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DOE-HDBK-1014/1-92JUNE 1992DOE FUNDAMENTALS HANDBOOKMATHEMATICSVolume 1 of 2U.S. Department of EnergyFSC-6910Washington, D.C. 20585Distribution Statement A. Approved for public release; distribution is unlimited.

This document has been reproduced directly from the best available copy.Available to DOE and DOE contractors from the Office of Scientific andTechnical Information. P. O. Box 62, Oak Ridge, TN 37831; (615) 576-8401.Available to the public from the National Technical Information Service, U.S.Department of Commerce, 5285 Port Royal Rd., Springfield, VA 22161.Order No. DE92019794

MATHEMATICSABSTRACTThe Mathematics Fundamentals Handbook was developed to assist nuclear facilityoperating contractors provide operators, maintenance personnel, and the technical staff with thenecessary fundamentals training to ensure a basic understanding of mathematics and itsapplication to facility operation. The handbook includes a review of introductory mathematicsand the concepts and functional use of algebra, geometry, trigonometry, and calculus. Wordproblems, equations, calculations, and practical exercises that require the use of each of themathematical concepts are also presented. This information will provide personnel with afoundation for understanding and performing basic mathematical calculations that are associatedwith various DOE nuclear facility operations.Key Words: Training Material, Mathematics, Algebra, Geometry, Trigonometry, CalculusRev. 0MA

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MATHEMATICSFOREWORDThe Department of Energy (DOE) Fundamentals Handbooks consist of ten academicsubjects, which include Mathematics; Classical Physics; Thermodynamics, Heat Transfer, and FluidFlow; Instrumentation and Control; Electrical Science; Material Science; Mechanical Science;Chemistry; Engineering Symbology, Prints, and Drawings; and Nuclear Physics and ReactorTheory. The handbooks are provided as an aid to DOE nuclear facility contractors.These handbooks were first published as Reactor Operator Fundamentals Manuals in 1985for use by DOE category A reactors. The subject areas, subject matter content, and level of detailof the Reactor Operator Fundamentals Manuals were determined from several sources. DOECategory A reactor training managers determined which materials should be included, and servedas a primary reference in the initial development phase. Training guidelines from the commercialnuclear power industry, results of job and task analyses, and independent input from contractorsand operations-oriented personnel were all considered and included to some degree in developingthe text material and learning objectives.The DOE Fundamentals Handbooks represent the needs of various DOE nuclear facilities'fundamental training requirements. To increase their applicability to nonreactor nuclear facilities,the Reactor Operator Fundamentals Manual learning objectives were distributed to the NuclearFacility Training Coordination Program Steering Committee for review and comment. To updatetheir reactor-specific content, DOE Category A reactor training managers also reviewed andcommented on the content. On the basis of feedback from these sources, information that appliedto two or more DOE nuclear facilities was considered generic and was included. The final draftof each of the handbooks was then reviewed by these two groups. This approach has resulted inrevised modular handbooks that contain sufficient detail such that each facility may adjust thecontent to fit their specific needs.Each handbook contains an abstract, a foreword, an overview, learning objectives, and textmaterial, and is divided into modules so that content and order may be modified by individual DOEcontractors to suit their specific training needs. Each subject area is supported by a separateexamination bank with an answer key.The DOE Fundamentals Handbooks have been prepared for the Assistant Secretary forNuclear Energy, Office of Nuclear Safety Policy and Standards, by the DOE TrainingCoordination Program. This program is managed by EG&G Idaho, Inc.Rev. 0MA

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MATHEMATICSOVERVIEWThe Department of Energy Fundamentals Handbook entitled Mathematics was preparedas an information resource for personnel who are responsible for the operation of theDepartment's nuclear facilities. A basic understanding of mathematics is necessary for DOEnuclear facility operators, maintenance personnel, and the technical staff to safely operate andmaintain the facility and facility support systems. The information in the handbook is presentedto provide a foundation for applying engineering concepts to the job. This knowledge will helppersonnel more fully understand the impact that their actions may have on the safe and reliableoperation of facility components and systems.The Mathematics handbook consists of five modules that are contained in two volumes.The following is a brief description of the information presented in each module of thehandbook.Volume 1 of 2Module 1 - Review of Introductory MathematicsThis module describes the concepts of addition, subtraction, multiplication, anddivision involving whole numbers, decimals, fractions, exponents, and radicals.A review of basic calculator operation is included.Module 2 - AlgebraThis module describes the concepts of algebra including quadratic equations andword problems.Volume 2 of 2Module 3 - GeometryThis module describes the basic geometric figures of triangles, quadrilaterals, andcircles; and the calculation of area and volume.Module 4 - TrigonometryThis module describes the trigonometric functions of sine, cosine, tangent,cotangent, secant, and cosecant. The use of the pythagorean theorem is alsodiscussed.Rev. 0MA

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MATHEMATICSModule 5 - Higher Concepts of MathematicsThis module describes logarithmic functions, statistics, complex numbers,imaginary numbers, matrices, and integral and derivative calculus.The information contained in this handbook is by no means all encompassing. An attemptto present the entire subject of mathematics would be impractical. However, the Mathematicshandbook does present enough information to provide the reader with a fundamental knowledgelevel sufficient to understand the advanced theoretical concepts presented in other subject areas,and to better understand basic system and equipment operations.Rev. 0MA

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Department of EnergyFundamentals HandbookMATHEMATICSModule 1Review of Introductory Mathematics

Review of Introductory MathematicsTABLE OF CONTENTSTABLE OF CONTENTSLIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivLIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vREFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viOBJECTIVES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiTERMINOLOGY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3CALCULATOR OPERATIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4FOUR BASIC ARITHMETIC OPERATIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6Calculator Usage, Special Keys . . . . .The Decimal Numbering System . . . .Adding Whole Numbers . . . . . . . . . .Subtracting Whole Numbers . . . . . . .Multiplying Whole Numbers . . . . . .Dividing Whole Numbers . . . . . . . . .Hierarchy of Mathematical OperationsSummary . . . . . . . . . . . . . . . . . . . .667911131619AVERAGES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20Average Value . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23FRACTIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24Proper and Improper Fractions . . . . . . . .Equivalent Fractions . . . . . . . . . . . . . . .Addition and Subtraction of Fractions . . .Least Common Denominator Using PrimesAddition and Subtraction . . . . . . . . . . . .Multiplication . . . . . . . . . . . . . . . . . . . .Division . . . . . . . . . . . . . . . . . . . . . . . .Summary . . . . . . . . . . . . . . . . . . . . . . .Rev. 0Page i.2425263133343437MA-01

TABLE OF CONTENTSReview of Introductory MathematicsTABLE OF CONTENTS (Cont.)DECIMALS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38Fraction to Decimal Conversion . . . .Decimal to Fraction Conversion . . . .Addition and Subtraction of DecimalsMultiplying Decimals . . . . . . . . . . . .Dividing Decimals . . . . . . . . . . . . . .Rounding Off . . . . . . . . . . . . . . . . .Summary . . . . . . . . . . . . . . . . . . . .38404242434447SIGNED NUMBERS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48Calculator Usage, Special KeysAddition . . . . . . . . . . . . . . . .Subtraction . . . . . . . . . . . . . .Multiplication . . . . . . . . . . . .Division . . . . . . . . . . . . . . . .Summary . . . . . . . . . . . . . . .484850515152SIGNIFICANT DIGITS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53Calculator Usage, Special Keys . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53Significant Digits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55PERCENTAGES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56Calculator Usage, Special Keys . . . . . . . . . . . . . . . . . . . . .Changing Decimals to Percent . . . . . . . . . . . . . . . . . . . . . .Changing Common Fractions and Whole Numbers to PercentChanging a Percent to a Decimal . . . . . . . . . . . . . . . . . . . .Percent Differential . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .MA-01Page ii.56565757585960Rev. 0

Review of Introductory MathematicsTABLE OF CONTENTSTABLE OF CONTENTS (Cont.)EXPONENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61Calculator Usage, Special KeysExponents . . . . . . . . . . . . . . .Basic Rules for Exponents . . .Zero Exponents . . . . . . . . . . .Negative Exponents . . . . . . . .Fractional Exponents . . . . . . .Summary . . . . . . . . . . . . . . .61616264646566SCIENTIFIC NOTATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67Calculator Usage . . . . . . . . . . . . . . . . .Writing Numbers in Scientific Notation .Converting Scientific Notation to IntegersAddition . . . . . . . . . . . . . . . . . . . . . . .Subtraction . . . . . . . . . . . . . . . . . . . . .Multiplication . . . . . . . . . . . . . . . . . . .Division . . . . . . . . . . . . . . . . . . . . . . .Summary . . . . . . . . . . . . . . . . . . . . . .6767697071717273RADICALS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74Calculator Usage, Special Keys .The Radical . . . . . . . . . . . . . .Simplifying Radicals . . . . . . . .Addition and Subtraction . . . . .Multiplication . . . . . . . . . . . . .Division . . . . . . . . . . . . . . . . .Dissimilar Radicals . . . . . . . . .Changing Radicals to ExponentsChanging Exponents to RadicalsSummary . . . . . . . . . . . . . . . .74747576767677777879Appendix ATI-30 Calculator Keypad Layout . . . . . . . . . . . . . . . . . . . . . . .