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From Counting to Addition 11 Figure 1-9 When adding three numbers, it doesn’t matter which two are added first. At A, we add three and five, and then add seven more. At B, we add seven and five, and then add three more. Either way we get fifteen. Figure 1-10 Big numbers are added in the same way as single-digit numbers. Here, we see that 125 added to 324 equals 449.
12 Mastering Technical Mathematics CARRYING In the preceding example, we deliberately chose numbers in each place that did not add up to over 10, to make it easy. If any number group or place adds to over 10, we must “carry” it to the next higher group or place. Three examples follow; it will help if you write the digits down in the columns for powers of 10 (that is, the ones place, the tens place, the hundreds place, and so on) as you read these descriptions. Suppose we want to add 27 35. We take the ones first: 7 5 12. The numeral 1 belongs in the tens place. Now, instead of just 2 and 3 to add in the tens place, you have an extra 1 that appears in the tens place from adding 7 and 5. This extra 1 is said to be carried from the ones place. This carrying process goes on any time the total at a certain place goes over 10. The final result of this addition process, called the sum, is therefore 27 35 62, because we have 1 2 3 6 in the tens place. Now what if we want to add 7,358 and 2,763? Start with the ones: 8 3 11, so we write 1 in the ones place and carry 1 to the tens place. Now the tens: 5 6 11, and the 1 carried from the ones makes 12. We write 2 in the tens place and carry 1 to the hundreds place. Now the hundreds: 7 3 10 and 1 carried from the tens makes 11 hundreds. We write 1 in the hundreds place and carry 1 to the thousands place. Now the thousands: 7 2 9 and 1 carried from the hundreds make 10 thousands. The final sum is therefore 10,121. Another example: suppose we have to add 7,196 and 15,273. We start with the ones: 6 3 9. We write 9 in the ones place and nothing is left to carry to the tens. Next, 9 7 16. We write the 6 and carry the 1 to the hundreds. Now the hundreds: 1 2 3, and the 1 carried makes 4. We have nothing to carry to the thousands. So in the thousands: 7 5 12. Now we carry 1 to the ten-thousands place, where only one number already has 1. We add 1 1 to get 2 in the ten-thousands place. The final sum in this case is 22,469. CHECKING ANSWERS Before calculators made things easy, bookkeepers would use two methods to add long columns of numbers. First, they would add starting at the top and working down. Then they would add the same numbers starting at the bottom and working up. They kept right on doing this once they got calculators because it was a good way for them to check their work! Adding long lists of numbers can get tedious, and it’s easy to make a mistake. You don’t want to do that when you’re dealing with important things such as tax returns. You see the advantage of using more than one method. The partial sums that you move through on the way are different when you go upward, as compared to when you work down. But you should always reach the same answer at the end. It’s not likely that you would enter the same wrong number twice under these conditions. If you get different answers, work each one again until you find where you made your mistake. (And be happy you found it before the tax man did!)
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 12 and 13: Contents xi Navigator’s Polar Vs.
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
Page 60 and 61: Division 43 Figure 4-2 A pictorial
Page 62 and 63: Division 45 MULTIPLICATION CHECKS D
Page 64 and 65: Division 47 Figure 4-6 At A, we can
Page 66 and 67: Division 49 In the example of Fig.
Page 68 and 69: Division 51 7/10 0.7 17/20 0.85 3
Page 70 and 71: Division 53 QUESTIONS AND PROBLEMS
Page 72 and 73: 5Fractions Chapter 4 presented some
Page 74 and 75: Fractions 57 This line of reasoning
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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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