EASA Module 1 Mathematics PDF

Module FOR LEVEL B-1 & B-2 CERTIFICATION 01 MATHEMATICS Aviation Maintenance Technician Certification Series - Arithmetic - Algebra - Geometry ...

Module FOR LEVEL B-1 & B-2 CERTIFICATION 01 MATHEMATICS Aviation Maintenance Technician Certification Series - Arithmetic - Algebra - Geometry  EASA Part-66 Aviation Maintenance Technician Certification Series NO COST REVISION/UPDATE SUBSCRIPTION PROGRAM Complete EASA Part-66 Aviation Maintenance Technician Certification Series NO COST REVISION/UPDATE PROGRAM Aircraft Technical Book Company is offering a revision/update program to our customers who purchase an EASA Module from the EASA Aviation Maintenance Technician Certification Series. The update is good for two (2) years from time of registration of any EASA Module or EASA bundled kits. If a revision occurs within two (2) years from date of registration, we will send you the revised pages FREE of cost to the registered email. Go to the link provided at the bottom of this page and fill out the form to be included in the EASA Revision/Update Subscription Program. In an effort to provide quality customer service please let us know if your email you register with changes so we can update our records. If you have any questions about this process please send an email to: [email protected] HERE’S HOW IT WORKS 1. All EASA Module Series textbooks contain an EASA subscription page explaining the subscription update process and provide a web site link to register for the EASA Revision/Update Subscription Program. 2. Go to the link provided below and fill out the web based form with your first and last name, current email address, and school if applicable. 3. From the time of purchase, if a revision occurs to the Module you have registered for, a revised PDF file containing the pages with edits will be sent to the registered email provided. 4. Please note that we try to keep our records as current as possible. If your email address provided at time of registration changes please let us know as soon as possible so we can update your account. 5. This service is FREE of charge for two (2) years from date of registration. LINK TO REGISTER FOR REVISION/UPDATE PROGRAM http://www.actechbooks.com/easasub/m01  MODULE 01 FOR LEVEL B-1 & B-2 CERTIFICATION MATHEMATICS Aviation Maintenance Technician Certification Series 72413 U.S. Hwy 40 Tabernash, CO 80478-0270 USA www.actechbooks.com +1 970 726-5111  AVAILABLE IN Printed Edition and Electronic (eBook) Format AVIATION MAINTENANCE TECHNICIAN CERTIFICATION SERIES Author Collete Clarke Layout/Design Shellie Hall / Michael Amrine Copyright © 2016 — Aircraft Technical Book Company. All Rights Reserved. No part of this publication may be reproduced, stored in a retrieval system, transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior written permission of the publisher. To order books or for Customer Service, please call +1 970 726-5111. www.actechbooks.com Printed in the United States of America  WELCOME The publishers of this Aviation Maintenance Technician Certification Series welcome you to the world of aviation maintenance. As you move towards EASA certification, you are required to gain suitable knowledge and experience in your chosen area. Qualification on basic subjects for each aircraft maintenance license category or subcategory is accomplished in accordance with the following matrix. Where applicable, subjects are indicated by an "X" in the column below the license heading. For other educational tools created to prepare candidates for licensure, contact Aircraft Technical Book Company. We wish you good luck and success in your studies and in your aviation career! EASA LICENSE CATEGORY CHART A1 B1.1 B1.2 B1.3 B2 Module number and title Airplane Airplane Airplane Helicopter Avionics Turbine Turbine Piston Turbine 1 Mathematics X X X X X 2 Physics X X X X X 3 Electrical Fundamentals X X X X X 4 Electronic Fundamentals X X X X 5 Digital Techniques / Electronic Instrument Systems X X X X X 6 Materials and Hardware X X X X X 7A Maintenance Practices X X X X X 8 Basic Aerodynamics X X X X X 9A Human Factors X X X X X 10 Aviation Legislation X X X X X 11A Turbine Aeroplane Aerodynamics, Structures and Systems X X 11B Piston Aeroplane Aerodynamics, Structures and Systems X 12 Helicopter Aerodynamics, Structures and Systems X 13 Aircraft Aerodynamics, Structures and Systems X 14 Propulsion X 15 Gas Turbine Engine X X X 16 Piston Engine X 17A Propeller X X X Module 01 - Mathematics iii  FORWARD PART-66 and the Acceptable Means of Compliance (AMC) and Guidance Material (GM) of the European Aviation Safety Agency (EASA) Regulation (EC) No. 1321/2014, Appendix 1 to the Implementing Rules establishes the Basic Knowledge Requirements for those seeking an aircraft maintenance license. The information in this Module of the Aviation Maintenance Technical Certification Series published by the Aircraft Technical Book Company meets or exceeds the breadth and depth of knowledge subject matter referenced in Appendix 1 of the Implementing Rules. However, the order of the material presented is at the discretion of the editor in an effort to convey the required knowledge in the most sequential and comprehensible manner. Knowledge levels required for Category A1, B1, B2, and B3 aircraft maintenance licenses remain unchanged from those listed in Appendix 1 Basic Knowledge Requirements. Tables from Appendix 1 Basic Knowledge Requirements are reproduced at the beginning of each module in the series and again at the beginning of each Sub-Module. How numbers are written in this book: This book uses the International Civil Aviation Organization (ICAO) standard of writing numbers. This methods displays large numbers by adding a space between each group of 3 digits. This is opposed to the American method which uses commas and the European method which uses periods. For example, the number one million is expressed as so: ICAO Standard 1 000 000 European Standard 1.000.000 American Standard 1,000,000 SI Units: The International System of Units (SI) developed and maintained by the General Conference of Weights and Measures (CGPM) shall be used as the standard system of units of measurement for all aspects of international civil aviation air and ground operations. Prefixes: The prefixes and symbols listed in the table below shall be used to form names and symbols of the decimal multiples and submultiples of International System of Units (SI) units. MULTIPLICATION FACTOR PReFIx SyMbOL 1 000 000 000 000 000 000 = 101⁸ exa E 1 000 000 000 000 000 = 101⁵ peta P 1 000 000 000 000 = 1012 tera T 1 000 000 000 = 10⁹ giga G 1 000 000 = 10⁶ mega M 1 000 = 103 kilo k 100 = 102 hecto h 10 = 101 deca da 0.1 =10-1 deci d 0.01 = 10-2 centi c 0.001 = 10-3 milli m 0.000 001 = 10-⁶ micro µ 0.000 000 001 = 10-⁹ nano n 0.000 000 000 001 = 10-12 pico p 0.000 000 000 000 001 = 10-1⁵ femto f 0.000 000 000 000 000 001 = 10-1⁸ atto a International System of Units (SI) Prefixes iv Module 01 - Mathematics  PREFACE This module contains an examination of basic mathematics including arithmetic, algebra and geometry. For most students, this module is a review of principles learned previously in life. Those being exposed to these mathematic principles for the first time may require remedial training. There are many applications for mathematics in aviation maintenance. The goal of Module 01 is to review the mathematic knowledge a certified EASA maintenance professional will need. Module 01 Syllabus as outlined in PART-66, Appendix 1. LEVELS CERTIFICATION CATEGORY ¦ B1 B2 Sub-Module 01 - Arithmetic Arithmetical terms and signs, methods of multiplication and division, fractions and 2 2 decimals, factors and multiples, weights, measures and conversion factors, ratio and proportion, averages and percentages, areas and volumes, squares, cubes, square and cube roots. Sub-Module 02 - Algebra a) Evaluating simple algebraic expressions, addition, subtraction, multiplication and 2 2 division, use of brackets, simple algebraic fractions. b) Linear equations and their solutions; Indices’s and powers, negative and fractional 1 1 indices’s; Binary and other applicable numbering systems; Simultaneous equations and second degree equations with one unknown; Logarithms. Sub-Module 03 - Geometry a) Simple geometrical constructions. 1 1 b) Graphical representation, nature and uses of graphs of equations/functions. 2 2 c) Simple trigonometry, trigonometrical relationships, use of tables and rectangular 2 2 and polar coordinates. Module 01 - Mathematics v  REVISION LOG VERSION ISSUE DATE DESCRIPTION OF CHANGE MODIFICATION DATE 001 2015 10 Module Creation and Release 002 2016 04 Module Rewrite and Release 2016 05 ACKNOWLEDGMENTS vi Module 01 - Mathematics  CONTENTS MATHEMATICS Subtraction Of Decimal Numbers‥‥‥‥‥‥‥‥‥‥‥‥‥‥ 1.14 Welcome‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥ iii Multiplication Of Decimal Numbers‥‥‥‥‥‥‥‥‥‥‥‥ 1.14 Forward‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥ iv Division Of Decimal Numbers‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥ 1.15 Preface‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥ v Rounding Off Decimal Numbers‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥ 1.16 Revision Log‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥ vi Converting Decimal Numbers To Fractions‥‥‥‥‥‥ 1.16 Acknowledgments‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥ vi Decimal Equivalent Chart ‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥ 1.17 Ratio‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥ 1.18 SUB-MODULE 01 Aviation Applications Of Ratios‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥ 1.18 ARITHMETIC Proportion‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥ 1.19 Knowledge Requirements‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥ 1.1 Solving Proportions‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥ 1.19 Arithmetic In Aviation Maintenance‥‥‥‥‥‥‥‥‥‥‥‥‥‥ 1.2 Average Value‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥ 1.19 Arithmetic‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥ 1.2 Percentage‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥ 1.20 The Integers‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥ 1.2 Expressing a Decimal Number as a Percentage‥‥‥‥ 1.20 Whole Numbers‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥ 1.2 Expressing a Percentage as a Decimal Number‥‥‥‥ 1.20 Addition Of Whole Numbers‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥ 1.2 Expressing a Fraction as a Percentage‥‥‥‥‥‥‥‥‥‥‥‥ 1.20 Subtraction Of Whole Numbers‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥ 1.3 Finding a Percentage of a Given Number‥‥‥‥‥‥‥‥‥ 1.20 Multiplication Of Whole Numbers‥‥‥‥‥‥‥‥‥‥‥‥‥ 1.3 Finding What Percentage One Number Division Of Whole Numbers‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥ 1.4 Is of Another‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥ 1.20 Factors And Multiples‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥ 1.4 Finding A Number When A Percentage Lowest Common Multiple (LCM) and Highest Of It Is Known‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥ 1.21 Common Factor (HCF)‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥ 1.5 Powers And Indices‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥ 1.21 Prime Numbers‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥ 1.5 Squares And Cubes‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥ 1.21 Prime Factors‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥ 1.5 Negative Powers‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥ 1.21 Lowest Common Multiple using Prime Factors Law Of Exponents‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥ 1.22 Method‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥ 1.6 Powers of Ten‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥ 1.22 Highest Common Factor using Prime Factors Roots‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥ 1.22 Method‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥ 1.7 Square Roots‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥ 1.22 Precedence‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥ 1.7 Cube Roots‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥ 1.24 Use of Variables‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥ 1.8 Fractional Indicies‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥ 1.24 Reciprocal‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥ 1.8 Scientific And Engineering Notation‥‥‥‥‥‥‥‥‥‥‥‥‥‥ 1.24 Positive And Negative Numbers (Signed Numbers)‥‥‥ 1.9 Converting Numbers From Standard Notation To Addition Of Positive And Negative Numbers‥‥‥‥‥ 1.9 Scientific Or Engineering Notation‥‥‥‥‥‥‥‥‥‥‥‥‥ 1.24 Subtraction Of Positive And Negative Numbers‥‥‥ 1.9 Converting Numbers From Scientific or Engineering Multiplication of Positive and Negative Numbers‥‥ 1.9 Notation To Standard Notation‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥ 1.25 Division Of Positive And Negative Numbers‥‥‥‥‥‥ 1.9 Addition, Subtraction, Multiplication, Division Of Fractions‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥ 1.9 Scientific and Engineering Numbers‥‥‥‥‥‥‥‥‥‥‥‥ 1.25 Finding The Least Common Denominator (LCD)‥ 1.10 Denominated Numbers‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥ 1.25 Reducing Fractions‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥ 1.11 Addition Of Denominated Numbers‥‥‥‥‥‥‥‥‥‥‥‥ 1.25 Mixed Numbers‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥ 1.11 Subtraction Of Denominated Numbers‥‥‥‥‥‥‥‥‥‥ 1.26 Addition and Subtraction Of Fractions‥‥‥‥‥‥‥‥‥‥ 1.11 Multiplication Of Denominated Numbers‥‥‥‥‥‥‥‥ 1.27 Multiplication Of Fractions‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥ 1.12 Division Of Denominated Numbers‥‥‥‥‥‥‥‥‥‥‥‥‥ 1.27 Division Of Fractions‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥ 1.13 Area and Volume‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥ 1.28 Addition Of Mixed Numbers‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥ 1.13 Rectangle‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥ 1.28 Subtraction Of Mixed Numbers‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥ 1.13 Square‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥ 1.28 The Decimal Number System‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥ 1.14 Triangle‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥ 1.29 Origin And Definition‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥ 1.14 Parallelogram‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥ 1.29 Addition Of Decimal Numbers‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥ 1.14 Trapezoid‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥ 1.29 Module 01 - Mathematics vii  CONTENTS Circle‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥ 1.30 Using Rules of Exponents to Solve Equations‥‥‥‥‥ 2.15 Ellipse‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥ 1.30 Logarithms‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥ 2.15 Wing Area‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥ 1.30 Transposition of Formulae‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥ 2.17 Volume‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥ 1.31 Number Bases‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥ 2.18 Rectangular Solids‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥ 1.31 The Binary Number System‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥ 2.18 Cube‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥ 1.32 Place Values‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥ 2.18 Cylinder‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥ 1.32 The Octal Number System‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥ 2.19 Sphere‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥ 1.33 The Hexadecimal Number System‥‥‥‥‥‥‥‥‥‥‥‥‥‥ 2.20 Cone‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥ 1.33 Number System Conversion‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥ 2.20 Weights And Measures‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥ 1.34 Binary, Octal and Hexadecimal to base 10 (Decimal)‥ 2.20 Questions‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥ 1.37 Decimal to either Binary, Octal or Hexadecimal‥‥‥ 2.20 Answers‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥ 1.38 Questions‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥ 2.23 Answers‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥ 2.24 SUB-MODULE 02 ALGEBRA SUB-MODULE 03 Knowledge Requirements‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥ 2.1 GEOMETRY Algebra In Aviation Maintenance‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥ 2.2 Knowledge Requirements‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥ 3.1 Algebra‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥ 2.2 Geometry In Aviation Maintenance‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥ 3.2 Evaluating Simple Algebraic Expressions‥‥‥‥‥‥‥‥‥‥‥ 2.3 Simple Geometric Constructions‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥ 3.2 Addition‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥ 2.3 Angles‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥ 3.2 Subtraction‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥ 2.3 Radians‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥ 3.2 Multiplication ‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥ 2.4 Converting Between Degrees And Radians‥‥‥‥ 3.3 Division‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥ 2.4 Degrees To Radians‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥ 3.3 Retaining The Relationship During Manipulation‥ 2.4 Radians To Degrees‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥ 3.3 Addition and Subtraction Of Expressions With Properties of Shapes‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥ 3.3 Parentheses/Brackets‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥ 2.5 Triangles‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥ 3.3 Order Of Operations‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥ 2.5 Four Sided Figures ‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥ 3.4 Simple Algebraic Fractions‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥ 2.5 Square‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥ 3.4 Algebraic Equations‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥ 2.6 Rectangle‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥ 3.4 Linear Equations ‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥ 2.6 Rhombus‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥ 3.4 Expanding Brackets‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥ 2.7 Parallelogram‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥ 3.4 Single Brackets‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥ 2.7 Trapezium‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥ 3.4 Multiplying Two Bracketed Terms‥‥‥‥‥‥‥‥‥‥‥‥ 2.7 Kite‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥ 3.4 Solving Linear Equations‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥ 2.8 Graphical Representations‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥ 3.4 Quadratic Equations‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥ 2.9 Interpreting Graphs And Charts‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥ 3.4 Finding Factors‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥ 2.9 Graphs With More Than Two Variables‥‥‥‥‥‥‥‥‥‥ 3.5 Method 1: Using Brackets‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥ 2.9 Cartesian Coordinate System‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥ 3.7 Method 2: The 'Box Method'‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥ 2.9 Graphs Of Equations And Functions‥‥‥‥‥‥‥‥‥‥‥‥ 3.8 Method 3: Factorize out Common Terms‥‥‥‥‥‥ 2.10 What is a Function?‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥ 3.8 Method 4: Difference of Two Squares.‥‥‥‥‥‥‥‥ 2.10 Linear Functions and their Graphs‥‥‥‥‥‥‥‥‥‥‥ 3.8 Look at the following quadratic equations:‥‥‥‥‥ 2.10 Slope of a Line‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥ 3.9 Solving Quadratic Equations‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥ 2.11 y-axis Intercept‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥ 3.10 Solving Using Factors‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥ 2.11 Quadratic Functions‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥ 3.11 Using the Quadratic Formula‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥ 2.11 Trigonometric Functions‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥ 3.14 Simultaneous Equations‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥ 2.12 Right Triangles, Sides And Angles‥‥‥‥‥‥‥‥‥‥‥‥‥‥ 3.14 More Algebraic Fractions‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥ 2.13 Trigonometric Relationships, Sine, Cosine And Tangent‥ 3.14 Indices And Powers In Algebra‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥ 2.13 Using Sine, Cosine And Tangent Tables‥‥‥‥‥‥‥‥‥ 3.14 viii Module 01 - Mathematics  CONTENTS Trigonometric Ratios for Angles Greater Than 90°‥ 3.16 Inverse Trigonometric Ratios‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥ 3.16 Inverse Sine‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥ 3.16 Inverse Cosine‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥ 3.16 Inverse Tangent‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥ 3.16 Pythagoras' Theorem‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥ 3.16 Graphs of Trigonometric Functions‥‥‥‥‥‥‥‥‥‥‥‥‥ 3.17 Polar Coordinates‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥ 3.18 Questions‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥ 3.21 Answers‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥ 3.22 Reference‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥ R.i Glossary‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥ G.i Index‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥‥ I.i Module 01 - Mathematics ix x Module 01 - Mathematics  ARITHMETIC PART-66 SYLLABUS LEVELS CERTIFICATION CATEGORY ¦ B1 B2 Sub-Module 01 ARITHMETIC Knowledge Requirements 1.1 - Arithmetic Arithmetical terms and signs, methods of multiplication and division, fractions and decimals, 2 2 factors and multiples, weights, measures and conversion factors, ratio and proportion, averages and percentages, areas and volumes, squares, cubes, square and cube roots. Level 1 Level 2 A familiarization with the principal elements of the subject. A general knowledge of the theoretical and practical aspects of the subject and an ability to apply that knowledge. Objectives: (a) The applicant should be familiar with the basic elements of the Objectives: subject. (a) The applicant should be able to understand the theoretical (b) The applicant should be able to give a simple description of the fundamentals of the subject. whole subject, using common words and examples. (b) The applicant should be able to give a general description of the (c) The applicant should be able to use typical terms. subject using, as appropriate, typical examples. (c) The applicant should be able to use mathematical formula in conjunction with physical laws describing the subject. (d) The applicant should be able to read and understand sketches, drawings and schematics describing the subject. (e) The applicant should be able to apply his knowledge in a practical manner using detailed procedures. Module 01 - Mathematics 1.1 ARITHMETIC IN AVIATION MAINTENANCE Arithmetic is the branch of mathematics dealing the weight and balance for the installation of new avionics with the properties and manipulation of numbers. and more. A sound knowledge and manipulation of Performing arithmetic calculations with success mathematic principles is used on a regular basis during requires an understanding of the correct methods and nearly all aspects of aircraft maintenance. procedures. Arithmetic may be thought of as a set of tools. The aviation maintenance professional will need The level to which an aviation maintenance student is these tools to successfully complete the maintenance, required to have knowledge of arithmetic is listed on repair, installation, or certification of aircraft equipment. page 1.1, according to the certification being sought. A description of the applicable knowledge levels are Arithmetic is the basis for all aspects of mathematics. presented and will be included at the beginning of each Math is used in measuring and calculating serviceability sub-module in the module. of close tolerance engine components, when calculating ARITHMETIC In this sub-module, arithmetic terms, positive, negative Therefore, always maintain the digits in the order and denominated numbers, and fractions and decimals they are given within a number, and whenever adding are explained. The use of ratios and proportions, subtracting, multiplying, or dividing numbers, be sure calculation of averages and percentages and the to maintain the place value of the digits. Figure 1-1 calculation of squares, cubes, and roots are also explained. illustrates the place value of digits within whole number. THE INTEGERS S D Integers are all positive and negative whole numbers N A S S S D U D including 0. N O E A R TH S PLACE VALUES D U S S N E O N N N U TH TE TE O H WHOLE NUMBERS 35 ➞ 3 5 Whole numbers are the numbers 0, 1, 2, 3, 4, 5, 6, 9, 8, 9, 269 ➞ 2 6 9 10, 11, 12, 13… and so on. 285 ➞ 2 8 5 852 ➞ 8 5 2 Digits are the whole numbers between 0 and 9. 12 749 ➞ 1 2 7 4 9 Figure 1-1. Place Values. When a large whole number is written, each digit within the number has a specific place and cannot be moved. This is known as place value. Each digit occupies a place ADDITION OF WHOLE NUMBERS within the whole number which determines the value of Addition is the process in which the value of one number is the entire number and must be maintained. added to the value of another. The result is called the sum. Example: 2 + 5 = 7 or, 2 The number 285 contains 3 digits: 2, 8 and 5. The +5 2 in the position shown represents 200, the 8 in 7 the position shown represents 80 and, the 5 in the position shown represents 5. Thus, the number 285. If the digit positions are changed, say 852, the number is not the same. Each digit now has a different place within the number and the place value has changed. Now the 8 represents 800, the 5 represents 50 and the 2 represents 2. 1.2 Module 01 - Mathematics When adding several whole numbers, such as 4 314, Example: 122, 93 132, and 10, align them into columns according How many hydraulic system filters are in the supply to place value and then add. room if there are 5 cartons and each carton contains ARITHMETIC 4 filters? By repeated addition: 4 314 122 4 + 4 + 4 + 4 + 4 = 20 or 4 × 5 = 20 93 132 + 10 97 578 ¥ This is the sum of the four Multiplication of large numbers involves the memorization whole numbers. of all combinations of the multiplication of digits. Figure 1-2 illustrates the products of all digits when multiplied. If the sum in a column is greater than 9, write the right digit at the bottom of the column and add the left digit into the sum of the column immediately to the left. 0 1 2 3 4 5 6 7 8 9 The sum of the right-most 10 column (4 + 8 + 0) is equal 0 0 0 0 0 0 0 0 0 0 0 234 to 12. So the 2 is put in 1 0 1 2 3 4 5 6 7 8 9 708 the right most column 40 total space and the 1 is 2 0 2 4 6 8 10 12 14 16 18 982 ¥ added into the sum of the 3 0 3 6 9 12 15 18 21 24 27 column immediately to the left. (1 + 3 + 0 + 4 = 8) 4 0 4 8 12 16 20 24 28 32 36 5 0 5 10 15 20 25 30 35 40 45 SUBTRACTION OF WHOLE NUMBERS Subtraction is the process in which the value of one 6 0 6 12 18 24 30 36 42 48 54 number is taken from the value of another. The answer 7 0 7 14 21 28 35 42 49 56 63 is called the difference. When subtracting two whole 8 0 8 16 24 32 40 48 56 64 72 numbers, such as 3 461 from 97 564, align them into columns according to place value and then subtract. 9 0 9 18 27 36 45 54 63 72 81 97 564 Figure 1-2. A multiplication table that includes the products of all digits. -3 461 This is the difference of 94 103 ¥ the two whole numbers. When multiplying large numbers, arrange the numbers to be multiplied vertically. Be sure to align the numbers When subtracting in any column, if the digit above in columns to protect the place value of each number. the digit to be subtracted is smaller than the digit to be subtracted, the digit directly to the left of this smaller Beginning with the right-most digit of the bottom number can be reduced by 1 and the small digit can be number, multiply it by the digits in the top number from increased by 10 before subtraction takes place. right to left. Record each product in the proper place value column. If a product is greater than 9, record the right 31 ¦ 2 11 most digit of the number in the proper place value column -7 -0 -7 and add the left digit into the product of the bottom digit 24 2 4 and the next digit to the left in the top number. MULTIPLICATION OF WHOLE NUMBERS Then multiply the next right-most digit in the bottom Multiplication is the process of repeated addition. The number by the digits in the top number and record the result is called the product. For example, 4 × 3 is the products in the proper place value columns. Once all same as 4 + 4 + 4. digits from the bottom number have been multiplied by all of the top digits, add all of the products recorded to obtain the final answer. Module 01 - Mathematics 1.3  When dividing small numbers, the multiplication table 18 shown in Figure 1-2 can be used. Think of any of the ×35 numbers at the intersection of a column and row as a 90 dividend. The numbers at the top and the left side of the 540 table column and row that intersect are the divisor and 630 ¥ This is the product of 18 × 35. the quotient. A process known as long division can also be used to calculate division that involves large numbers. When solving this problem, the first step is to It is shown in the following example. multiply the 5 in the number 35 by both digits in the number 18. Long division involves repeated division of the divisor into place value holders of the dividend from left to right. The second step is to multiply the 3 in the number 35 by both digits in the number 18. By doing this, both Example: place value holders in the number 35 are multiplied 239 landing gear bolts need to be divided between 7 by both place value holders in the number 18. Then, aircraft. How many bolts will each aircraft receive? the result from steps 1 and 2 are added together. 34 ¥ quotient Note that recording of the products in each step divisor ¦ 7 239 ¥ dividend must begin directly under the digit being multiplied -210 on the second line and proceeds to the left. 29 -28 1 ¥ remainder In the following example, the same procedure is used. Each place value holder in the numbers to be multiplied are multiplied by each other. Note that the divisor is greater than the left most place value holder of the dividend. The two left 321 most place value holders are used in the first step of ×43 repeated division. 963 1 2840 When the divisor is greater than a single digit number, 13 803 ¥ This is the product of 43 × 321. the number of left most place value holders in the dividend used to begin the repeated division process is DIVISION OF WHOLE NUMBERS adjusted until the divisor is smaller. In the following Division is the process of finding how many times one example, the 3 left most place value holders are used. number (called the divisor) is contained in another 73 number (called the dividend). The result is the quotient, 18 1325 and any amount left over is called the remainder. A -1260 division problem may be written as follows: 65 -54 quotient dividend 11 or = quotient divisor dividend divisor FACTORS AND MULTIPLES or Any number that is exactly divisible by a given number dividend ÷ divisor = quotient is a multiple of the given number. Using numbers, say 12 divided by 3, the problem could For example, 24 is a multiple of 2, 3, 4, 6, 8, and 12, look like this: since it is divisible by each of these numbers. Saying that 24 is a multiple of 3, for instance, is equivalent to saying or 12 or that 3 multiplied by some whole number will give 24. 3 12 3 12 ÷ 3 Any number is a multiple of itself and also of 1. 1.4 Module 01 - Mathematics Any number that is a multiple of 2 is an even number. Example: The even numbers begin with 2 and progress by 2's. The factors of 12 are 1, 2, 3, 4, 6, 12. Any number that is not a multiple of 2 is an odd number. The factors of 48 are 1, 2, 3, 4, 6, 8, 12, 16, 24, 48. ARITHMETIC Any number that can be divided into a given number From the list the highest common factor is 12. without a remainder is a factor of the given number. The given number is a multiple of any number that is Using lists as shown above to find the LCM and HCF of one of its factors. For example, 2, 3, 4, 6, 8, and 12 are numbers can be very laborious, even for relatively small factors of 24. The following equalities show various numbers. An easier way is to use Prime Factors. combinations of the factors of 24: First we need to know what prime numbers and prime 1 × 24 = 24 2 × 12 = 24 3 × 8 = 24 factors are. 4 × 6 = 24 6 × 4 = 24 8 × 3 = 24 12 × 2 = 24 24 × 1 = 24 PRIME NUMBERS A prime number is a number which has only two factors All of these factors of 24 can be found in the and those factors are 1 and the number itself. multiplication table in Figure 1-2, except 12. For example, 7 is a prime number. Its only factors are 1 LOWEST COMMON MULTIPLE (LCM) and 7. 19 is a prime number. Its only factors are 1 and AND HIGHEST COMMON FACTOR 19. Here is a list of the first eight prime numbers: 2, 3, 5, (HCF) 7, 11, 13, 17 and 19. The lowest common multiple of two or more whole numbers is the smallest positive number which is a common multiple PRIME FACTORS of the numbers. Every whole number can be written as a product of prime factors. The easiest way to f ind the LCM of two or more numbers is to list the multiples and pick out the smallest To write a number as a product of prime numbers, we common multiple. continuously divide the number by a prime as shown in the following examples. Example: The multiples of 4 are 4, 8, 12, 16, 20, 24,.. Example: The multiples of 6 are 6, 12, 18, 24… Write 924 as a product of prime. Looking at both lists gives us 12 as the lowest Solution: common multiple. 924 ÷ 2 = 462 462 ÷ 2 = 231 Example: The LCM of 7 and 8 is 56. Using your knowledge of basic multiplication tables, This means if we listed all the multiples of both 7 and we know 3 × 7 = 21 and since 231 ends in 1 we try 8, 56 would be the smallest common number. dividing by 3 next: The highest common factor of two or more numbers is the 231 ÷ 3 = 77 highest number that will divide in to the numbers. The next step is simple: The easiest way to find the highest common factor of two or more numbers is to list all the factors and read off 77 ÷ 7 = 11 the highest value. We conclude: 924 = 2 × 2 × 3 × 7 × 11 Module 01 - Mathematics 1.5  Example: Write the number as a product of primes. Write 240 as a product of primes. 342 = 2 × 3 × 3 × 19 Solution: 240 ÷ 2 = 120 Example: 120 ÷ 2 = 60 Write 630 as a product of primes. 60 ÷ 2 = 30 30 ÷ 2 = 15 Solution: 15 ÷ 3 = 5 Start by dividing the number by its lowest prime factor. In this case the lowest prime factor is 2. We conclude: Then continue dividing by 3, 5, 7 etc until the only remainder is a prime number. 240 = 2 × 2 × 2 × 2 × 3 × 5 630 Example: Here's few more numbers written in terms of prime factors. 2 315 40 = 2 × 2 × 2 × 5 231 = 3 × 7 × 11 1 638 = 2 × 3 × 3 × 7 × 13 3 105 Another way to write numbers in terms of prime factors is shown in the following example. 3 35 Example: Write 342 as a product of primes. 5 7 Solution: Start by dividing the number by its lowest prime Write the number as a product of primes. factor. In this case the lowest prime factor is 2. Then continue dividing by 3, 5, 7 etc until the only 630 = 2 × 3 × 3 × 5 × 7 remainder is a prime number. LOWEST COMMON MULTIPLE USING 342 PRIME FACTORS METHOD To find the LCM of two or more whole numbers: 1. Write the numbers in terms of prime factors. 2. Multiply each factor the greatest number of times it 2 171 occurs in either number. Example: 3 57 Find the LCM of 12 and 80. Solution: List the prime factors of each. 3 19 12 = 2 × 2 × 3 80 = 2 × 2 × 2 × 2 × 5 1.6 Module 01 - Mathematics  Multiply each factor the greatest number of times it PRECEDENCE occurs in either number. We h av e i nt ro duc e d whole nu mb er s a nd s e en examples of adding, subtracting, multiplying and ARITHMETIC 12 has one 3, and 80 has four 2's and one 5, so we dividing whole numbers. These are called operations multiply 2 four times, 3 once, and five once. in mathematics. We will use these operations again in sub-module 02 - Algebra. LCM = 2 × 2 × 2 × 2 × 3 × 5 There is an accepted order in which these operations must This gives us 240, the smallest number that can be be carried out otherwise there could be many answers to divided by both 12 and 80. one simple calculation. Consider the following example: Evaluate: Example: 8-2×4+9 Find the LCM of 9, 15 and 40 If we simply carried out the operations as they are Solution: written down we would get the answer as follows: 9=3×3 8-2=6 15 = 3 × 5 6 × 4 = 24 40 = 2 × 2 × 2 × 5 24 + 9 = 33 2 appears at most three times. This however is not correct! If you input 8-2×4+9 in 3 appears at most twice. to an electronic calculator and you will find that: 5 appears at most once. 8-2×4+9=9 LCM = 2 × 2 × 2 × 3 × 3 × 5 = 360 This is correct, of course, because a calculator uses the HIGHEST COMMON FACTOR USING PRIME rules of precedence. Figure 1-3 shows the order in which FACTORS METHOD calculations are carried out. 1. Write the numbers in terms of prime factors. 2. The product of all common prime factors is the Priority Order of Calculation HCF of the given numbers. First Brackets Second Powers and Roots Example: Third Multiplication and Division Find the Highest Common Factor (HCF) of 240 Fourth Addition and Subtraction and 924. Figure 1-3. Order in which calculations are carried out table. Solution: List the prime factors of each number. Let's look at the example above again, step by step. 240 = 2 × 2 × 2 × 2 × 3 × 5 8-2×4+9=8-8+9=9 924 = 2 × 2 × 3 × 7 × 11 Here we must multiply before we add or subtract. The factors that are common to both lists are 2 and 2 and 3. The example above could be written in a clearer manner as follows using brackets. Therefore the Highest Common Factor of 240 and 924 is: 8 - (2 × 4) + 9 2 × 2 × 3 = 12 Module 01 - Mathematics 1.7  Here's few more examples: Example: Calculate the value of 2 (x + 3y) - 4xy 5 + 3 × 3 = 5 + 9 = 14 When x = 5 and y = 2 24 ÷ (4 + 2) = 24 ÷ 6 = 4 5 × 22 = 5 × 4 = 20 Solution: (3 × 4 - 4)2 ÷ 2 = (12 - 4)2 ÷ 2 = 82 ÷ 2 = 64 ÷ 2 = 32 2 (x + 3y) - 4xy If there is more than one set of brackets, we start = 2 (5 + 3 × 2) - 4 × 5 × 2 calculating from the innermost first. = 2 (5 + 6) - 20 × 2 = 2 (11) - 40 Example: = 22 - 40 What is the value of: = -18 34 - 2 × (3 × (6 - 2) + 3) Before moving on to fractions you will need to know what the reciprocal of a number is. Solution: RECIPROCAL 34 - 2 × (3 × (6 - 2) + 3) The reciprocal of a number is one divided by the number. = 34 - 2 × (3 × (4) + 3) = 34 - 2 × (3 × 4 + 3) Example: = 34 - 2 × (12 + 3) The reciprocal of 6 is 1/6 = 34 - 2 × (15) The reciprocal of 45 is 1/45 = 34 - 30 That's easy! =4 How would the reciprocal of a fraction be written? At this stage it is worth mentioning the following. In mathematics we often leave out the multiplication, ×, Let's take the fraction 1/9. It's reciprocal is 1/(1/9). symbol when using brackets (or when using Algebra as This is really asking how many times does 1/9 divide we will see in the next section). in to 1 and of course the answer is 9! So, the reciprocal of 1/9 is 9. So, for example, 2 × (4 + 2), can be written as 2 (4 + 2). A few more examples: The reciprocal of 1/6 is 6 (or 6/1). USE OF VARIABLES The reciprocal of 8 is 1/8. Letters are frequently used in mathematics, this allows us The reciprocal of 1/20 is 20 (or 20/1). to calculate quantities using formulae over and over again. It can be seen from the above examples that the Common letters used are x, y, t, r, v, h. Because these reciprocal of a fraction can be found by inverting the letters can have different values they are called variables. fraction (i.e. flipping it upside down). Numbers such as 3, 45, 19 etc. are known as constants What is the reciprocal of 3/7? Easy, its 7/3. since they cannot have any other value. We will be mentioning reciprocals again when we see Example: how to divide fractions and in Sub-Module 02 Algebra. Calculate the value of u + at When u = 2, a = 4 and t = 9 Solution: u + at = 2 + (4) (9) = 2 + 36 = 38 1.8 Module 01 - Mathematics POSITIVE AND NEGATIVE NUMBERS (SIGNED NUMBERS) Positive numbers are numbers that are greater than zero. (that is, change a positive number to a negative number ARITHMETIC Negative numbers are numbers less than zero. (Figure or vice versa). Finally, add the two numbers together. 1-4) Signed whole numbers are also called integers. Example: -5 -4 -3 -2 -1 0 +1 +2 +3 +4 +5 The daytime temperature in the city of Denver was 6° below zero (−6°). An airplane is cruising Figure 1-4. Scale of signed numbers. at 15 000 feet above Denver. The temperature at 15 000 feet is 20° colder than in the city of Denver. What is the temperature at 15 000 feet? ADDITION OF POSITIVE AND NEGATIVE NUMBERS Subtract 20 from −6: −6 – 20 = −6 + −20 = −26 The sum (addition) of two positive numbers is positive. The sum (addition) of two negative numbers is negative. The temperature is −26°, or 26° below zero at The sum of a positive and a negative number can be 15 000 feet above the city. positive or negative, depending on the values of the numbers. A good way to visualize a negative number MULTIPLICATION OF POSITIVE AND is to think in terms of debt. If you are in debt by $100 NEGATIVE NUMBERS (or, −100) and you add $45 to your account, you are now The product of t wo negative numbers is a lways only $55 in debt (or −55). Therefore: −100 + 45 = −55. positive. The product of a positive and a negative number is always negative. Example: The weight of an aircraft is 2 000 pounds. A radio Examples: rack weighing 3 pounds and a transceiver weighing 10 pounds are removed from the aircraft. What is 3 × 6 = 18, −3 × 6 = −18, −3 × −6 = 18, the new weight? For weight and balance purposes, 3 × −6 = −18 all weight removed from an aircraft is given a minus sign, and all weight added is given a plus sign. DIVISION OF POSITIVE AND NEGATIVE NUMBERS 2 000 + −3 + −10 = 2 000 + −13 = 1 987 The quotient of t wo positive numbers is a lways positive. The quotient of two negative numbers is Therefore, the new weight is 1 987 pounds. always positive. The quotient of a positive and negative number is always negative. SUBTRACTION OF POSITIVE AND NEGATIVE NUMBERS Examples: To subtract positive and negative numbers, first change the "­­-" (subtraction symbol) to a "+" (addition symbol), 6 ÷ 3 = 2, −6 ÷ 3 = −2, −6 ÷ −3 = 2, 6 ÷ −3 = −2 and change the sign of the second number to its opposite FRACTIONS A fraction represents part of something, or, part of For example, the capacity of a fuel tank is 4 metric tons of a whole thing. Anything can be divided into equal fuel. If we divide the fuel tank into 4 equal parts (4 metric parts. We can refer to the number of these equal parts tons ÷ 4), each part has a capacity of 1 metric ton of fuel. when we are expressing how much of the whole item we are talking about. We use fractions to express how much of the whole tank capacity to which we are referring. Module 01 - Mathematics 1.9 If one of the equal parts of the tank is filled with fuel, An improper fraction is a fraction in which the numerator the tank has 1 metric ton of fuel in it. A fraction that is equal to or larger than the denominator. expresses this states the amount we are talking about compared to the total capacity of the tank. 4/4, 9/8, 26/21, and 3/2 are all examples of improper fractions. This is written as follows: A whole number and a fraction together are known as a 1 or 1 4 mixed number. Manipulating mixed numbers is discussed 4 in greater detail in the next section of this chapter. Therefore, when there is 1 metric ton of fuel in a 4 metric 11/2, 51/4, 103/4, and 81/8 are all examples of mixed ton tank, the tank is 1/4 full. A completely full tank numbers. would have all 4 parts full, or 4/4. Fractions can be added, subtracted, multiplied and The bottom number represents the number of equal divided. But to do so, certain rules must be applied to parts into which a whole item has been divided. It is arrive at the correct result. called the denominator. FINDING THE LEAST COMMON The top number represents the specific amount of the DENOMINATOR (LCD) whole item about which we are concerned. It is called To add or subtract fractions, the fractions must have the numerator. a common denominator; that is, the denominators of the fractions to be added or subtracted must be the The line between the numerator and the denominator is same. If the denominators are not the same initially, the a division line. It shows that the numerator is divided by fractions can be converted so that they have the same the denominator. denominator. When adding and subtracting fractions, the lowest, or least common denominator (LCD) is often Some examples of fractions are: used because it often simplifies the answer. The LCD is the same as the LCM. We can therefore use either of the 17 2 5 1 13 8 methods shown above to find the LCD. 18 3 8 2 16 13 Example: The denominator of a fraction cannot be 0. This would be To add 1/5 + 1/10 the LCD can be found as follows: like saying the whole item we are talking about has been List the multiples of 5 ¦ 5, 10, 15, 20, 25, 30, etc… divided into 0 pieces. List the multiples of 10 ¦ 10, 20, 30, 40, 50, etc… As a result, 1 can be expressed as a fraction as any number Choose the smallest multiple common to each list. 10 is a over itself, or, any number divided by itself is equal to 1. multiple common to both 5 and 10 and it is the smallest. Note that 20 and 30 are also common to each list of multiples. These could be used as common denominators 1 = 22 = 3 = 4 = 5 = 100 = 9.5 3 4 5 100 9.5 as well. But the least common denominator, 10, is usually chosen to keep the numbers simple. 1 = 17.598 = 5 282 17.598 5 = -8 282 -8 Example: There are names for various kinds of fractions. A proper To add 1/4 + 1/2 + 1/3 the LCD can be found as follows: fraction has a smaller numerator than the denominator. List the multiples of 4 ¦ 4, 8, 12, 16, 20, 24, etc… 1/4, 1/2, 3/ 16, and 7/8 are all examples of proper List the multiples of 2 ¦ 2, 4, 6, 8, 10, 12, 14, 16, fractions. 18, 20, 22, 24, etc... List the multiples of 3 ¦ 3, 6, 9, 12, 15, 18, 21, 24, etc… 1.10 Module 01 - Mathematics Choose the smallest multiple common to each list. 12 is Mixed numbers can be written as improper fractions. a multiple common to 4, 2, and 3 and it is the smallest. Example: ARITHMETIC Note that 24 is also common to each list of multiples. The number 34/5 represents 3 full parts and four fifths. It could be used as a common denominator to solve the addition problem. But 12 is the least common Three full parts is equivalent to 15 fifths. denominator. It is used to keep the numbers involved small and to simplify the answer to the problem. We So: can multiply both the numerator and denominator of a fraction by the same number without changing its value. 3 54 = 15 + 4 = 15+4 = 19 5 5 5 5 Example: ADDITION AND SUBTRACTION OF 5 = 5×2 = 5×4 = 5×8 = 5×9 FRACTIONS 3 3×2 3×4 3×8 3×9 As stated, in order to add or subtract fractions, we use a common denominator. Of course, we can also divide the numerator and denominator of a fraction by the same number without Example: changing its value. Find the value of: 1 + 1 + 3 Example: 5 6 10 24 = 24÷3 = 8 = 8÷2 = 4 = 4÷2 = 2 36 36÷3 12 12÷2 6 6÷2 3 Solution: The LCD in this case is 30 (list the multiples of 5, 6 You can easily verify each of the examples using a and 10 if necessary). calculator but remember calculators are not allowed in EASA examinations! Change each fraction so that its denominator is 30: REDUCING FRACTIONS 1 = 1×6 = 6 A fraction needs to be reduced when it is not in "lowest 5 5×6 30 terms." Lowest terms means that the numerator and 1 = 1×5 = 5 denominator do not have any factors in common. 6 6×5 30 That is, they cannot be divided by the same number (or factor). To reduce a fraction, determine what the 3 = 3×3 = 9 common factor(s) are and divide these out of the 10 10×3 30 numerator and denominator. For example, when both the numerator and denominator are even numbers, So the question becomes: they can both be divided by 2. 6 + 5 + 9 = 6+5+9 = 20 30 30 30 30 30 6 = 6 ÷2 = 3 10 10÷2 5 It's good practice to reduce the fraction as far as possible. MIXED NUMBERS So our final step is: A mixed number is a combination of a whole number and a fraction. 20 = 20 ÷10 = 2 30 30 ÷10 3 Each of the following are examples of mixed numbers: Example: Find the value of: 34/5, 29/8, 65/7 3 + 2 + 9 5 7 2 Module 01 - Mathematics 1.11  Solution: Note that 41/ 15 is an improper fraction and since 15 divides in to 41 twice with 11 over, we can write: The LCD of 2, 5 and 7 is 70 41 = 2 11 15 15 Writing each fraction with 70 as its denominator gives: MULTIPLICATION OF FRACTIONS 3 = 3×14 = 42 There is a very simple rule for multiplying fractions: 5 5×14 70 multiply the numerators and multiply the denominators. An example will show how easy this process is. 9 = 9×35 = 315 2 2×35 70 Example: 2 = 2×10 = 20 Find the value of: 7 7×10 70 2 ×3 5 7 So the question becomes: 42 + 20 + 315 = 42+20+315 = 377 Solution: 70 70 70 70 70 2 × 3 = 2×3 = 6 5 7 5×7 35 It is worth noting that 377 is a prime number and Example: since its only factors are 1 and itself, 377 70 cannot be Perform the following operation: 27 reduced further, but it can be written as 5 70. 4 ×5 7 9 The following example involves both addition and subtraction. Solution: 4 × 5 = 4×5 = 20 Example: 7 9 7×9 63 Find the value of: Example: 4 + 7 - 6 Calculate the following. 5 3 15 2 −5×5 3 9 6 And reduce the answer it to its lowest form. Solution: Solution: Be very careful with this type is question. It is very The LCD is 15. tempting to calculate 23 − 59 first, but you must follow the precedence rules. 4 = 4×3 = 12 5 5×3 15 We first calculate: 5 × 5 = 5×5 = 25 7 = 7×5 = 35 9 6 9×6 54 3 3×5 15 Then we use a common denominator to compute: Now we calculate: 2 − 25 12 + 35 - 6 = 41 3 54 15 15 15 15 1.12 Module 01 - Mathematics  The LCD is 54, therefore: Example: 2 − 25 = 36 − 25 = 36-25 = 11 What is the length of the grip of the bolt shown 3 54 54 54 54 54 in Figure 1-5? The overall length of the bolt is 31⁄2 ARITHMETIC inches, the shank length is 31⁄8 inches, and the DIVISION OF FRACTIONS threaded portion is 15⁄16 inches long. To find the To divide fractions, multiply by the reciprocal of the grip, subtract the length of the threaded portion lower fraction. from the length of the shank. Example: To subtract, start with the fractions. A common Divide 7⁄8 by 4⁄ 3: denominator must be used. 16 is the LCD. So the 7 4 7 3 7×3 21 problem becomes; 8 3 8 × 4 8×4 32 2 5 3 16 - 1 16 = grip length in inches ADDITION OF MIXED NUMBERS To add mi xed numbers, add the whole numbers Borrowing will be necessary because 5/16 is larger together. Then add the fractions together by finding a than 2/16. From the whole number 3, borrow 1, common denominator. The final step is to add the sum which is actually 16/16. After borrowing, the first of the whole numbers to the sum of the fractions for the mixed number is expressed as 218/16 and the equation final result. Note the following example uses feet as the is now as follows: unit of measurement which can commonly result in the use of fractions. 1 foot is equal to 30.49 centimeters. 18 5 2 16 - 1 16 = grip length in inches Example: The cargo area behind the rear seat of a small Again, to subtract, start with the fractions: airplane can handle solids that are 43⁄4 feet long. If 18 − 5 = 13 the rear seats are removed, then 21⁄3 feet is added to 16 16 16 the cargo area. What is the total length of the cargo area when the rear seats are removed? Then, subtract the whole numbers: 2−1=1 3 1 3 1 9 4 4 2 (4 + 2) + 6 4 3 4 3 12 12 Therefore, the grip length of the bolt is 113⁄16 inches. 13 1 6 7 12 12 ¥ feet of cargo room. Note: The value for the overall length of the bolt was given in the example, but it was not needed to solve SUBTRACTION OF MIXED NUMBERS the problem. This type of information is sometimes To s u bt r a c t m i x e d n u m b e r s , f i n d a c o m m o n referred to as a "distractor" because it distracts from the denominator for the fractions. Subtract the fractions information needed to solve the problem. from each other (it may be necessary to borrow from the larger whole number when subtracting the fractions). SHANK Subtract the whole numbers from each other. The f inal step is to combine the f inal whole number GRIP OVERALL LENGTH with the final fraction. Note the following example THREADS uses inches as the unit of measurement which can commonly result in the use of fractions. 1 inch is equal to 2.54 centimeters. THREADS OVERALL LENGTH Figure 1-5. Bolt dimensions. Module 01 - Mathematics 1.13 THE DECIMAL NUMBER SYSTEM ORIGIN AND DEFINITION The number system that we use every day is called the S TH S D D N N S A decimal system. The prefix in the word decimal is a Latin A TH S S S U U D D T S O O N IN N S TH H TH A A TH S