Axes, 371-373. See also Cartesian
coordinate system
Axioms, 258 259
A
Abscissa, 372
Absolute coordinates, 433-436
Absolute error, 104-105
Absolute positioning, 432-436
Absolute value, 163-171
Acute angle, 277, 347, 349-351, 372
Acute triangle. 375-378
Addendum, 246
Addition
in algebra, 157, 176-177, 185
in angular measurement., 263-264
on calculator, 28-30, 68
in combined operations, 23-24
of decimal fractions, 41, 68
of fractions, 8, 28 – 30
of fractions and mixed and whole
numbers, 9
of mixed numbers, 9, 28-30
in order of operations, 63
powers and. 52
of signed numbers, 164—165
Addition principles of equality,
202-203
Adjacent angles, 278
Adjacent side (in triangle), 334,
347-351 , 372
Algebra, 155-251
addition in, 176-177, 202-203
combined operations in , 185-186,
216-218
combined with trigonometry, 333
common letter designations in, 193
for cutting speed, 237-242
for cutting time, 238 -242
definitions in, 176
direct proportions in, 232-233
division in, 182-183, 203-204
ratios in, 224-225
equations in, 192-251
formulas in, 218—221. See also
Formulas
for gears, 246-251
inverse proportions in, 233—234
multiplication in, 178-179, 209
parentheses in, 185
powers in, 183- 184, 210-211
principles of equality in, 199-204
proportions in, 226-227, 231-234
rearranging formulas in, 219-221
for revolutions per minute, 238-242
roots in, 184 -185, 209-210
scientific notation in, 186-188
B
Base (in percentage problem), 77-84
Base angles, 284-285, 356
isosceles triangle and, 292
Bench micrometers, 134
Bevel protractor, 273-274
Bevels, 355-356
Bilateral tolerance, 107-109
Binary numeration system, 439-442
Bisecting
of angles, 322-323
of circles, 303, 304, 321
of triangles, 292-293
477478 MATHEMATICS FOR MACHINE TECHNOLOGY
Bisector, perpendicular, 320-321 Corresponding angles, 278 » 279, 286,
Bit, 440
Boring See Drilling/boring holes
Bow micrometers, 135
Brackets, 63, 157
Bureau of Standards, 95
Byte. 440
Circular tooth thickness, 247-251
Circumference, 301, 301- 305
in arc length formula, 310-311
Clearance, 108-109, 247-251
cutting/forming tools and, 420-422
CNC. See Computer numerical
control (CNC)
Cofunctions, 335-338, 342-385
Combined operations
in algebra, 185-186, 216-218
on calculator, 28, 30-31, 71-74
of common fractions and mixed
numbers, 22-25
of decimal fractions, 63-65
powers and, 52
of signed numbers, 170-171
Common factors, dividing by, 15-16
Comparison measurement, 103
Complementary angles, 274, 344
Complex fractions, 2, 24—25
Compound angles, 390—422
angles of rotation/tilt and, 393—4 ) 5
clearance and, 420-422
cutting/forming tools and, 418-422
diagonal of rectangular solid and,
390-391
drilling/boring holes and, 393—401
holes and, 393-401
machining surfaces and, 404-409
made by intersecting surfaces,
surfaces 412and —415 , 404-422
formulas for, 390. See also
Formulas
true lengths/angles and, 391-392,
412^22
Computer aided drafting/design
(CAD), 439
Computer aided manufacturing
(CAM), 439
Computer integrated manufacturing
(CIM), 439
Computer numerical control (CNC),
162, 428-442
absolute binary numeration positioningsystem in, 432in—,436
439-442
incremental positioning in, 432,
436
—437
language-based vs. graphic-based
programs for, 428-429
programming three-axis system for, in 428 , 429 —429, 432
two-axis system in , 429 —430
Computers, bits and bytes in, 440
Congruent triangles, 290
Continuous path, 432
Coordinates
absolute, 433-436
incremental, 436-437
See also Cartesian coordinate
system
291
congruent triangles and, 290
Corresponding sides, 286, 291- 292
Cos. See Cosine (cos)
Cosecant, 334, 336-338, 342-385
Cartesian coordinate system and,
373
Cosine (cos), 334, 335-338, 342-385
Cartesian coordinate system and,
372-373
Law of Cosines and, 378 380
Cotangent (tan), 334, 336-338,
342-385
Cartesian coordinate system and,
CAD. See Computer aided
drafting/design (CAD)
Calculator
angular measurement on, 262-266,
375- 338, 347-351
clearing, 28
combined operations and, 28,
30-31, 71-74
decimals and, 68-74
fractions and, 28-30
mixed numbers and, 27-30
positive vs. negative numbers on ,
164—171, 186-187
powers and, 69, 167-168
roots and, 59, 70, 169-170
scientific notation and, 187-188
trigonometry on , 335-338, 347-351
Calipers
dial, 103
gage blocks and, 146
vernier, 103, 121-124, 129-131
CAM. See Computer aided
manufacturing (CAM)
Cancellation , 15- 16
Cartesian coordinate system , 371-373
absolute positioning and, 432- 436
and computer numerical control
(CNC), 428, 429^37
incremental positioning and, 432,
436
—4-37
Law of Cosines and, 378, 380
Centimeters, 97-98, 99
Central angle. 301 , 303, 309-311
Chord, 301, 303, 305. 310-312
perpendicular bisector of , 321
CIM. See Computer integrated
manufacturing (CIM)
Circle, 301-315
angle formed on, 311
angle formed outside, 312
angles of, 260, 309-310
area of, 157-158
definitions involving, 301
degrees and, 260. See also Degrees
polygon inscribed in , 302
tangent circle to another, 301, 304,
313-315, 366-367
tangent lines to, 323-324
Circular pitch. 247-251
Circular sections, computing areas of,
51, 57. See also Arcs; Powers;
Roots
373
Cross multiplying , 226
Cutting speed, 237-242
Cutting time, 238-242
Cntting/forming tools, 418-422
D
Decimal degrees, 260, 261-263
Decimal-inches
on steel rules, 117
on vernier calipers, 121-124
on vernier height gages, 121-122,
124-125
Decimals, 32-49
addition of , 41
binary numbers and , 439 442
on calculator, 68-74
combined operations of, 63-65
division of, 47—49, 69
equivalent, 62 – 63
expressing as common fractions, 38
expressing as percents, 74-75
expressing common fractions as,
33, 37-38
expressing percents as, 75
multiplication of, 45-46, 69
reading and writing, 32-33
rounding, 36-38, 68
in scientific notation, 186-188
on steel rules, 117
structure of system of, 440
subtraction of , 42
terminating vs. nonterminating, 37
Decimeters, 97-98
Dedendum, 246
Degrees, 260-266
in arc length formula, 3 ) 0-311
calculator and, 335 338
protractors and, 271-274
trigonometry and, 335-338,
342-385
See also Angles
Degrees of precision, 103-104, 320
angles and, 260
gage blocks and, 146
pi and, 302INDEX 479
for cutting time, 238-242
definition of, 192
for front-clearance-angle
applications, 421—422
for gears, 246-25 i
Law of Cosines and, 378-380
Law of Sines and, 375-378
for percentages, 78-84
rearranging , 175, 219-221, 248, 380
for revolutions per minute, 238-242
for spur gears, 246-251
square roots and, 57, 58
substituting values in , 218-219, 248
iSee also Equations
Fraction bar, 63, 157
Fractional degrees, 260. See also
Degrees
Fractional exponents, 170-171. See
also Exponents
Fractions, 1-49
addition of, 6-11, 28-30
calculator and, 28-30
combined operations involving,
22-25
common, 1-38
complex, 2, 24-25
decimal. See Decimals
definitions of , 2
division of, 19-20, 28-30
equivalent, 3, 7-8
expressing as percents, 75
expressing common fractions as
decimal, 33, 37-38
expressing decimal fractions as
common , 38
expressing percents as common, 76
improper, 2, 3^1
inverting, 19-20
multiplication of, 15-17, 28-30
reduced to lowest tenns, 3
roots of, 58
subtraction of, 11- 12, 28-30
Functions
angles and , 335-385. See also
Cartesian coordinate system
Co-, 335-338, 342-385
of complementary angles, 344
inverse, 337-338
in trigonometry, 335-338, 342 – 385
variations of, 342-344
Equal sign, 192-193
Equality
expression of, 192-193
principles of, 199-204, 209-211
Equations, 175, 192-251
checking, 196
combined operations in, 216-218
definition of, 192
and principles of equality,
199-204, 209-211
solving, 199-204
transposition and, 201-202
writing from word statements,
193- 196
See also Formulas
Equidistant lines, 259
Equilateral triangle. 285, 292 -293
Equivalent decimals, 62-63
Equivalent fractions, 3, 7-8
Errors (in measurement), 104-105
gage blocks and, 147
Exponents, 51
algebra and, 176-179, 182-184
fractional, 170-171
positive vs. negative, 167-171,
186-187, 440
scientific notation and, 186-188
Extremes, 226
steel rules and, 115
vernier calipers and, 121, 129
Denominate numbers, 78, 95
Denominator, 2
cancellation and, 15-16
least common, 6-8
in parentheses, 22, 52, 63
Depth gage attachment, 124
Depth micrometers, 134, 135
Diagonals, 390 392
Dial calipers, 103
Dial indicators, 103, 124, 129
gage blocks and, 146
Diameter, 246, 247-251
circumference and, 302
definition of, 301
Diametral pitch, 247-251
Disc and blade micrometers, 134
Dividers, 320
Division
in algebra, 157, 182-183, 185, 187
in angular measurement, 265-266
on calculator, 28-30, 69
in combined operations, 23-24
by common factors, 15-16
of decimal fractions, 47-49, 69
of fractions, 19-20, 28-30
of mixed numbers, 20, 28-30
in order of operations, 63
powers and, 52
scientific notation and, 187
of signed numbers, 167
Division principles of equality,
203-204
Dovetails, 357-358
Drill press, culling speed of, 237-239
Drilling/boring holes, 393-401
Dual dimensioning, 98
F
Factors
in algebra, 176-179, 182-185
conversion, 99
dividing by common, 15-16
literal , 176 -179, 182-184
powers and, 51
Feet, 96-97, 99
Flame cutting, 428
Forming tools. See Cutting/forming
tools
Formulas, 51
algebraic, 155-158
for angle of rotation in relation to
frontal plane used in drilling,
399
—401
for angle of rotation in relation to
frontal plane used in
machining, 408^109
for angle of rotation/tilt. 399—415
for angle of tilt used in drilling,
399-401
for angle of tilt used in machining,
408-409
for angles of intersecting angular
surfaces, 414-415
arc length, 310-311
circumference, 302-305
compound angles and, 390,
399-L22
for cutting/forming tools, 419-422
for cutting speed, 237-242
£
Electrical discharge machines, 428
Electronic comparators, 103
Ellipse, perimeter of , 158
English measurement
angular, 260, 261, 335, 347
conversion between metric and,
98-99
cutting speed using, 237-239
expressing larger units as smaller
in, 96
expressing smaller units as larger
in, 96-97
gage blocks and, 146, 147
metric vs., 95
micrometers and, 134-138
revolutions per minute using, 238
on steel rules, 115 117
units of, 96
vernier calipers and , 121-124
vernier height gages and, 121-122,
124
—125

G
Gage blocks, 146-148
trigonometry and, 355
Gages, vernier. See Vernier height
gages
Gearing-diametral pitch system,
247-249
Gearing-metric module system,
249-251480 MATHEMATICS FOR MACHINE TECHNOLOGY
auxiliary, 254-257 » 361-367,
398
—401, 405-422
dividing into equal parts, 324
geometric constructions of, 320-324
oblique, 259
parallel . See Parallel lines
perpendicular. See Perpendicular
lines
projected, 354-358. 361-367,
Literal factors 398—401 , 176 , 405 -179 -422 , 182-184
Literal numbers, 155
Location of points, 259, 429-430
Geais, 246-251
Geometric constructions, 319-324
Geometric principles, 278-280
applications of, 354-358
circles and, 303-305
Geometry, 51, 258-328
algebra and, 155, 192
angles in, 259-266
axioms and postulates in, 258-259
circles in, 301-315
combined with trigonometry, 333
plane, 258-328
points and lines in, 259
polygons in, 284-295
triangles in, 284—295
Greatest possible error, 104
Grinder
computer numerical control (CNC)
for, 428
cutting speed of , 237-239
Grouping symbols, 63, 64. See also
Fraction bar; Parentheses;
Radical symbol
Inverse functions, 337-338
Inversion , 19-20, 440
Involute curve, 246, 248
Irrational number, 302
Isosceles triangle, 284, 285, 292-293
applications of, 356-357
K
Kilometers, 97-98, 99
L
Lathe
computer numerical control (CNC)
for, 428, 429-430
cutting speed of, 237-239
Law of Cosines, 378 -380
Law of Sines , 375-378
Least common denominator, 6-8
Legs (of triangle), 284, 285
Lengths
computing angles using given ,
393-395
true, and compound angles, 390.
See also Compound angles
See also Linear measurement;
Powers; Roots; Unknown
sides/angles
Like terms, 176-179, 182-185,
225-227
Limits (in tolerance), 107-109
Linear measurement
absolute error in, 104-105
allowance in, 109, 115
basic dimension in, 107-108
clearance in, 108-109. See also
Clearance
conversion between English and
metric, 98-99
decimal-inch rules for, 117-118
definition of , 95
degrees of precision in, 103-104.
See also Degrees of precision
English, 96-97. See also English
measurement
errors in, 104-105
fit of mating parts in, 108-109
fractional-inch rules for, 116-117
gage blocks and, 146-148
greatest possible error in, 104
interference in , 109
limits in, 107-109
mating parts and, 108-109
mean dimension in, 107-108
metric, 97-98. See also Metric
measurement
relative error in, 104—105
tolerance in, 107-109. See also
Tolerance
M
Machining, of compound-angular
surfaces, 404-409
Machining centers, 428
Marking tools, 320-324
Mating parts, 108-109
Mean , 226
in angles, 337-338
in dimensions, 107-108
Measurement
absolute error in, 104-105
angular, 260-284, 335, 347. See
also Angles
calipers for, 103, 121-124, 129-131
comparison, 103
decimal-inch rules for, 117-118
definitions of, 95-96
degrees of precision in, 103-104.
See also Degrees of precision
English vs. metric, 95. See also
English measurement; Metric
measurement
fractional-inch rules for, 116-117
greatest possible error in, 104
height gages for, 121-122,
124-125, 130-131
high amplification comparators
for, 103
instruments of, 103
of length. See Linear measurement
limitations in, 103
linear. Sec Linear measurement
micrometers for, 103, 134-138,
141-143, 146
protractors for, 271-274
relative error in, 104-105
of squares. See Squares, computing
areas of
of volume. See Volumes, computing
steel rules for, 103, 115-118
vernier bevel protractor for,
273-274
vernier calipers for, 103
vernier height gages for, 121-122,
124-125. 130-131
vernier micrometers for, 137-138,
142-143
H
Hectometers, 97-98
Hexagon, 294
High amplification comparators, 103
Holes
absolute positioning system for,
433-436
compound-angular, 393-401
distance between, 356
drilling/boring, 393—401
Horizontal coordinates, 371-373. See
also Cartesian coordinate system
Hypotenuse, 285, 293, 334, 347-351
Cartesian coordinate system and,
372-373
sine bar/plate and, 355
I
Improper fractions, 2, 3-4
Inches, 96-97, 99, 347
computer numerical control (CNC)
and, 429, 430
decimal-. See Decimal-inches
degrees of precision and, 104
on steel rules, 115-1 17
Incremental coordinates, 436-437
Incremental positioning, 432, 436—437
Indicators, dial. See Dial indicators
Infinity, 344
Inscribed angle, 301, 302, 310
Inside micrometers, 134, 135
Inspection, 428
Intercepted arc, 309-312
Interference, 109
International System of Units (SI), 95
Intersecting surfaces, 412-415 Lines, 259INDEX 481
Mechanical comparators, 103
Meters, 97-98, 99
Metric measurement
angular, 260, 261-263, 335, 347
conversion between English and,
98-99
cutting speed using, 240
English vs., 95
gage blocks and, 147, 148
micrometers and, 141-143
revolutions per minute using, 240
on steel rules, 115-118
units of linear, 97
vernier calipers and, 129-131
Micrometers, 103
English, 134-138
gage blocks and, 146
metric, 141-143
Miles. 96-97, 99
Millimeters, 97-98, 99, 347
computer numerical control (CNC)
and, 429, 430, 432
Milling machine
computer numerical control (CNC)
for, 428, 429
continuous-path , 432
cutting speed of, 237-239
point-to-point, 432, 433-436
Minus sign, 163-171
Minutes (angles), 260-266
Mixed numbers
addition of, 9, 28-30
calculator and , 27-30
combined operations involving ,
22-25
definition of, 2
division of, 20, 28-30
expressed as improper fractions, 3-4
expressed as percents, 75
improper fractions expressed as, 4
multiplication of, 16-17
subtraction of, 12-13
Module system, 249-251
Multiplication
in algebra, 155, 157, 178-179, 185
m order of operations, 25, 63-65
parentheses within , 63
powers and, 51
Percentages
calculating, 77-84
practical applications involving,
82-84
types of complex problems
involving, 84
types of simple problems
involving, 77-78
Percents, 74-84
definition of, 74
expressing common fractions as, 75
expressing decimal fractions as,
74-75, 75
expressing mixed numbers as, 75
Penmeter
calculating, 157, 158
circumference and, 302
Perpendicular bisector, 320-321
Perpendicular lines, 259. 279, 292
circles and, 303, 304
geometric constructions and,
320-321
Pi
, 157 158, 302
Pinions, 246 251
Pitch, 246, 247-25 1
Pitch circles, 246
Pitch diameter, 246
Plane geometry, 258-328
circles in , 301-315
polygons in , 284-295
triangles in, 284-295
Planes
in angles of rotation/tilt., 404-415
compound angles and, 390^-09
cutting/forming tools and, 421-422
See also Surfaces
Plotting points, 259, 429^130
Plus sign , 163-171
Pneumatic comparators. 103
Point-to-point system, 432, 433-436
Points (geometric), 259
computer numerical control (CNC)
and, 429-430
Polygons, 284, 290, 294-295
inscribed in circles, 302
regular, 294, 302
similar, 290
types of, 294
Posihoning
absolute, 432-436
incremental, 432, 436 437
for machining, 407
system for hole, 433-436
system for tool , 432 437
Positive exponents, 167-171,
186′ 187. 440
Positive functions/values, 372-373.
See also Cartesian coordinate
system
N
NC . See Numerical control (NC)
Negative exponents, 167-171,
186-187, 440
Negative functions/values, 372-373.
See also Cartesian coordinate
system
Negative numbers, 163-171. 186-187
Number scale, 163-171
Numbers
arithmetic vs. literal, 155
binary, 439^142
degree of precision of, 103-104
directed, 163
inational , 302
positive vs. negative, 163-1.7 j ,
186-187
signed, 162-171
whole. See Whole numbers
Numerator, 2
cancellation and, 15-16
in parentheses, 22, 52, 64
Numerical coefficients, 176 -179,
182-184 ,
Numerical control (NC), 162, 439.
See also Computer numerical
control (CNC)
O
Oblique lines, 259
Oblique triangles, 371, 375-380
Obtuse angle, 277, 375-378
Obtuse triangles, 375-378
ON/OFF, 439-442
Opposite side (in triangle), 334,
347-351, 372
gage blocks and, 355
Law of Sines and , 375-378
Optical comparators, 103
Order of operations. 22-23, 63
in algebra, 157-158, 185-186
calculator and, 30
decimal fractions and, 63-65
powers and, 52, 63
radical symbol and, 58
roots and, 58, 63
signed numbers and, 170
Order of terms, 225
Ordinate, 372
Origin , 371-373, 433
Outside diameter, 246
187
in angular measurement, 265
on calculator, 28-30, 69
in combined operations, 23—24
cross, 226
of decimal fractions, 45-46, 69
decimal places and, 65
division as inverse of, 19-20
of fractions, 15-17, 28-30
of mixed numbers, 16-17
in order of operations, 63
powers and, 52, 65
scientific notation and , 187
of signed numbers. 166
Multiplication principles of equality,
P
Parallel lines, 259, 292
geometric constructions and, 322
transversals and, 279-280
Parallelogram, 294
Parentheses, 22-23
in algebra, 155, 157, 185
209 in combined operations, 25482 MATHEMATICS FOR MACHINE TECHNOLOGY
Solid, rectangular, 390-392, 398^422
Solid trigonometry, 390. See also
Compound angles
Spur gears, 246-251
Square, 294
computing area of, 51-53, 57. See
also Powers; Roots
Square roots, 57-59, 69, 70. See also
Roots
Squared numbers, 51-53, 69. See also
Powers
Steel rules, 103, 115-118
Straight angle, 277
Subtraction
in algebra, 157, 177 178 , 185,
200-201
in angular measurement, 264-265
on calculator, 28-30, 68
in combined operations, 23 -24
of decimals, 42, 68
of fractions, 11-12, 28-30
of mixed numbers, 12-13, 28-30
in order of operations, 63
powers and, 52
of signed numbers, 165-166
Subtraction principles of equality,
200 201
Supplementary angles, 274, 279
Surfaces, compound -angular, 404-422
angles made by intersecting,
412-415
Rectangular solid, 390-392, 398-422
Reference angle, 334, 372
Reflex angle, 277
Regular hexagon, 294
Regular polygons, 294, 302
Relative error, 104-105
Revolutions per minute, 238-242
Right angle, 277
greater than, 371, 375. See also
Obtuse angle
less than, 375. See also Acute angle
Right triangle, 285, 292-293
complex applications of , 361-367
compound angles and, 397,
405-415
ratio of sides of, 333, 347-351
simple applications of , 354-358
terms for sides of, 334
trigonometry and, 333-334,
347-367
Root circle, 246
Root diameter, 246
Root principles of equality, 209-210
Roots, 57-59
algebra and, 157
on calculator, 59, 70, 169-170
in order of operations, 63
principal , 169
of signed numbers, 169-170
that are not whole numbers, 59
Rotation, double, 404
Rotation, angle of. 292-415
Rounding, 36-38, 68
Rules, steel , 103, 115-118
Positive numbers, 163-171, 186-187
Postulates, 258-259, 278. See also
Geometric principles
Power principles of equality, 210-211
Powers, 51-53
algebra and, 157, 183-184
on calculator, 69, 167-168
decimal places and, 65
in order of operations, 63
of signed numbers, 167 – 168
Precision, degrees of . See Degrees of
precision
Principal root, 169
Principles, geometric. See Geometric
principles
Principles of equality, 199-204,
209-211
addition , 202-203
division, 203-204
multiplication, 209
power, 210-211
root, 209-210
subtraction, 200-201
Projected lines, 354-358, 361-367,
398-401, 405-422
Proportions, 226-227
circles and, 303
direct, 232-233
inverse. 233-234
Protractors, 271-274
bevel, with vernier scale, 273-274
simple semicircular, 271-273
Pyramids, 405
Pythagorean Theorem, 293, 304, 351
Symbol(s)
in algebra, 155-158
fraction bar, 63, 157
grouping, 63, 64
infinity, 344
parentheses as. See Parentheses
radical
, 57, 58, 157
S
Q Scalene triangle, 284
Scientific notation, 186-188
Screw thread micrometers, 134, 135
Scriber, 124
Secant , 301, 312
Cartesian coordinate system and, 373
in trigonometry, 334, 336-338,
342-385
Seconds (angles), 260-266
Sector, 301
Segment, 301, 305
SI. See International System of Units
Quadrants, 371-373. See also
Cartesian coordinate system T
Tangent (tan), 301, 304, 311-315
R Cartesian coordinate system and,
373
Radical symbol, 57, 58, 157
in order of operations, 63
Radius, 301. 303
in arc length formula. 311
Rate (percent), 77-84
Ratio
to circle, 323-324, 357-358
circles that are, 301, 304, 313-315 ,
366-367
constructing , 323-324
in trigonometry, 334—338, 342-385
Tapers, 355-356
Terms (in algebra), 176-179, 182-185
order of, 225
ratios and, 225-227
Thread cutting. 420-422
Thread wire checking dimensions,
357-358
Tilt, angle of , 393- 415
Tilt
, double, 404
Tolerance, 107-109, 320
Tool -positioning systems, 432 -437
Tools/instruments
calipers, 103, 121-124, 129-131
(SI)
in arc length formula, 310-311
description of , 224-225
expressed in lowest terms. 225
proportion in, 226, 231-234
right triangle and, 333
trigonometry and, 333-335
Rectangle, 294
perimeter of, 157
solid, 390-392, 398-422
Rectangular coordinate system,
371-373. See also Cartesian
coordinate system
Signed numbers, 162-171
calculator and, 164-165
Signs of operation , 155, 163-171
Similar figures, 290
Similar polygons, 290
Similar triangles, 291, 292
Sine (sin), 335-338, 348-385
Cartesian coordinate system and,
372-373
Law of Sines and, 375
—378
Sine bar, 355
Sine plate, 355INDEX 483
cutting/forming, 418-422
depth gage attachment, 124
dial indicators, 103, 124, 129
gage blocks. See Gage blocks
height gages, 121-122, 124-125,
130-131
high amplification comparators, 103
marking , 320- 324
measuring, 103
micrometers, 103, 134—138,
141-143, 146
positioning systems for, 432-437
protractors for, 271-274
steel rules, 103, 115-118
vernier bevel protractor, 273-274
vernier calipers, 103
vernier height gages, 121 -122,
124-125, 130—131
vernier micrometers, 137-138,
142-143
Tooth thickness, 247-251
Trammels
, 320
Transposition, 201 202
Transversal , 278-280
Triangles, 284-295
acute, 375-378
adjacent side in , 334 , 347-351, 372
coordinates and, 371-373. See also
Cartesian coordinate system
congruent, 290
equilateral, 285, 292-293
hypotenuse of. See Hypotenuse
isosceles. See Isosceles triangle
legs of, 284, 285
oblique, 371, 375-380
obtuse, 375-378
opposite side in, 334, 347-351, 372
right . See Right triangle
scalene, 284
trigonometry and , 333- 385
types of, 284-285
Trigonometry, 333-385
algebra and, 155, 192
analysis of functions in , 342-344
Cartesian coordinate system in.
371-373. See also Cartesian
coordinate system
complex applications of, 361-367
powers and, 51
ratio method in, 334-335
rectangular coordinate system in,
371-373. See also Cartesian
coordinate system
right-angle. 354-358. See also
Right triangle
signed numbers and, 162
simple applications of, 354-358
solid, 390. See also Compound
angles
three-dimensional, 390. See also
Compound angles
unity or unit circle method in, 334
True length/angle, 391-392
cutting/forming tools and, 418-422
True value, 104-105
Turning center, 428
Vernier height gages
English, 121-122, 124-125
metric, 130-131
Vernier micrometers
English, 137-138
metric, 142-143
Vertex, 271, 292, 372
Vertical coordinates, 371-373. See
also Cartesian coordinate system
Volumes, computing, 51-53
English vs. metric measurements
for, 95. See also English
measurement ; Metric
measurement
W
Whole depth. 246
Whole numbers
addition of fractions and mixed
and, 9
dividing decimals by, 48-49
roots of, 57-59
roots that are not, 59
Word statements (writing equations
from), 193-196
Working depth, 247-251
U
Unilateral tolerance, 107-109
Unity/unit circle method, 334
Unknown quantity, 193, 226
gears and, 248-251
Unknown sides/angles, 291-292, 333
347-367
compound angles and, 390-422
Law of Cosines and, 378-380
Law of Sines and , 375-378
Unlike terms, 176-179, 182-185,
225-227
X saxis, 371-373, 429-430, 432. See
also Cartesian coordinate system
y yaxis, 371-373, 429-430, 432. See
also Cartesian coordinate system
Yards, 96-97, 99
V V
-blocks, 357-358
V-slots, distance between , 356-357
Vernier bevel protractor, 273-274
Vernier calipers, 103
English , 121-124
gage blocks and, 146
metric, 129-131
Z zaxis, 429^130, 432
Zero/one (binary system), 439^442
Zero point, 433. See also Origin
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