The Handbook Of Essential Mathematics
For Public Release: Distribution UnlimitedThe Air Force Research LaboratoryThe Handbook ofEssential MathematicsFormulas, Processes, and TablesPlus Applications in Personal FinanceXXYXX2Y2XYYY(X Y)2 X2 2XY Y2Compilation and Explanations: John C. SparksEditors: Donald D. Gregory and Vincent R. MillerFor Pubic Release: Distribution Unlimited
The Handbook ofEssential MathematicsAir Force Publication 2006For Public Release: Distribution Unlimited2
ForwardWright-Patterson Air Force Base (WPAFB) has enjoyed alengthy and distinguished history of serving the Greater-Daytoncommunity in a variety of ways. One of these ways is through theWPAFB Educational Outreach (EO) Program, for which the AirForce Research Laboratory (AFRL) is a proud and continuoussupporter, providing both technical expertise (from over 2000practicing scientists and engineers) and ongoing resources for thevarious programs sponsored by the WPAFB EducationalOutreach. The mission of the WPAFB EO program isTo form learning partnerships with the K-12 educationalcommunity in order to increase student awareness and excitementin all fields of math, science, aviation, and aerospace; ultimatelydeveloping our nation’s future scientific and technical workforce.In support of this mission, the WPAFB EO aspires to bethe best one-stop resource for encouragement and enhancementof K-12 science, math and technology education throughout theUnited States Air Force. It is in this spirit that AFRL offers TheHandbook of Essential Mathematics, a compendium ofmathematical formulas and other useful technical information thatwill well serve both students and teachers alike from early gradesthrough early college. It is our sincere hope that you will use thisresource to either further your own education or the education ofthose future scientists and engineers so vital to preserving ourcherished American freedoms.LESTER MCFAWN, SESExecutive DirectorAir Force Research Laboratory3
IntroductionFormulas! They seem to be the bane of every beginningmathematics student who has yet to realize that formulas areabout structure and relationship—and not about memorization.Granted, formulas have to be memorized; for, it is partly throughmemorization that we eventually become ‘unconsciouslycompetent’: a true master of our skill, practicing it in an almosteffortless, automatic sense. In mathematics, being ‘unconsciouslycompetent’ means we have mastered the underlying algebraiclanguage to the same degree that we have mastered our nativetongue. Knowing formulas and understanding the reasoningbehind them propels one towards the road to mathematicalfluency, so essential in our modern high-tech society.The Handbook of Essential Mathematics contains threemajor sections. Section I, “Formulas”, contains most of themathematical formulas that a person would expect to encounterthrough the second year of college regardless of major. Inaddition, there are formulas rarely seen in such compilations,included as a mathematical treat for the inquisitive. Section I alsoincludes select mathematical processes, such as the process forsolving a linear equation in one unknown, with a supportingexamples. Section II, “Tables”, includes both ‘pure math’ tablesand physical-science tables, useful in a variety of disciples rangingfrom physics to nursing. As in Section I, some tables are includedjust to nurture curiosity in a spirit of fun. In Sections I and II, eachformula and table is enumerated for easy referral. Section III,“Applications in Personal Finance”, is a small textbook within abook where the language of algebra is applied to that everydayfinancial world affecting all of us throughout our lives from birth todeath. Note: The idea of combining mathematics formulas withfinancial applications is not original in that my father had a similartype book as a Purdue engineering student in the early 1930s.I would like to take this opportunity to thank Mr. AlGiambrone—Chairman of the Department of Mathematics, SinclairCommunity College, Dayton, Ohio—for providing requiredmemorization formula lists for 22 Sinclair mathematics coursesfrom which the formula compilation was partially built.John C. SparksMarch 20064
DedicationThe Handbook of Essential Mathematics is dedicated to allAir Force familiesO Icarus I ride high.With a whoosh to my backAnd no wind to my face,Folded handsIn quiet rest—Watching.O Icarus.The clouds glide by,Their fields far belowOf gold-illumed snow,Pale yellow, tranquil moonTo my right—Evening sky.And Wright.O Icarus.Made it so—Silvered chariot streakingOn tongues of fire leaping—And I will soon be sleepingAbove your dreams.August 2001: John C. Sparks100th Anniversary of Powered Flight1903—2003M5
Table of ContentsSection I: Formulas withSelect ProcessesIndex to ProcessesPage 061. Algebra131.1. What is a Variable?1.2. Field Axioms1.3. Divisibility Tests1.4. Subtraction, Division, Signed Numbers1.5. Rules for Fractions1.6. Partial Fractions1.7. Rules for Exponents1.8. Rules for Radicals1.9. Factor Formulas1.10. Laws of Equality1.11. Laws of Inequality1.12. Order of Operations1.13. Three Meanings of ‘Equals’1.14. The Seven Parentheses Rules1.15. Rules for Logarithms1.16. Complex Numbers1.17. What is a Function?1.18. Function Algebra1.19. Quadratic Equations & Functions1.20. Cardano’s Cubic Solution1.21. Theory of Polynomial Equations1.22. Determinants and Cramer’s Rule1.23. Binomial Theorem1.24. Arithmetic Series1.25. Geometric Series1.26. Boolean Algebra1.27. Variation 7394041414243
Table of Contents cont2. Classical and Analytic Geometry442.1. The Parallel Postulates2.2. Angles and Lines2.3. Triangles2.4. Congruent Triangles2.5. Similar Triangles2.6. Planar Figures2.7. Solid Figures2.8. Pythagorean Theorem2.9. Heron’s Formula2.10. Golden Ratio2.11. Distance and Line Formulas2.12. Formulas for Conic Sections2.13. Conic Sections4444454647474950525354553. Trigonometry573.1. Basic Definitions: Functions & Inverses573.2. Fundamental Definition-Based Identities583.3. Pythagorean Identities583.4. Negative Angle Identities583.5. Sum and Difference Identities583.6. Double Angle Identities603.7. Half Angle Identities603.8. General Triangle Formulas603.9. Arc and Sector Formulas623.10. Degree/Radian Relationship623.11. Addition of Sine and Cosine633.12. Polar Form of Complex Numbers633.13. Rectangular to Polar Coordinates643.14. Trigonometric Values from Right Triangles 644. Elementary Vector Algebra4.1. Basic Definitions and Properties4.2. Dot Products4.3. Cross Products4.4. Line and Plane Equations4.5. Miscellaneous Vector Equations7656565666767
Table of Contents cont5. Elementary Calculus685.1. What is a Limit?5.2. What is a Differential?5.3. Basic Differentiation Rules5.4. Transcendental Differentiation5.5. Basic Antidifferentiation Rules5.6. Transcendental Antidifferentiation5.7. Lines and Approximation5.8. Interpretation of Definite Integral5.9. Fundamental Theorem of Calculus5.10. Geometric Integral Formulas5.11. Select Elementary Differential Equations5.12. Laplace Transform: General Properties5.13. Laplace Transform: Specific Transforms6. Money and Finance68697070717273737575767778806.1. What is Interest?806.2. Simple Interest816.3. Compound and Continuous Interest816.4. Effective Interest Rates826.5. Present-to-Future Value Formulas826.6. Present Value of a “Future Deposit Stream” 826.7. Present Value of a “ “ with Initial Lump Sum 836.8. Present Value of a Continuous “ “836.9. Types or Retirement Savings Accounts846.10. Loan Amortization856.11. Annuity Formulas866.12. Markup and Markdown866.13. Calculus of Finance867. Probability and Statistics7.1. Probability Formulas7.2. Basic Concepts of Statistics7.3. Measures of Central Tendency7.4. Measures of Dispersion7.5. Sampling Distribution of the Mean7.6. Sampling Distribution of the Proportion887878889909192
Table of Contents contSection II: Tables1. Numerical941.1. Factors of Integers 1 through 1921.2. Prime Numbers less than 10001.3. Roman Numeral and Arabic Equivalents1.4. Nine Elementary Memory Numbers1.5. American Names for Large Numbers1.6. Selected Magic Squares1.7. Thirteen-by-Thirteen Multiplication Table1.8. The Random Digits of PI1.9. Standard Normal Distribution1.10. Two-Sided Student’s t Distribution1.11. Date and Day of Year2. Physical Sciences9496969797971011021031041051062.1. Conversion Factors in Allied Health2.2. Medical Abbreviations in Allied Health2.3. Wind Chill Table2.4. Heat Index Table2.5. Temperature Conversion Formulas2.6. Unit Conversion Table2.7. Properties of Earth and Moon2.8. Metric System2.9. British System106107108108109109112113114Section III: Applications inPersonal Finance1. The Algebra of Interest1181.1. What is Interest?1.2. Simple Interest1.3. Compound Interest1.4. Continuous Interest1.5. Effective Interest Rate1181201221241299
Table of Contents cont2. The Algebra of the Nest Egg2.1. Present and Future Value2.2. Growth of an Initial Lump Sum Deposit2.3. Growth of a Deposit Stream2.4. The Two Growth Mechanisms in Concert2.5. Summary3. The Algebra of Consumer Debt3.1. Loan Amortization3.2. Your Home Mortgage3.3. Car Loans and Leases3.4. The Annuity as a Mortgage in Reverse4. The Calculus of Finance4.1. Jacob Bernoulli’s Differential Equation4.2. Differentials and Interest Rate4.3. Bernoulli and Money4.4. 5187188191AppendicesA. Greek Alphabet200B. Mathematical Symbols201C. My Most Used Formulas20410
Section IFormulas withSelect Processes11
Index to ProcessesProcessWhere in Section I1. Complex Rationalization Process2. Quadratic Trinomial Factoring Process3. Linear Equation Solution Process4. Linear Inequality Solution Process5. Order of Operations6. Order of Operations with Parentheses Rules7. Logarithmic Simplification Process8. Complex Number Multiplication9. Complex Number Division10. Process of Constructing Inverse Functions11. Quadratic Equations by Formula12. Quadratic Equations by Factoring13. Cardano’s Cubic Solution Process14. Cramer’s Rule, Two-by-Two System15. Cramer’s Rule, Three-by-Three System16. Removal of xy Term in Conic Sections17. The Linear First-Order Differential Equation18. Median .22.42.12.65.11.77.3.6
1. Algebra1.1.What is a Variable?In the fall of 1961, I first encountered the monster calledx in my high-school freshman algebra class. The letter x is still amonster to many, whose real nature has been confused by suchwords as variable and unknown: perhaps the most horrifyingdescription of x ever invented! Actually, x is very easilyunderstood in terms of a language metaphor. In English, we haveboth proper nouns and pronouns where both are distinct anddifferent parts of speech. Proper nouns are specific persons,places, or things such as John, Ohio, and Toyota. Pronouns arenonspecific persons or entities such as he, she, or it.To see how the concept of pronouns and nouns applies toalgebra, we first examine arithmetic, which can be thought of as aprecise language of quantification having four action verbs, a verbof being, and a plethora of proper nouns. The four action verbs areaddition, subtraction, multiplication, and division denotedrespectively by , , , . The verb of being is called equals or is,denoted by . Specific numbers such as 12 , 3.4512 , 23 53 ,123769,0.00045632 , 45 , , serve as the arithmetical equivalent toproper nouns in English. So, what is x ? x is merely a nonspecificnumber, the mathematical equivalent to a pronoun in English.English pronouns greatly expand our capability to describe andinform in a general fashion. Hence, pronouns add increasedflexibility to the English language.Likewise, mathematicalpronouns—such as x, y , z , see Appendix B for a list of symbolsused in this book—greatly expand our capability to quantify in ageneral fashion by adding flexibility to our language of arithmetic.Arithmetic, with the addition of x, y, z and other mathematicalpronouns as a new part of speech, is called algebra.In Summary: Algebra can be defined as a generalized arithmeticthat is much more powerful and flexible than standard arithmetic.The increased capability of algebra over arithmetic is due to theinclusion of the mathematical pronoun x and its associates y , z ,etc. A more user-friendly name for variable or unknown ispronumber.13
1.2.Field AxiomsThe field axioms decree the fundamental operatingproperties of the real number system and provide the basis for alladvanced operating properties in mathematics. Let a, b & c beany three real numbers (pronumbers). The field axioms are asfollows.Addition Multiplication .a b is a unique reala b is a uniquenumberreal numberCommutativea b b aa b b aAssociative( a b) c a (b c)(ab)c a (bc)0 a 0 a1 a 1 aPropertiesClosureIdentitya a ( a) 0 ( a) a 0InverseDistributive orLinking PropertyTransitivitya 0 a 1 a 1aa (b c) a b a ca b&b c a ca b&b c a ca b&b c a cNote: ab a (b) ( a )b are alternaterepresentations of a b141 1a
1.3.Divisibility ion That Makes it SoThe last digit is 0,2,4,6, or 8The sum of the digits is divisible by 3The last two digits are divisible by 4The last digit is 0 or 5The number is divisible by both 2 and 3The number formed by adding five times the lastdigit to the “number defined by” the remainingdigits is divisible by 7**The last three digits are divisible by 8The sum of the digits is divisible by 9The last digit is 011 divides the number formed by subtracting twotimes the last digit from the “ “ remaining digits**The number is divisible by both 3 and 413 divides the number formed by adding fourtimes the last digit to the “ “ remaining digits**The number is divisible by both 2 and 7The number is divisible by both 3 and 517 divides the number formed by subtracting fivetimes the last digit from the “ ” remaining digits**19 divides the number formed by adding twotimes the last digit to the “ “ remaining digits**23 divides the number formed by adding seventimes the last digit to the “ “ remaining digits**29 divides the number formed by adding threetimes the last digit to the remaining digits**31 divides the number formed by subtractingthree times the last digit from the “ “ remainingdigits**37 divides the number formed by subtractingeleven times the last digit from the “ “ remainingdigits****These tests are iterative tests in that you continue to cyclethrough the process until a number is formed that can beeasily divided by the divisor in question.15
1.4.Subtraction, Division, Signed Numbers1.4.1.Definitions:Subtraction:Division:a b a ( b)1a b a b1.4.2.Alternate representation of a b : a b 1.4.3.Division Properties of ZeroZero in numerator: a 0 Zero in denominator:Zero in both:1.4.4.ab0 0aais undefined00is undefined0Demonstration that division-by-zero is undefineda c a b c for all real numbers aba c , then a 0 c a 0 for all real numbers a ,If0an algebraic impossibility1.4.5.Demonstration that attempted division-by-zero leads toerroneous results.Letx y ; then multiplying both sides by x givesx 2 xy x 2 y 2 xy y 2 ( x y )( x y ) y ( x y )Dividing both sides by x y where x y 0 givesx y y 2y y 2 1.The last equality is a false statement.16
1.4.6.Signed Number Multiplication:1.4.7.Table for Multiplication of Signed Numbers: the italicizedwords in the body of the table indicate the resulting sign ofthe associated product.( a ) b ( a b)a ( b) (a b)( a) ( b) (a b)Multiplication of a bSign of bSign of ration of the algebraic reasonableness of thelaws of multiplication for signed numbers. In both columns,both the middle and rightmost numbers decrease in theexpected logical fashion.(4) (5) 20(4) (4) 16(4) (3) 12( 4) ( 2) 8( 5) (4) 20( 5) (3) 15( 5) (2) 10( 5) (1) 5(4) (1) 4( 4) ( 0) 0(4) ( 1) 4(4) ( 2) 8(4) ( 3) 12(4) ( 4) 16(4) ( 5) 20( 5) (0) 0( 5) ( 1) 5( 5) ( 2) 10( 5) ( 3) 15( 5) ( 4) 20( 5) ( 5) 25( 5) ( 6) 3017
1.5.LetRules for Fractionsacandbe fractions with b 0 and d 0 .bda c ad bcb da ac ca1.5.2. Fractional Equivalency: c 0 b bc cba c a c1.5.3. Addition (like denominators): b bb1.5.1.Fractional Equality:1.5.4.Addition (unlike denominators):1.5.6.Subtraction (unlike denominators):a c ad cb ad cb b d bd bdbdNote: bd is the common denominatora c a c 1.5.5. Subtraction (like denominators):b bb1.5.7.1.5.8.1.5.9.1.5.10.1.5.11.a c ad cb ad cb b d bd bdbda c acMultiplication: b d bda c a d adDivision: c 0 b d b c bcaa c a 1 aDivision (missing quantity): c bb 1 b c bcab a c adReduction of Complex Fraction:c b d bcda aaPlacement of Sign: bb b18
1.6.Partial FractionsLet P (x) be a polynomial expression with degree less than thedegree of the factored denominator as shown.1.6.1.Two Distinct Linear Factors:P( x)AB ( x a )( x b) x a x bThe numerators A, B are given byA 1.6.2.P(a)P(b),B a bb aThree Distinct Linear Factors:P( x)ABC ( x a)( x b)( x c) x a x b x cThe numeratorsA, B, C are given byP(a )P (b),B ,(a b)(a c)(b a )(b c)P (c )C (c a )(c b)A 1.6.3.N Distinct Linear Factors:nP( x)n (x a )i i 1Aiwith Ai x aiP(ai )n (aj 1j ii 119i aj)
1.7.Rules for Exponents1.7.1.Addition: a anm a n mann m1.7.2. Subtraction: m aan mnm1.7.3. Multiplication: ( a ) a1.7.4.Distributed over a Simple Product: ( ab) a b1.7.5.Distributed over a Complex Product: ( a b ) annmp nnmnb pnnan a 1.7.6. Distributed over a Simple Quotient: nb b n am a mn1.7.7. Distributed over a Complex Quotient: p pnb b 11.7.8. Definition of Negative Exponent: a nna1a an11.7.10. Definition when No Exponent is Present: a a01.7.11. Definition of Zero Exponent: a 11.7.9.Definition of Radical Expression:n1.7.12. Demonstration of the algebraic reasonableness of the n0definitions for a and a via successive divisions by 2 .Notice the power decreases by 1 with each division.16 32 2 2 2 2 2 2 48 16 2 2 2 2 2 34 8 2 2 2 22 4 2 2212141811161 2 2 2020 1 2 121 [12 ] 2 [14 ] 2 [18 ] 2 2 1122 2 2123 2 3124 2 4
1.8.Rules for Radicals1.8.1.Basic Definitions:na a n and1.8.2.Complex Radical:nam a n1.8.3.Associative:1.8.4.Simple Product:1.8.5.Simple Quotient:1.8.6.Complex Product:1.8.7.Complex Quotient:1.8.8.Nesting:121a a a2mm(n a )m n a m a nna n b n abna n a bbna m b nm a mb nna nm a m mbnbn1.8.9.n ma nm aRationalizing Numerator for n m :n1.8.10. Rationalizing Denominator for n m :ama nbb an mbnam bn a n ma1.8.11. Complex Rationalization Process:aa (b c ) b c (b c )(b c )aa (b c ) b2 cb ca ca2 cNumerator: bb( a c )1.8.12. Definition of Surd Pairs: If a b is a radical expression,then the associated surd is given by a m b .21
1.9.Factor Formulas1.9.1.1.9.2.Simple Common Factor: ab acGrouped Common Factor:1.9.3.1.9.4.Difference of Squares: a b ( a b)(a b)Expanded Difference of Squares: a(b c) (b c)aab ac db dc (b c)a d (b c) (b c)a (b c)d (b c)(a d )22(a b) 2 c 2 (a b c)(a b c)221.9.5. Sum of Squares: a b ( a bi )(a bi ) i complex2221.9.6. Perfect Square: a 2ab b (a b)1.9.7.General Trinomial:x 2 (a b) x ab ( x 2 ax) (bx ab) ( x a) x ( x a)b ( x a)( x b)1.9.8.Sum of Cubes: a b (a b)(a ab b )33221.9.9. Difference of Cubes: a b (a b)(a ab b )1.9.10. Difference of Fourths:332a 4 b 4 (a 2 b 2 )(a 2 b 2 ) a 4 b 4 (a b)(a b)(a 2 b 2 )1.9.11. Power Re
The Handbook of Essential Mathematics contains three major sections. Section I, “Formulas”, contains most of the mathematical formulas that a person would expect to encounter through the second year of college regardless of major. In addition, there are formulas rarely seen in such compilations,
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