Algebra know it ALL

Algebra Know-It-ALL About the Author Stan Gibilisco is an electronics engineer, researcher, and mathematician who has authored a number of titles for the McGraw-HillDemystified series, along with more than 30 other books and dozens of magazine articles His work has been published in several languages Copyright © 2008 by The McGraw-Hill Companies, Inc Click here for terms of use Algebra Know-It-ALL Beginner to Advanced, and Everything in Between Stan Gibilisco New York Chicago San Francisco Lisbon London Madrid Mexico City Milan New Delhi San Juan Seoul Singapore Sydney Toronto Copyright © 2008 by The McGraw-Hill Companies, Inc All rights reserved Manufactured in the United States of America Except as permitted under the United States Copyright Act of 1976, no part of this publication may be reproduced or distributed in any form or by any means, or stored in a database or retrieval system, without the prior written permission of the publisher 0-07-154618-9 The material in this eBook also appears in the print version of this title: 0-07-154617-0 All trademarks are trademarks of their respective owners Rather than put a trademark symbol after every occurrence of a trademarked name, we use names in an editorial fashion only, and to the benefit of the trademark owner, with no intention of infringement of the trademark Where such designations appear in this book, they have been printed with initial caps McGraw-Hill eBooks are available at special quantity discounts to use as premiums and sales promotions, or for use in corporate training programs For more information, please contact George Hoare, Special Sales, at george_hoare@mcgraw-hill.com or (212) 904-4069 TERMS OF USE This is a copyrighted work and The McGraw-Hill Companies, Inc (“McGraw-Hill”) and its licensors reserve all rights in and to the work Use of this work is subject to these terms Except as permitted under the Copyright Act of 1976 and the right to store and retrieve one copy of the work, you may not decompile, disassemble, reverse engineer, reproduce, modify, create derivative works based upon, transmit, distribute, disseminate, sell, publish or sublicense the work or any part of it without McGraw-Hill’s prior consent You may use the work for your own noncommercial and personal use; any other use of the work is strictly prohibited Your right to use the work may be terminated if you fail to comply with these terms THE WORK IS PROVIDED “AS IS.” McGRAW-HILL AND ITS LICENSORS MAKE NO GUARANTEES OR WARRANTIES AS TO THE ACCURACY, ADEQUACY OR COMPLETENESS OF OR RESULTS TO BE OBTAINED FROM USING THE WORK, INCLUDING ANY INFORMATION THAT CAN BE ACCESSED THROUGH THE WORK VIA HYPERLINK OR OTHERWISE, AND EXPRESSLY DISCLAIM ANY WARRANTY, EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO IMPLIED WARRANTIES OF MERCHANTABILITY OR FITNESS FOR A PARTICULAR PURPOSE McGraw-Hill and its licensors not warrant or guarantee that the functions contained in the work will meet your requirements or that its operation will be uninterrupted or error free Neither McGraw-Hill nor its licensors shall be liable to you or anyone else for any inaccuracy, error or omission, regardless of cause, in the work or for any damages resulting therefrom McGraw-Hill has no responsibility for the content of any information accessed through the work Under no circumstances shall McGraw-Hill and/or its licensors be liable for any indirect, incidental, special, punitive, consequential or similar damages that result from the use of or inability to use the work, even if any of them has been advised of the possibility of such damages This limitation of liability shall apply to any claim or cause whatsoever whether such claim or cause arises in contract, tort or otherwise DOI: 10.1036/0071546170 Professional Want to learn more? 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If you’d like more information about this book, its author, or related books and websites, please click here To Samuel, Tim, and Tony This page intentionally left blank For more information about this title, click here Contents Preface xiii Acknowledgment xiv Part Numbers, Sets, and Operations Counting Methods Fingers and Sticks Roman Numerals Hindu-Arabic Numerals The Counting Base 11 Practice Exercises 17 The Language of Sets 19 The Concept of a Set 19 How Sets Relate 22 Set Intersection 27 Set Union 30 Practice Exercises 33 Natural Numbers and Integers How Natural Numbers are Made Special Natural Numbers 38 Natural Number Nontrivia 42 The Integers 45 Practice Exercises 49 35 35 vii viii Contents Addition and Subtraction Moving Up and Down 51 51 Identity, Grouping, and Signs 55 The Commutative Law for Addition 57 The Associative Law for Addition 59 Practice Exercises 63 Multiplication and Division 65 Moving Out and In 65 Identity, Grouping, and Signs 70 The Commutative Law for Multiplication 73 The Associative Law for Multiplication 75 The Distributive Laws 78 Practice Exercises 81 Fractions Built of Integers “Messy” Quotients 83 83 “Reducing” a Fraction or Ratio 87 Multiplying and Dividing Fractions 89 Adding and Subtracting Fractions 91 Practice Exercises 94 Decimal Fractions 95 Powers of 10 95 Terminating Decimals 99 Endless Decimals 101 Conversions 104 Practice Exercises 107 Powers and Roots 109 Integer Powers 109 Reciprocal-of-Integer Powers 112 Multiplying and Dividing with Exponents Multiple Powers 119 Practice Exercises 123 Irrational and Real Numbers The Number Hierarchy 124 124 More About Irrationals and Reals 129 How Real Variables Behave 133 More Rules for Real Variables 136 Practice Exercises 140 10 Review Questions and Answers 142 117 APPENDIX D Answers to Final Exam Questions 11 16 21 26 31 36 41 46 51 56 61 66 71 76 81 86 91 96 101 106 111 116 121 126 131 136 141 146 e a b b d c c d b c a a c a e a a c d b c c a d b d e a d b 12 17 22 27 32 37 42 47 52 57 62 67 72 77 82 87 92 97 102 107 112 117 122 127 132 137 142 147 e d a d c a c e d b d e a b a e a e a a a e d c e a a b c a 13 18 23 28 33 38 43 48 53 58 63 68 73 78 83 88 93 98 103 108 113 118 123 128 133 138 143 148 b d b c c a d a a e b c c d b c d c c a e b b a c d e a e d 14 19 24 29 34 39 44 49 54 59 64 69 74 79 84 89 94 99 104 109 114 119 124 129 134 139 144 149 c c b e b d e a a d c d e e e b e a e b b b d e c a e a a c 716 Copyright © 2008 by The McGraw-Hill Companies, Inc Click here for terms of use 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 105 110 115 120 125 130 135 140 145 150 b a e b a b b b c e d e a b b a b c b a c d a a d d c c b a APPENDIX E Special Characters in Order of Appearance Symbol I V X L C D M K ∞ ω ℵ ℵ0 = + ∈ ∉ {} ∅ / ⊆ ⊂ = ≡ ∩ ∪ × − () First use Chapter Chapter Chapter Chapter Chapter Chapter Chapter Chapter Chapter Chapter Chapter Chapter Chapter Chapter Chapter Chapter Chapter Chapter Chapter Chapter Chapter Chapter Chapter Chapter Chapter Chapter Chapter Chapter Chapter Meaning Roman numeral for Roman numeral for Roman numeral for 10 Roman numeral for 50 Roman numeral for 100 Roman numeral for 500 Roman numeral for 1,000 Alternative Roman numeral for 1,000 Lemniscate symbol for infinity Lowercase Greek omega symbol for infinity Uppercase Hebrew aleph symbol for infinity Aleph-null, the number of whole numbers Ellipsis, indicating repetition of a sequence or pattern Conventional symbol for numerical equality Conventional symbol for addition Set symbol meaning “is an element of ” Set symbol meaning “is not an element of ” Braces for enclosing list of set elements Symbol for the null (empty) set Conventional symbol for division, fraction, or ratio Set symbol meaning “is a subset of ” Set symbol meaning “is a proper subset of ” Set symbol meaning “is congruent to” Alternative set symbol meaning “is congruent to” Set symbol meaning “intersect” Set symbol meaning “union” Conventional symbol for multiplication Conventional symbol for negative numerical value Parentheses for grouping of quantities 717 Copyright © 2008 by The McGraw-Hill Companies, Inc Click here for terms of use 718 Special Characters in Order of Appearance Symbol || − [] · ÷ ≠ : > < ≥ ≤ {} Σ ± √ ⇒ ⇔ Δ f −1 First use Chapter Chapter Chapter Chapter Chapter Chapter Chapter Chapter Chapter Chapter Chapter Chapter Chapter Chapter Chapter Chapter Chapter 11 Chapter 11 Chapter 15 Chapter 17 Meaning Symbol for absolute value (of quantity between bars) Conventional symbol for subtraction Brackets for grouping of quantities Alternative symbol for multiplication Alternative symbol for division Symbol meaning “is not equal to” Symbol for a ratio or proportion Inequality symbol meaning “is strictly larger than” Inequality symbol meaning “is strictly smaller than” Symbol meaning “is larger than or equal to” Symbol meaning “is smaller than or equal to” Braces for grouping of quantities Decimal point Uppercase Greek sigma symbol for sum Plus-or-minus sign Surd symbol for square root Symbol meaning “implies” or “if/then” Symbol meaning “if and only if ” Uppercase Greek delta symbol for difference Notation for inverse of a function f Suggested Additional Reading • • • • • • • • Bluman, A Math Word Problems Demystified New York: McGraw-Hill, 2005 Bluman, A., Pre-Algebra Demystified New York: McGraw-Hill, 2004 Gibilisco, S Everyday Math Demystified New York: McGraw-Hill, 2004 Gibilisco, S Technical Math Demystified New York: McGraw-Hill, 2006 Huettenmueller, R Algebra Demystified New York: McGraw-Hill, 2003 Huettenmueller, R College Algebra Demystified New York: McGraw-Hill, 2004 Huettenmueller, R Pre-Calculus Demystified New York: McGraw-Hill, 2005 Olive, J Maths: A Student’s Survival Guide, 2d ed Cambridge, England: Cambridge University Press, 2003 • Prindle, A Math the Easy Way, 3d ed Hauppauge, N.Y.: Barron’s Educational Series, 1996 • Shankar, R Basic Training in Mathematics: A Fitness Program for Science Students New York: Plenum Publishing Corporation, 1995 719 Copyright © 2008 by The McGraw-Hill Companies, Inc Click here for terms of use This page intentionally left blank Index A absolute maximum, 399–402, 404–407, 409, 510–515 minimum, 398, 400–403, 407–408, 410–411, 510–515 absolute value definition of, 51–52 of complex number, 358–359 of imaginary number, 352–353 of integer, 150 orders of magnitude and, 98 addend, definition of, 57 addition as displacement, 52–53 associative law for, 59–63, 134, 151–153 commutative law for, 57–59, 134, 151–152 method of solving linear system, 255–258, 267–270, 329–330 of complex numbers, 357, 359–361 of exponents, 117–119 of fractions, 91–93, 158 of imaginary numbers, 353–354 signs in, 56 additive identity element, 55, 133 inverse, 62, 133 aleph, as infinity symbol, 10, 129, 169 aleph-null, 129, 169 antecedent, definition of, 178 antilogarithm, 488 Arabic numeral, 8–10 arithmetic mean, 124 Arithmetic, Fundamental Theorem of, 45 associative law for addition, 59–63, 134, 151–153 for multiplication, 75–78, 134, 155–157 average, 124 axis increments, 225–226 B base in numeration system, 11–17, 95, 143 of exponential, 487 of logarithm, 479–480 base-10 antilogarithm, 488 exponential, 487–488, 541–543 logarithm, 480, 538–540 base-e antilogarithm, 488 exponential, 488, 541–543 logarithm, 480–481, 540–541 billion definition of, English, 143 U.S., 143 721 Copyright © 2008 by The McGraw-Hill Companies, Inc Click here for terms of use 722 Index bijection definition of, 212–213 example of, 213–214, 316–317 binary numeral, 14–17, 143–144 binomial complex number expressed as, 357 cubed form, 413–415 factor form, 367–371, 391–395, 415–419, 435–437, 517 factor rule, 423, 518 to the nth form, 432–435, 519–520 binomial-trinomial form of cubic equation, 419–424, 428 braces, grouping with, 93 brackets, grouping with, 61 C calculus, differential, 410 cardinal, transfinite, 131 cardinality of set, 129, 169 Cartesian n-space, 292 Cartesian plane assembly of, 223–225 function graphed in, 232–234 origin in, 223 quadrants of, 223–225, 317–318 relation graphed in, 226–231 Cartesian three-space, 290–292 Celsius temperature scale, 45 cipher, as numeral for zero, 100 co-domain of mapping, 209–211 coefficient in quadratic equation, 367 leading, 443 coincident sets, 24–25 common antilogarithm, 488 denominator, 91–92 exponential, 487–488, 541–543 factor, 87 logarithm, 480, 538–540 prime factor, 88 commutative law for addition, 57–59, 134, 151–152 improper use of, 77 for multiplication, 73–75, 134, 155–157 completing the square, 371–375 complex number absolute value of, 358–359 addition, 357, 359–361 as root of quadratic equation, 381–395 conjugates, 358, 384–386, 391–395, 501 definition of, 355 division, 357–358 expressed as binomial, 357, 500–501 multiplication, 357 notation, 355–356 plane, 356–361 relationship to other numbers, 359 subtraction, 357 composite number, definition of, 40–41 number, negative, 48–49 compound fraction, 90, 159 congruent sets, 24–25, 27, 30–31, 146 conjugates, complex, 358, 384–386, 391–395, 501 consequent, definition of, 178 constant in quadratic equation, 367 letter, 192–193 coordinates Cartesian, 223–235 rectangular, 225, 233 corollary, definition of, 44 counterexample, 60 counting methods, 3–18 number, 7, credit, 58 cross-multiplication, 74 cube geometric, 113–114 root, 113–114, 164 cubic/cubic system, 459–461 cubic equation binomial-cubed form of, 413–415 binomial-factor form of, 415–419, 517 binomial-trinomial form of, 419–424, 428 polynomial standard form of, 422–430, 515–516 real roots of, 420–421 curve fitting, 229 Index D E debit, 58 decillion, definition of, decimal endless, 101–104, 121, 160–161 fraction, 95–108 nonterminating, 101–104, 160–161 numeral, 11–17, 143–144 point, 96–98 terminating, 99–101, 160–161 to ratio conversion, 104–107 denominator common, 91–92 definition of, 84 zero as, 135 denumerable set, 129, 168 dependent variable, definition of, 215 Descartes, Rene, 223 diagonal form of matrix, 301–304, 341, 343–344 difference vs ratio, 482–483, 492 differential calculus, 410 digit, definition of, discriminant, 378, 381–382, 504–506 disjoint sets, 25, 28, 31, 146 displacement addition as, 52–53 definition of, 51 division as, 66–67 multiplication as, 65–66 subtraction as, 53–54 distributive laws, 78–80, 134–135, 156 dividend, definition of, 67 dividing through, 175–176 division as displacement, 66–67 by zero, 67–68 notation for, 70 of complex numbers, 357–358 of fractions, 90–91, 159 signs in, 72 synthetic, 423–427 divisor, definition of, 67 domain definition of, 208–209 of mapping, 208–211 double elimination, 255–258, 267–270, 285 echelon form of matrix, 301–303, 341–343 element of set, 19–21, 144 ellipsis, 148 empty set (see null set) endless decimal nonrepeating, 102–103, 160–161 repeating, 101–104, 160–161 equal sets, 24–25 equation first degree, 192–207 higher-degree, 432–446 manipulating or morphing, 68, 173–176 polynomial, 432–446 second-degree, 363 equivalence, logical, 179 equivalence relation, 180, 310 essential domain of mapping, 208–211 even number, definition of, 39 exponent definition of, 44–45 reciprocal within, 484 exponential base, 95, 487 base-10, 487–488, 541–543 base-e, 488, 541–543 common, 487–488, 541–543 constant, 192, 487 definition of, 487 vs logarithm, 488–489 natural, 488, 541–543 in product, 494 in ratio, 494–495 exponents addition of, 117–119 irrational-number, 139–140 multiplication of, 120–121 subtraction of, 118–119 extremum, 399–402 Euler’s constant, 479 F factor common, 87 imaginary numbers in, 387–391 723 724 Index factor (Cont.) of natural number, 39–42 of quadratic equation, 370, 387–395 prime, 40–42, 88, 148–150 factoring, 370 Fahrenheit temperature scale, 45 finite set, 20–21 first-degree equation combinations of operations in, 198–208 constants in, 192–196 differences in, 192–196 in one variable, 192–207 number games involving, 206–207 products in, 196–198 ratios in, 196–198 standard form of, in one variable, 201, 313–314 sums in, 192–196 word problems involving, 203–207 formula, definition of, 92 fraction as ratio of integers, 83–94 compound, 90, 159 decimal, 95–108, 161–162 improper, 84 lowest form of, 87–89, 157–159 proper, 84–86, 158 reducing, 87–89 simple, 84 fractions addition of, 91–93, 158 division of, 90–91, 159 multiplication of, 89–90, 159 subtraction of, 91–93, 159 function definition of, 218 examples of, 218–220 graphed in Cartesian plane, 232–234 notation, 399 vertical-line test for, 234, 396–397 Fundamental Theorem of Arithmetic, 45 G grammar, mathematical, 56 graphs Cartesian, 223–231 of quadratic functions, 396–412 of two-by-two linear systems, 264–280 grouping with braces, 93 with brackets, 61 improper use of, 77 with parentheses, 56 H hexadecimal numeral, 12–13, 16–17, 143–144 higher-degree equation, 432–446 Hindu-Arabic numeral, 8–10 horizontal-line test for inverse function, 234, 321 hypercube, 114 I identity element additive, 55, 133 multiplicative, 70–71, 133 if and only if statement, 179 iff statement, 179 if/then statement, 178–179 imaginary number absolute value of, 352–353 addition, 353–354 definition of, 349–351 in factor of quadratic equation, 387–391 line, 351–355 pure, 357 as root of quadratic equation, 382–385, 387–391 subtraction, 354 unit, 350–351, 498–501 implication, logical, 178–179 improper fraction, 84 inconsistent linear system, 275–277, 345 increment, in slope, 236 independent variable, definition of, 215 inequality behavior of, 179–183, 308–310 definition of, 176 manipulation or morphing, 183–190 solving, 189–191 types of, 176–178 Index infinite ordinal, 37–38, 148 set, 20–21 infinity, 10, 37–38, 68, 129–132, 148 inflection point, 474–476 injection definition of, 211–212 example of, 213–214, 315–317 integer absolute value of, 150 definition of, 45 generation of, 47–49 implied list of, 149 in hierarchy, 167–168 negative, 45–47 nonnegative, 51 powers, 109–112 root, 112–117 rules for squaring, 131–132 intercept, definition of, 238 intersection of sets, 27–30, 145–146 inverse additive, 62, 133 function, horizontal-line test for, 234 logarithm, 488 multiplicative, 90, 110, 134 relation, 219–220, 273–275 irrational number as exponent, 139–140 expression of, 124–125 in hierarchy, 167–168 impossibility of listing, 129–132 pi as example of, 103 irrational roots, 444 linear equation from graph, 244–249 graph of, 236–250 point-slope form of, 242–244, 324–325 slope-intercept form of, 236–242, 323–326 two-point form of, 248–249 linear/quadratic system, 447–451, 463–466 linear relation graph of, 236–250 linear system general, 290–294 n-by-n, 292 three-by-three-281–289, 296–307 three-by-two, 293–294 two-by-two, 251–280, 326–331 local maximum, 474–476 minimum, 474–476 logarithm common, 480, 538–540 conversions, 484–485 base-e, 480–481, 540–541 base-10, 480, 538–540 base of, 479–480 definition of, 479 vs exponential, 488–489 inverse, 488 natural, 480–481, 540–541 logical equivalence, 179 implication, 178–179, 182–183 lower bound for real roots, 441–443 lowest form of fraction, 87–89, 100, 157–159 M J j operator, definition of, 350 L law, mathematical, 60 leading coefficient, 443 lemma, definition of, 64 lemniscate, as infinity symbol, 10 letter constant, 192–193 linear/cubic system, 456–459, 471–475 725 magnitude, order of, 95–99, 160, 162–163 many-to-one relation, 219 mapping bijective, 212–214, 316–317 co-domain of, 209–211, 314–315 definition of, 208–209 domain of, 208–211, 314–315 maximal domain of, 209–211, 314–315 essential domain of, 208–211 injective, 211–214, 315–317 one-to-one, 212 726 Index mapping (Cont.) onto, 212 range of, 208–211, 314–315 surjective, 212, 315–317 mathematical grammar, 56 law, 60 matrix diagonal form of, 301–304, 341, 343–344 echelon form of, 301–303, 341–343 for solving linear systems, 296–307, 340–345 operations with, 298–300 unit diagonal form of, 300–301, 304–305, 341, 344 maximal domain of mapping, 209–211 maximum absolute, 399–402, 404–407, 409, 510–515 local, 474–476 member of set, 19–21, 144 million, definition of, minimum absolute, 398, 400–403, 407–408, 410–411, 510–515 local, 474–476 minus sign for negative number, 54 monomial, definition of, 363 morph and mix, 251–255, 264–267, 286, 326–329 multiple powers, 119–122 multiplicand, definition of, 66 multiplication as displacement, 65–66 associative law for, 75–78, 134, 155–157 by zero, 135 commutative law for, 73–75, 134, 155–157 notation for, 70 of complex numbers, 357 of exponents, 120–121 of fractions, 89–90, 159 signs in, 72 multiplicative identity element, 70–71, 133 inverse, 90, 110 multiplicity of root, 372, 433–437, 459, 503–504 multiplier, definition of, 66 multiplying through, 174–175 Murphy’s law, 189 mutant quadratic, 364–367 N natural antilogarithm, 488 exponential, 488, 541–543 logarithm, 480–481, 540–541 natural number definition of, 35 divisibility of, 43 factor of, 39–42 generation of, 35–38, 147–150 in hierarchy, 167–168 n-by-n linear system, 292 negative changing reciprocal to, 484 integer power, 95, 163–164 negative number definition of, 45–47 minus sign in, 54 subtraction of, 56–57 negative power, 138 nondenumerable set, 130–131, 168 nondisjoint sets, 26 nonillion, definition of, nonterminating decimal nonrepeating, 102–103, 160–161 repeating, 101–102, 160–161 n-space, Cartesian, 292 null set, 20, 27, 31, 35–36, 146 number comparison with numeral, 142 composite, 40–41, 48–49 conversions, 104–107 even, 39 counting, definition of, games, 206–207 hierarchy, 124–129 integer, 45–49, 51, 109–117, 131–132, 149–153, 167–168 irrational, 103, 124–125, 129–132, 167–168 natural, 35–45, 147–150, 167–168 negative, 45–47, 54, 56–57 odd, 39 Index number (Cont.) prime, 40–45, 48–49, 148–150 rational, 91 124–125, 129–130, 166–168 real, 125–126, 133–140, 167–168 whole, numeral Arabic, 8–10, 143 binary, 14–17, 143–144 comparison with number, 142 decimal, 11–17, 143–144 definition of, 3, hexadecimal, 12–13, 16–17, 143–144 Hindu-Arabic, 8–10, 143 octal, 12–13, 143–144 Roman, 6–7, 11–12, 142 toothpick, 4–6 numeration systems, 11–17 numerator definition of, 84 zero as, 135 O octal numeral, 12–13, 16–17, 143–144 octillion, definition of, odd number, definition of, 39 omega, as infinity symbol, 10 one-to-many relation, 219 one-to-one correspondence, 24, 212 mapping, 212 onto mapping, 212 order of magnitude, 95–99, 160, 162–163 ordered n-tuple, definition of, 293 pair, definition of, 209–210 triple, definition of, 290 ordinal infinite, 37–38, 148 transfinite, 37–38, 148 origin, in Cartesian plane, 223 overlapping sets, 26, 28–29, 31–32 P parabola, as graph, 396–412, 510–515 parallelogram, 359 parentheses grouping with, 56 in simple products, 71 in simple quotients, 71 perfect square, 41, 149, 371–372, 374 pi, 103 plus-or-minus sign, use of, 116 point-slope form, 242–244, 324–325 polynomial equation, 432–446 second-degree, 363–367 standard form, cubic, 422–430, 515–516 standard form, higher-degree, 438, 519 standard form, quadratic, 364–367, 501–502, 515 positive integer power, 95, 160 power irrational-number, 139–140 negative, 138 negative integer, 95, 110–112, 163–164 negative reciprocal, 116–117 of power, 493–494 positive integer, 95, 109–112, 160 vs product, 493–494 rational-number, 120–122, 138 reciprocal-of-integer, 112–117 zeroth, 95, 110–112, 135, 160 powers multiple, 119–122 rational number, 120–122 precedence, rules of, 71–72, 154 prime factor, 40–42, 88, 148–150 number, definition of, 40–41, 148 number, largest, 43–45 number, negative, 48–49 product definition of, 66 exponentials in, 494 vs power, 483 vs sum, 482 proper fraction, 84–86, 158 subset, 24, 146–147 PS form (see point-slope form) 727 728 Index pure imaginary number, 357 real number, 357 Q Q.E.D., definition of, 63 quadrants of Cartesian plane, 223–225, 317–318 quadratic equation, binomial factor form, of 367–371, 391–395 equation, mutant, 364–366 equation, polynomial standard form of, 364–367, 501–502, 515 equation, with complex roots, 381–395 equation, with real roots, 363–380 formula, 375–377, 381–386, 391–392, 504 function, graph of, 396–412 function, with no real zeros, 407–412 function, with one real zero, 402–407 function, with two real zeros, 396–402 quadratic/quadratic system, 451–456, 466–471 quadrillion, definition of, quintillion, definition of, Quod erat demonstradum, 63 quotient, definition of, 67 R radical notation, 113–115 radix, 11–17, 95 range of mapping, 208–211 ratio as equivalent of fraction, 85–87 definition of, 67 vs difference, 482–483, 492 exponentials in, 494–495 in exponent, 493 to decimal conversion, 104 rational number density, 124–125, 166 definition of, 91 in hierarchy, 167–168 implied list of, 129–130 line, 124 powers, 120–122 rational root, 443–445, 522–523 real number definition of, 124–125 in hierarchy, 167–168 line, 125 pure, 357 real root, 439–445, 516–519, 520–523 real variables, behavior of, 133–140 reciprocal behavior of, 134 changing to negative, 484 definition of, 90 of integer, 90, 110 vs negative exponent, 491–492 within exponent, 484 reciprocal-of-integer powers, 112–117 reductio ad absurdum, 44 redundant linear system, 277–278, 345 reflexive property, 179–180, 310 relation bijective, 216 as set of ordered pairs, 215 definition of, 179 graphed in Cartesian plane, 226–231 injective, 215 inverse, 219–220, 229–231, 273–275, 316, 320–321 many-to-one, 219 one-to-many, 219 surjective, 215–216 remainder definition of, 67 in quotient of integers, 83 rename and replace, 258–262, 271–275, 286, 330–331 rise over run, 237 Roman numeral, 6–7 root cube, 113–114, 164 even, of negative number, 117, 165 integer, 112–117, 164 multiplicity of, 372, 433–437, 459, 503–504 rational, 443–445, 522–523 real, 439–445, 516–519, 520–523 square, 112–113, 164–165 rotate-and-mirror method, 334–335 rounding error, 483, 485–486 Index S second-degree equation, definition of, 363 polynomial, 363–367 set concept of, 19–22 denumerable, 129, 168 element of, 19–21, 144 empty, 20 finite, 20–21 infinite, 20–21 member of, 19, 144 nondenumerable, 130–131, 168 null, 20, 27, 31 properties of, 144–147 universal, 23–24 within a set, 21 sets coincident, 24–25, 146 congruent, 24–25, 27, 30–31, 146 disjoint, 25, 28, 31, 146 equal, 24–25 intersection of, 27–30, 145–146 nondisjoint, 26, 28 overlapping, 26, 28–29, 31–32 union of, 30–33, 145–146 sextillion, definition of, SI form (see slope-intercept form) sign in division, 72 in multiplication, 72 sign-changing element, 71 significant figures, 100 simple fraction, 84 slope definition of, 236–238 determination of, 321–323 slope-intercept form, 236–242, 253, 323–326 solution set definition of, 278 of quadratic equation, 369 of two-by-two linear system, 278 square completing, 371–375 geometric, 112–113 perfect, 41, 149, 371–372, 374 square (Cont.) root, 112–113, 164–165 unit, 127–128 squaring, definition of, 41 subset, 23–24, 146–147 substitution, in linear system, 258–262, 271–275, 330–331 subtraction as displacement, 53–54 of complex numbers, 357 of exponents, 118–119 of fractions, 91–93, 159 of imaginary numbers, 354 of negative number, 56–57 signs in, 56 sum vs product, 482, 492 summation symbol, 99 surd symbol, 113 surjection definition of, 212 example of, 315–317 symmetric property, 179–180, 310 synthetic division, 423–427 T tangent to axis, 402 terminating decimal, 99–101, 160–161 tesseract, 114 thousand, definition of, three-by-three geometry, 290–292 three-by-three linear system eliminating variables in, 281–285 equation form of, 296 matrices and, 296–307 matrix form of, 297–298, 340 inconsistent, 345 redundant, 345 solving, 281–289, 296–307, 335–345 three-by-two linear system, 293–294 three-space, Cartesian, 290–292 toothpick numeral, 4–6 transfinite ordinal, 37–38, 148 transitive property, 179–180, 310 trillion, definition of, 729 730 Index two-by-two cubic/cubic system, 459–461 two-by-two general system graph of, 530–538 solution of, 523–530 two-by-two linear system addition, 255–258, 267–270, 329–330 double elimination, 255–258, 267–270, 285 graphs of, 264–280, 331–335 inconsistent, 275–277 morph and mix, 251–255, 264–267, 286, 326–329 redundant, 277–278 rename and replace, 258–262, 271–275, 286, 330–331 substitution, 258–262, 271–275, 330–331 two-by-two linear/cubic system, 456–459, 471–475 two-by-two linear/quadratic system, 447–451, 463–466 two-by-two quadratic/quadratic system, 451–456, 466–471 two-point form, 248–249 V variable definition of, 52 dependent, 215 independent, 215 Venn diagram, 22–23 vertex of parabola, 510–515 vertical-line test for function, 234, 239, 319, 396–397 W whole number, word problems, 203–207 X x-intercept, 332–335 Y y-intercept definition of, 238 determination of, 321–323 Z U union of sets, 30–33, 145–146 unit diagonal form of matrix, 300–301, 304–305, 341, 344 unit square, 127–128 universal quantifier, 180 set, 23–24 universe, 23–24 upper bound for real roots, 440–443 zero as placeholder, 8–9, 143 denominator, 135 division by, 67–68 lack of, in Roman system, 6–7 multiplication by, 135 numerator, 135 of quadratic function, 396–412, 510–515 zeroth power, 95, 110–112, 135, 160 root, 168 .. .Algebra Know-It-ALL About the Author Stan Gibilisco is an electronics engineer, researcher, and mathematician... several languages Copyright © 2008 by The McGraw-Hill Companies, Inc Click here for terms of use Algebra Know-It-ALL Beginner to Advanced, and Everything in Between Stan Gibilisco New York Chicago... your own pace When you’ve finished this book, I highly recommend McGraw-Hill’s Algebra Demystified and College Algebra Demystified, both by Rhonda Huettenmueller, for further study Stan Gibilisco