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Contents xi Navigator’s Polar Vs. Cartesian Coordinates 313 Latitude and Longitude 314 Celestial Coordinates 315 Cartesian Three-Space 316 Cylindrical Coordinates 317 Spherical Coordinates 318 Questions and Problems 320 22 Complex Numbers 323 Part 4 Imaginary Quantities 323 The Complex Number Plane 325 Multiplying Complex Quantities 325 Three Cube Root Examples 326 Reciprocal of a Complex Quantity 329 Division of Complex Quantities 330 Rationalization 330 Checking Results 331 Operation Summary 332 Use of a Complex Plane 332 Roots by Complex Quantities 334 Questions and Problems 335 Tools of Applied Mathematics 23 Trigonometry in the Real World 339 Parallax 339 The Base-line 340 Accuracy 340 Distance to an Object 341 Stadimetry 342 Interstellar Distance Measurement 343 Distance Vs. Parallax 345 A Point of Confusion 346 Radar 346 Laws of Sines and Cosines 347 Use of Radar 348 How to Hunt a “Fox” 350 The Fox Finds Itself 351 Position of a Ship 352 Questions and Problems 356 24 Logarithms and Exponentials 359 The Logarithm Base 359 Common Logarithms 360 Natural Logarithms 361 Basic Properties of Logarithms 363 Conversion of Log Functions 366 Logarithm Comparisons 367 The Exponential Base 368 Common Exponentials 368 Natural Exponentials 369 Alternative Expressions 371 Logarithms Vs. Exponentials 371 Basic Properties of Exponentials 372 Exponential Comparisons 376 Numerical Orders of Magnitude 378 Questions and Problems 379 25 Scientific Notation Tutorial 381 Standard Form 381 Alternative Form 382 The “Times” Sign 382 Plain-Text Exponents 383 Orders of Magnitude 383 When to Use Scientific Notation 385 Prefix Multipliers 385 Multiplication 386 Division 387 Raising to a Power 388 Taking a Root 388 Addition 389 Subtraction 389 Significant Figures 390 How Accurate Are You? 390 What About 0? 391 What About Exact Values? 392 More About Addition and Subtraction 392 Questions and Problems 393 26 Vectors 395 Magnitude and Direction 395 Sum of Two Vectors 396 Multiplication of a Vector by a Scalar 397
xii Contents Dot Product of Two Vectors 397 Magnitude and Direction in the Polar Plane 398 Sum of Two Vectors in the Polar Plane 399 Multiplication of a Vector by a Scalar in the Polar Plane 400 Dot Product of Two Vectors in Polar Plane 401 Magnitude in XYZ Space 401 Direction in XYZ Space 402 Sum of two Vectors in XYZ Space 403 Multiplication of a Vector by a Scalar in XYZ Space 404 Dot Product of Two Vectors in XYZ Space 404 Cross Product of Two Vectors 405 Negative Vector Magnitudes 406 Questions and Problems 408 27 Logic and Truth Tables 411 Sentences 411 Negation (Not) 411 Conjunction (And) 412 Disjunction (Or) 413 Implication (If/Then) 414 Logical Equivalence (Iff) 415 Truth Tables 416 Tables for Basic Logic Functions 416 The Equals Sign 417 Q.E.D. 418 Precedence 419 Contradiction and Double Negation 420 Commutative Laws 420 Associative Laws 421 Law of Implication Reversal 421 De Morgan’s Laws 422 Distributive Law 422 Truth Table Proofs 423 Proofs Without Tables 429 Questions and Problems 430 28 Beyond Three Dimensions 433 Imagine That! 433 Time-Space 434 Position and Motion 435 Time as Displacement 437 Universal Speed 438 The Four-Cube 438 The Rectangular Four-Prism 440 Impossible Paths 441 General Time-Space Hypervolume 442 Cartesian N-Space 443 Distance Formulas 444 Vertices of a Tesseract 444 Hypervolume of a Four-Prism 445 Euclid’s Axioms 446 The Parallel Postulate 447 Geodesics 448 Modified Parallel Postulate 448 No Parallel Geodesics 449 More Than One Parallel Geodesic 450 Curved Space 450 The “Hyperfunnel” 451 Questions and Problems 452 29 A Statistics Primer 455 Experiment 455 Variable 455 Population 457 Sample, Event, and Census 457 Random Variable 458 Frequency 458 Parameter 459 Statistic 459 Distribution 459 Frequency Distribution 462 Cumulative Frequency Data 464 Mean 465 Median 467 Mode 468 Questions and Problems 470 30 Probability Basics 473 Sets and Set Notation 473 Event Vs. Outcome 474 Sample Space 475 Mathematical Probability 476 Empirical Probability 476
Page 2 and 3: Mastering Technical Mathematics
Page 4 and 5: Mastering Technical Mathematics Thi
Page 6 and 7: To Tim, Samuel, and Tony
Page 8 and 9: For more information about this tit
Page 10 and 11: Contents ix Real Vs. Imaginary Numb
Page 14 and 15: Contents xiii Law of Large Numbers
Page 16 and 17: Introduction This book is intended
Page 18 and 19: Copyright © 2008, 1999 by The McGr
Page 20 and 21: 1From Counting to Addition We’ve
Page 22 and 23: From Counting to Addition 5 Figure
Page 30 and 31: From Counting to Addition 13 WEIGHT
Page 34 and 35: 13. In the United States, common li
Page 36 and 37: 2Subtraction Just as addition is co
Page 38 and 39: Subtraction 21 the hundreds column,
Page 40 and 41: Subtraction 23 MAKING CHANGE This i
Page 42 and 43: Subtraction 25 Figure 2-6 Two metho
Page 44 and 45: 3Multiplication Suppose you go into
Page 46 and 47: Multiplication 29 In Fig. 3-2A, exa
Page 48 and 49: Multiplication 31 Figure 3-3 An exa
Page 50 and 51: Multiplication 33 Figure 3-6 It doe
Page 52 and 53: Multiplication 35 Figure 3-9 Subtra
Page 54 and 55: Multiplication 37 Figure 3-12 Multi
Page 56 and 57: Multiplication 39 3. Multiply the f
Page 58 and 59: 4Division Before the days of electr
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Division 45 MULTIPLICATION CHECKS D
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Division 47 Figure 4-6 At A, we can
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Division 49 In the example of Fig.
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Division 51 7/10 0.7 17/20 0.85 3
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Division 53 QUESTIONS AND PROBLEMS
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5Fractions Chapter 4 presented some
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Fractions 57 This line of reasoning
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Fractions 59 493/23 21.xxx . . . (
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Fractions 61 The numbers in the ori
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Fractions 63 5/12 25/60 3/10 18/6
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Fractions 65 “limiting values”
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Fractions 67 QUESTIONS AND PROBLEMS
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6Plane Polygons So far, this book h
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Plane Polygons 71 WHAT IS A SQUARE?
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Plane Polygons 73 inches is not the
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Plane Polygons 75 Figure 6-7 Two wa
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Plane Polygons 77 AREA OF AN OBTUSE
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Plane Polygons 79 Figure 6-11 Some
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Plane Polygons 81 QUESTIONS AND PRO
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7Time, Percentages, and Graphs You
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Figure 7-2 A racecar driver can cal
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When a percentage expresses a chang
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Time, Percentages, and Graphs 89 Fi
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Time, Percentages, and Graphs 91 Fi
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5-mile track. Each graph gives you
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Part 2 ALGEBRA, GEOMETRY, AND TRIGO
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8First Notions in Algebra As proble
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First Notions in Algebra 99 Figure
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First Notions in Algebra 103 (a b)
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To recap, here are the rules if you
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4. Suppose x 5 and y 7. Then what
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9Some “School” Algebra In this
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Some “School” Algebra 115 DIMEN
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Some “School” Algebra 117 4 u
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Some “School” Algebra 119 20x
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Some “School” Algebra 121 Figur
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Figure 9-9 Sometimes it’s easier
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9. Adding 1 to the numerator and de
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10Quadratic Equations Simultaneous
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Quadratic Equations 129 Figure 10-3
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What about the “negative solution
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11 Some Useful Shortcuts Have you n
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Some Useful Shortcuts 143 FINDING S
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used to derive square roots to seve
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Some Useful Shortcuts 147 Figure 11
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Some Useful Shortcuts 149 Figure 11
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Some Useful Shortcuts 151 which sim
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12. In each of the following pairs
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12 Mechanical Mathematics In this c
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Mechanical Mathematics 157 Figure 1
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is at (a times t) feet per second.
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If the mass is 1 kg, the force is 9
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13 Ratio and Proportion Thus far, t
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Ratio and Proportion 175 PROPORTION
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In any triangle, its three interior
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Ratio and Proportion 179 Figure 13-
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14 Trigonometric and Geometric Calc
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Trigonometric and Geometric Calcula
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PROPERTIES OF THE ISOSCELES TRIANGL
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Suppose you name the equal pairs of
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Copyright © 2008, 1999 by The McGr
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15 Systems of Counting The decimal
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Systems of Counting 205 Figure 15-2
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Systems of Counting 207 OCTAL NUMBE
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Systems of Counting 209 DECIMAL BIN
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are shortcut methods for performing
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1. Find the decimal equivalent of t
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16 Progressions, Permutations, and
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Progressions, Permutations, and Com
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17 Introduction to Derivatives The
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Figure 17-2 The derivative of the f
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is negative when x is negative, bec
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This rule holds true for all expres
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Introduction to Derivatives 243 CIR
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Introduction to Derivatives 245 Fig
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18 More about Differentiation The p
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More about Differentiation 253 Figu
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More about Differentiation 255 by t
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More about Differentiation 257 MULT
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In the powers table, 0 n is always
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DERIVATIVE OF FUNCTION MULTIPLIED B
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More about Differentiation 263 QUES
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19 Introduction to Integrals Now th
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Introduction to Integrals 267 Figur
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(it subtracts from itself), so you
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Introduction to Integrals 275 VOLUM
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1. Find the derivatives of the prod
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20Combining Calculus with Other Too
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plotting nine points, but to get a
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In a situation like this, you have
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Combining Calculus with Other Tools
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These principal axes can be regarde
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e reaches 1, the ellipse “breaks
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etween the plane and the double con
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21Alternative Coordinate Systems Th
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Alternative Coordinate Systems 303
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perpendicular to the other two. The
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22Complex Numbers Long ago, humanki
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Multiplying repeatedly by i works t
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Complex Numbers 327 Figure 22-4 The
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Complex Numbers 329 When you cube i
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Complex Numbers 331 Figure 22-9 The
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Complex Numbers 333 Figure 22-11 At
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Complex Numbers 335 QUESTIONS AND P
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Part 4 TOOLS OF APPLIED MATHEMATICS
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23Trigonometry in the Real World Tr
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The absolute accuracy (in fixed uni
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Trigonometry in the Real World 343
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objects, even when they are observe
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Now find the distance PV by applyin
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7. What is the range of the aircraf
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24Logarithms and Exponentials A log
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Logarithms and Exponentials 361 Fig
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Now try an example in which neither
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Let’s put a number into this form
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Logarithms and Exponentials 369 10
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Suppose x is some real number. The
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To demonstrate this, let b e, x 2
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“Inv” key followed by a “ln
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and so on. A decrease of 1 order of
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25Scientific Notation Tutorial In s
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Another alternative multiplication
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line that covers 3 orders of magnit
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Scientific Notation Tutorial 387 (3
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Scientific Notation Tutorial 389 (5
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equality symbols. It is universally
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Sometimes, the outcome of determini
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26Vectors You’ve already seen a f
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Vectors 397 MULTIPLICATION OF A VEC
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Vectors 399 The values of the angle
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Vectors 401 If a is multiplied by a
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Vectors 403 Figure 26-5 Direction a
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Vectors 405 This formula works in a
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Vectors 407 You are told that these
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Vectors 409 4. Consider the two vec
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27Logic and Truth Tables To gain a
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are true. If any one of the compone
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antecedent is (P & C). The “impli
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Logic and Truth Tables 417 X Y X &
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Logic and Truth Tables 419 (X 1 Y)
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Logic and Truth Tables 421 The same
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classes—a certain way that multip
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Table 27-9B Derivation of truth val
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Logic and Truth Tables 427 A X Y X
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Logic and Truth Tables 429 X Y Z X
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Logic and Truth Tables 431 X Y Z X
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28Beyond Three Dimensions As you ha
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Beyond Three Dimensions 435 Figure
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a little later! It “lives” for
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Beyond Three Dimensions 441 IMPOSSI
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Beyond Three Dimensions 443 Displac
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Beyond Three Dimensions 449 NO PARA
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such as the pyramid, cube, and sphe
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2. If there are no pairs of paralle
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29A Statistics Primer Statistics is
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A Statistics Primer 457 Figure 29-2
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A Statistics Primer 459 PARAMETER A
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A Statistics Primer 463 of dice, th
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To determine µ sc , check the tabl
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A Statistics Primer 471 making it m
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30Probability Basics Probability is
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Probability Basics 475 Event 1 Even
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Probability Basics 477 As with math
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Probability Basics 479 that both co
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Probability Basics 481 The notion o
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Probability Basics 483 To calculate
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Probability Basics 485 • 30 peopl
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Probability Basics 487 The resultin
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Probability Basics 489 Because the
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Probability Basics 491 Figure 30-11
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Probability Basics 493 views are re
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Final Exam Do not refer to the text
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Final Exam 497 (a) An even number t
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Final Exam 499 X Y Z ? F F F F F F
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Final Exam 501 (c) A harmonic progr
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Final Exam 503 (c) 1.8 10 6 km (d)
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Final Exam 505 47. Suppose an item
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Final Exam 507 Figure E-6 Illustrat
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Final Exam 509 (a) p q (b) p q 1
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Final Exam 511 (d) Roots of w (e) P
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Final Exam 513 82. Which of the fol
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Final Exam 515 (b) It decreases in
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Final Exam 517 95. In the situation
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Appendix 1 Solutions to End-of-Chap
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Solutions to End-of-Chapter Questio
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Figure A-30 Illustration for soluti
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Appendix 2 Answers to Final Exam Qu
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Appendix 3 Table of Derivatives Let
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Appendix 4 Table of Indefinite Inte
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Table of Indefinite Integrals 609 f
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Suggested Additional Reading Bluman
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Index absolute accuracy, 341 error,
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cosines, law of, 347-348 counter, m
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geometry elliptic, 449 Euclidean, 4
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exponential, 369-371 frequency, 255
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esonance, in system, 168-170, 255 r
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direction angles, 402-403 direction
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