NONRESIDENT
`TRAINING
`COURSE
`
`April 1995
`
`Engineering Aid 3
`
`NAVEDTRA 14069
`
`Notice: NETPDTC is no longer responsible for the content accuracy of the
`NRTCs.
`
`For content issues, contact servicing Center of Excellence: Center for Seabees and
`SONY Exhibit 1035
`Facilities Engineering (CSFE); (805) 982-2086 or DSN: 551-2086.
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`
`

`

`Although the words “he,” “him,” and
`“his” are used sparingly in this course to
`enhance communication,
`they are not
`intended to be gender driven or to affront or
`discriminate against anyone.
`
`DISTRIBUTION STATEMENT A: Approved for public release; distribution is unlimited.
`
`

`

`COMMANDING OFFICER
`NETPMSA
`6490 SAUFLEY FIELD ROAD
`P EN SA CO LA F L 32509-5237
`
`0 4 J u n 9 6
`
`ERRATA # 2
`
`S p e c i f i c I n s t r u c t i o n s a n d E r r a t a
`T r a i n i n g M a n u a l
`
`f o r
`
`ENGINEERING AID BASIC
`
`1 .
`
`T h i s e r r a t a s u p e r s e d e s a l l p r e v i o u s e r r a t a .
`
`2 . N o a t t e m p t h a s b e e n m a d e t o i s s u e c o r r e c t i o n s f o r e r r o r s i n
`t y p i n g , p u n c t u a t i o n , a n d s o f o r t h , t h a t d o n o t a f f e c t t e c h n i c a l
`a c c u r a c y o r r e a d a b i l i t y .
`
`3 . M a k e t h e f o l l o w i n g c h a n g e s :
`
`C h a n g e
`“ N a v a l P u b l i c a t i o n s a n d
`R e p l a c e
`F o r m s C e n t e r ” w i t h “ A v i a t i o n
`S u p p o r t O f f i c e ”
`
`R e p l a c e f i g u r e 3 - 1 4 w i t h f i g u r e
`3 - 1 4 A & B
`
`“ D O D - S T D - 1 O O C ” w i t h “ M I L -
`R e p l a c e
`STD-1OOE”
`
`R e p l a c e f i g u r e 5 - 1 2 w i t h f i g u r e s
`5 - 1 1 a n d 5 - 1 2
`
`R e p l a c e f i g u r e 7 - 2 2 w i t h f i g u r e s
`7 - 2 1 a n d 7 - 2 2
`
`R e p l a c e c a p t i o n i n f i g u r e 1 4 - 2 0
`w i t h “ S a m p l e f i e l d n o t e s f r o m
`c r o s s - s e c t i o n l e v e l i n g a t f i r s t
`t h r e e s t a t i o n s s h o w n i n f i g u r e
`1 4 - 1 7 . ”
`
`R e p l a c e 1 2 0 0 / 3 . 7 w i t h 3 . 7 / 1 2 0 0
`
`R e p l a c e 8 0 0 / - 5 . 0 w i t h
`
`- 5 . 0 / 8 0 0
`
`P a g e
`2 - 3
`
`C o l u m n
`1
`
`2
`
`1
`
`2 2
`
`3 - 1 1
`
`3 - 1 9
`
`5 - 7
`
`7 - 1 7
`
`1 4 - 1 9
`
`1 4 - 2 8
`
`1 4 - 2 8
`
`

`

`

`

`DRAWING FORMATS
`
`Drawing format is the systematic space
`arrangement of required information within the drafting
`sheet. This information is used to identify, process, and
`file drawings methodically. Standard sizes and formats
`for military drawings are arranged according to
`DoD-STD-100C, Engineering Drawing Practices, and
`MIL-HDBK- 1006/1, Policy and Procedures for Project
`Drawing and Specification Preparation. With the
`exception of specific local command requirements,
`DoD-STD-1OOC and MIL-HDBK-1006/1 are your
`guidelines for preparing SEABEE drawings.
`
`Most of the documents applicable to these
`standards have recently been revised and updated in
`order to gain like information and to share uniformity
`of form and language within the Naval Construction
`Force and between DoD organizations. Other
`
`influencing factors are the current widespread use of
`reduced-size copies of both conventional and
`computer-generated drawings and exchange of
`microfilm.
`
`SHEET SIZES
`
`Standard drawing sheet sizes are used to facilitate
`readability, reproduction, handling, and uniform filing.
`Blueprints produced from standard size drawing sheets
`are easily assembled in sets for project stick files and
`can readily be folded for mailing and neatly filed in
`project letter size or legal size folders. (Filing drawings
`and folding blueprints will be covered later in this
`training manual.)
`
`Finished format sizes for drawings shown in
`figure 3-14, view A, are according to ANSI Y14.1
`
`Figure 3-14.—Guide for preparing horizontal and vertical margins, sizes, and finished drawing format.
`
`45.857
`
`3-11
`
`

`

`

`

`Figure 5-11.—Alternative method of extending to top view projection lines.
`
`Figure 5-12.—American standard arrangement of views in a six-view third-angle multi-view projection.
`
`view always lies in the plane of the drafting surface and
`does not require any rotation. Notice that the front, right
`side, left side, and rear views line up in direct horizontal
`projection.
`
`Use the minimum number of views necessary to
`show an item. The three principal views are the top,
`front, and right-side. The TOP VIEW (also called a
`PLAN in architectural drawings) is projected to and
`drawn on an image plane above the front view of the
`
`object. The FRONT VIEW (ELEVATION) should
`show the most characteristic shape of the object or its
`most natural appearance when observed in its
`permanent or fixed position. The RIGHT-SIDE VIEW
`(ELEVATION) is located at a right angle to the front
`and top views, making all the views mutually
`perpendicular.
`
`SPACING OF VIEWS.— Views should be
`spaced on the paper in such a manner as
`
`5-7
`
`

`

`

`

`Figure 7-23.—Expansion joint for a wall.
`
`Figure7-21.—Construction joint between wall and
`footing with a keyway.
`
`Figure 7-22.—Use of a contraction joint.
`
`incident to shrinkage of the concrete. Atypical dummy
`contraction joint (fig. 7-22) is usually formed by cutting
`a depth of one third to one fourth the thickness of the
`section. Some contracting joints are made with no filler
`or with a thin coat of paraffin or asphalt and/or other
`materials to break the bond. Depending on the extent
`of local temperature, joints in reinforced concrete slabs
`may be placed at 15- to 25-ft intervals in each direction.
`
`Expansion Joints
`
`Figure 7-24.—Expansion joint for a bridge.
`
`Figure 7-25.—Expansion joint for a floor slab.
`
`Wherever expansion might cause a concrete slab to
`buckle because of temperature change, expansion joints
`(also called isolation joints) are required. An expansion
`joint is used with a pre-molded cork or mastic filler to
`separate sections from each other, thus allowing room
`for expansion if elongation or closing of the joint is
`anticipated. Figures 7-23, 7-24, and 7-25 show
`
`expansion joints for a variety of locations. Expansion
`joints may be installed every 20 ft.
`
`CONCRETE FORMS
`
`Most structural concrete is made by placing
`(also called CASTING) plastic concrete into
`
`7-17
`
`

`

`

`

`
`
`
`
`
`PREFACE
`
`By enrolling in this self-study course, you have demonstrated a desire to improve yourself and the Navy.
`Remember, however, this self-study course is only one part of the total Navy training program. Practical
`experience, schools, selected reading, and your desire to succeed are also necessary to successfully round
`out a fully meaningful training program.
`
`THE COURSE: This self-study course is organized into subject matter areas, each containing learning
`objectives to help you determine what you should learn along with text and illustrations to help you
`understand the information. The subject matter reflects day-to-day requirements and experiences of
`personnel in the rating or skill area. It also reflects guidance provided by Enlisted Community Managers
`(ECMs) and other senior personnel, technical references, instructions, etc., and either the occupational or
`naval standards, which are listed in the Manual of Navy Enlisted Manpower Personnel Classifications
`and Occupational Standards, NAVPERS 18068.
`
`THE QUESTIONS: The questions that appear in this course are designed to help you understand the
`material in the text.
`
`VALUE: In completing this course, you will improve your military and professional knowledge.
`Importantly, it can also help you study for the Navy-wide advancement in rate examination. If you are
`studying and discover a reference in the text to another publication for further information, look it up.
`
`1991 Edition Prepared by
`EAC Andres M. Embuido,
`EACS Reynaldo N. Azucena, and
`EACS Gary L. Davis
`1995 Revision
`EAC(SCW) Michael R. Mann
`
`NAVSUP Logistics Tracking Number
`0504-LP-026-7350
`
`i
`
`

`

`Sailor’s Creed
`
`“I am a United States Sailor.
`
`I will support and defend the
`Constitution of the United States of
`America and I will obey the orders
`of those appointed over me.
`
`I represent the fighting spirit of the
`Navy and those who have gone
`before me to defend freedom and
`democracy around the world.
`
`I proudly serve my country’s Navy
`combat team with honor, courage
`and commitment.
`
`I am committed to excellence and
`the fair treatment of all.”
`
`ii
`
`

`

`CONTENTS
`
`CHAPTER
`
`1. Mathematics and Units of Measurement. . . . . . . . . . . . .
`
`2. Drafting Equipment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
`
`3. Drafting: Fundamentals and Techniques;
`Reproduction Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
`
`4. Drafting: Geometric Construction . . . . . . . . . . . . . . . . . .
`
`5. Drafting: Projections and Sketching . . . . . . . . . . . . . . . .
`
`6. Wood and Light Frame Structures. . . . . . . . . . . . . . . . . .
`
`7. Concrete and Masonry . . . . . . . . . . . . . . . . . . . . . . . . . . . .
`
`8. Mechanical Systems and Plan. . . . . . . . . . . . . . . . . . . .
`
`9. Electrical Systems and Plan. . . . . . . . . . . . . . . . . . . . .
`
`10. Construction Drawings . . . . . . . . . . . . . . . . . . . . . . . . . . . .
`
`11. Elements of Surveying and Surveying
`Equipment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
`
`12. Direct Linear Measurements and Field Survey
`Safety . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
`
`13. Horizontal Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
`
`14. Direct Leveling and Basic Engineering Surveys . . . . . . .
`
`15. Materials Testing: Soil and Concrete . . . . . . . . . . . . . . . .
`
`16. Administration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
`
`APPENDIX
`
`I. Glossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
`
`II. Engineering Technical Library . . . . . . . . . . . . . . . . . . . . .
`
`III. Useful Mathematical Symbols, Formulas, and
`Constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
`
`IV. Useful Drafting Symbols.. . . . . . . . . . . . . . . . . . . . . . . . .
`
`V. Sample Survey Field Notes, . . . . . . . . . . . . . . . . . . . . . . .
`
`VI. References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
`
`Page
`
`1-1
`
`2-1
`
`3-1
`
`4-1
`
`5-1
`
`6-1
`
`7-1
`
`8-1
`
`9-1
`
`10-1
`
`11-1
`
`12-1
`
`13-1
`
`14-1
`
`15-1
`
`16-1
`
`I-1
`
`II-1
`
`III-1
`
`IV-1
`
`V-1
`
`VI-1
`
`INDEX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . INDEX-1
`
`iii
`
`

`

`CREDITS
`
`T h e i l l u s t r a t i o n i n d i c a t e d b e l o w i s i n c l u d e d i n t h i s e d i t i o n o f
`Engineering Aid Basic through the courtesy of the designated company.
`P e r m i s s i o n t o u s e t h i s i l l u s t r a t i o n i s g r a t e f u l l y a c k n o w l e d g e d .
`
`SOURCE
`
`E L E I n t e r n a t i o n a l , I n c .
`
`FIGURE
`
`15-28
`
`i v
`
`

`

`
`
`
`
`
`INSTRUCTIONS FOR TAKING THE COURSE
`
`ASSIGNMENTS
`
`The text pages that you are to study are listed at
`the beginning of each assignment. Study these
`pages carefully before attempting to answer the
`questions. Pay close attention to tables and
`illustrations and read the learning objectives.
`The learning objectives state what you should be
`able to do after studying the material. Answering
`the questions correctly helps you accomplish the
`objectives.
`
`SELECTING YOUR ANSWERS
`
`Read each question carefully, then select the
`BEST answer. You may refer freely to the text.
`The answers must be the result of your own
`work and decisions. You are prohibited from
`referring to or copying the answers of others and
`from giving answers to anyone else taking the
`course.
`
`SUBMITTING YOUR ASSIGNMENTS
`
`To have your assignments graded, you must be
`enrolled in the course with the Nonresident
`Training Course Administration Branch at the
`Naval Education and Training Professional
`Development and Technology Center
`(NETPDTC). Following enrollment, there are
`two ways of having your assignments graded:
`(1) use the Internet to submit your assignments
`as you complete them, or (2) send all the
`assignments at one time by mail to NETPDTC.
`
`Grading on the Internet:
`Internet grading are:
`
`Advantages to
`
`you may submit your answers as soon as
`you complete an assignment, and
`you get your results faster; usually by the
`next working day (approximately 24 hours).
`
`In addition to receiving grade results for each
`assignment, you will receive course completion
`confirmation once you have completed all the
`
`assignments. To submit your assignment
`answers via the Internet, go to:
`
`https://courses.cnet.navy.mil
`
`COMPLETION TIME
`
`Courses must be completed within 12 months
`from the date of enrollment. This includes time
`required to resubmit failed assignments.
`
`v
`
`

`

`PASS/FAIL ASSIGNMENT PROCEDURES
`
`If your overall course score is 3.2 or higher, you
`will pass the course and will not be required to
`resubmit assignments. Once your assignments
`have been
`graded you will receive course
`completion confirmation.
`
`If you receive less than a 3.2 on any assignment
`and your overall course score is below 3.2, you
`will be given the opportunity to resubmit failed
`assignments. You may resubmit failed
`Internet students will
`assignments only once.
`receive notification when they have failed an
`assignment--they may then resubmit failed
`assignments on the web site. Internet students
`may view and print results for failed
`assignments from the web site. Students who
`submit by mail will receive a failing result letter
`and a new answer sheet for resubmission of each
`failed assignment.
`
`COMPLETION CONFIRMATION
`
`After successfully completing this course, you
`will receive a letter of completion.
`
`STUDENT FEEDBACK QUESTIONS
`
`We value your suggestions, questions, and
`criticisms on our courses. If you would like to
`communicate with us regarding this course, we
`encourage you, if possible, to use e-mail. If you
`write or fax, please use a copy of the Student
`Comment form that follows this page.
`
`
`
`
`
`
`
`NAVAL RESERVE RETIREMENT CREDIT
`
`If you are a member of the Naval Reserve, you
`will receive retirement points if you are
`authorized to receive them under current
`directives governing retirement of Naval
`Reserve personnel. For Naval Reserve
`retirement, this course is evaluated at 20 points.
`These points will be credited in units as follows:
`
`Unit 1 - 12 points upon satisfactory
`completion of Assignments 1 through 8
`
`Unit 2 - 8 points upon satisfactory
`completion of Assignments 9 through 13
`
`(Refer to Administrative Procedures for Naval
`Reservists on Inactive Duty,
`BUPERSINST
`1001.39, for more information about retirement
`points.)
`
`COURSE OBJECTIVES
`
`In completing this nonresident training course,
`you will demonstrate a knowledge of the subject
`
`vi
`
`

`

`matter by correctly answering questions on the
`following subjects: Mathematics and Units of
`Measurement; Drafting Equipment; Drafting:
`Fundamentals
`and Techniques; Drafting:
`Geometric Construction; Drafting: Projections
`and Sketching; Reproduction Process; Wood and
`Light Frame Structures; Concrete and Masonry;
`Mechanical Systems and Plan; Electrical
`Systems and Plan; Construction Drawings;
`Elements
`of Surveying
`and Surveying
`Equipment; Direct Linear Measurements and
`Field Survey Safety; Horizontal Control; Direct
`Leveling and Basic Engineering Surveys;
`Materials Testing: Soil and Concrete; and
`Administration.
`
`vii
`
`

`

`

`

`Student Comments
`
`Course Title:
`
`Engineering Aid 3
`
`NAVEDTRA:
`
`14069
`
`Date:
`
`We need some information about you:
`
`Rate/Rank and Name:
`
`Street Address:
`
`SSN:
`
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`
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`
`State/FPO:
`
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`
`Your comments, suggestions, etc.:
`
`Privacy Act Statement: Under authority of Title 5, USC 301, information regarding your military status is
`requested in processing your comments and in preparing a reply. This information will not be divulged without
`written authorization to anyone other than those within DOD for official use in determining performance.
`
`NETPDTC 1550/41 (Rev 4-00)
`
`ix
`
`

`

`

`

`CHAPTER 1
`
`MATHEMATICS AND UNITS
`OF MEASUREMENT
`
`Mathematics is the Engineering Aid’s basic
`tool. The use of mathematics is found in every
`rating in the Navy, from the simple arithmetic
`of counting for inventory purposes to the
`complicated equations encountered in computer
`and engineering designs. In the Occupational Field
`13 ratings, the Engineering Aid is looked upon
`as superior in knowledge when it comes to the
`subject of mathematics, which generally is a
`correct assumption; however, to be worthy of this
`calling, you have the responsibility to learn more
`about this subject. Mathematics is a broad science
`that cannot be covered fully in formal service
`school training, so it is up to you to devote
`some of your own time to the study of this
`subject.
`The EA must have the ability to compute
`easily, quickly, systematically, and accurately.
`This requires a knowledge of the fundamental
`properties of numbers and the ability to estimate
`the accuracy of computations based on field
`measurements or collected field data. To compute
`rapidly, you need constant practice and should
`be able to use any available device to speed
`up and simplify computations. In solving a
`mathematical problem, you should take a
`different approach than you would if it were
`simply a puzzle you were solving for fun.
`Guesswork has no place in its consideration, and
`the statement of the problem itself should be
`devoid of anything that might obscure its true
`meaning. Mathematics is not a course in memory
`but one in reasoning. Mathematical problems
`should be read and so carefully analyzed that all
`conditions are well fixed in mind. Avoid all
`unnecessary work and shorten the solution
`wherever possible. Always apply some proof or
`check to your work. Accuracy is of the greatest
`importance; a wrong answer is valueless.
`This chapter covers various principles of
`mathematics. The instructions given will aid the
`EA in making mathematical computations in the
`field and the office. This chapter also covers units
`of measurement and the conversion from one
`
`system to the other; that is, from the English to
`the metric system.
`
`FUNDAMENTALS OF
`MATHEMATICS
`
`MATHEMATICS is, by broad definition, the
`science that deals with the relationships between
`quantities and operations and with methods by
`which these relationships can be applied to
`determine unknown quantities from given or
`measured data. The fundamentals of mathematics
`remain the same, no matter to what field they are
`applied. Various authors have attempted to
`classify mathematics according to its use. It has
`been subdivided into a number of major branches.
`Those with which you will be principally
`concerned are arithmetic, algebra, geometry, and
`trigonometry.
`ARITHMETIC is the art of computation by
`the use of positive real numbers. Starting with the
`review of arithmetic, you will, by diligent effort,
`build up to a study of algebra.
`ALGEBRA is the branch of mathematics that
`deals with the relations and properties of numbers
`by means of letters, signs of operation, and other
`symbols. Algebra includes solution of equations,
`polynomials, verbal problems, graphs, and so on.
`GEOMETRY is the branch of mathematics
`that investigates the relations, properties, and
`measurement of solids, surfaces, lines, and angles;
`it also deals with the theory of space and of figures
`in space.
`TRIGONOMETRY is the branch of mathe-
`matics that deals with certain constant relation-
`ships that exist in triangles and with methods by
`which they are applied to compute unknown
`values from known values.
`
`STUDY GUIDES
`
`Mathematics is an
`many books on the
`
`exact science, and there are
`subject. These numerous
`
`l-l
`
`

`

`books are the result of the mathematicians’
`efforts to solve mathematical problems with ease.
`Methods of arriving at solutions may differ, but
`the end results or answers are always the same.
`These different approaches to mathematical
`problems make the study of mathematics more
`interesting, either by individual study or as a
`group.
`You can supplement your study of mathe-
`matics with the following training manuals:
`
`1. Mathematics, Vol. 1, N A V E D T R A
`10069-D1
`
`2. Mathematics, Vol. 2-A, N A V E D T R A
`10062
`
`3. Mathematics, Vol. 2-B, N A V E D T R A
`10063
`
`4. Mathematics, Vol. 3, N A V E D T R A
`10073-A1
`
`TYPES OF NUMBERS
`
`Positive and negative numbers belong to the
`class called REAL NUMBERS. Real numbers and
`imaginary numbers make up the number system
`in algebra. However, in this training manual, we
`will deal only with real numbers unless otherwise
`indicated.
`A real number may be rational or irrational.
`The word rational comes from the word ratio. A
`number is rational if it can be expressed as the
`quotient, or ratio, of two whole numbers.
`Rational numbers include fractions like 2/7,
`whole numbers (integers), and radicals if the
`radical is removable. Any whole number is
`rational because it could be expressed as a
`quotient with 1 as its denominator. For instance,
`8 equals 8/1, which is the quotient of two integers.
`A number like
` is rational since it can be
`expressed as the quotient of the two integers in
`the form 4/1. An irrational number is a real
`number that cannot be expressed as the ratio of
`two integers. The numbers
`
` are examples of irrational numbers.
`and 3.1416
`An integer may be prime or composite. A
`number that has factors other than itself and
`1 is a composite number. For example, the
`number 15 is composite. It has the factors 5
`
`and 3. A number that has no factors except itself
`and 1 is a prime number. Since it is advantageous
`to separate a composite number into prime
`factors, it is helpful to be able to recognize a few
`prime numbers. The following are examples of
`prime numbers: 1, 2, 3, 5, 7, 11, 13, 17, 19, and
`23.
`
`A composite number may be a multiple of two
`or more numbers other than itself and 1, and it
`may contain two or more factors other than itself
`and 1. Multiples and factors of numbers are as
`follows: Any number that is exactly divisible by
`a given number is a multiple of the given number.
`For example, 24 is a multiple of 2, 3, 4, 6, 8, and
`12 since it is divisible by each of these numbers.
`Saying that 24 is a multiple of 3, for instance,
`is equivalent to saying that 3 multiplied by
`some whole number will give 24. Any number is
`a multiple of itself and also of 1.
`
`FRACTIONS, DECIMALS,
`AND PERCENTAGES
`
`The most general definition of a fraction states
`that “a fraction is an indicated division. ” Any
`division may be indicated by placing the dividend
`over the divisor with a line between them. By the
`above definition, any number, even a so-called
`“whole” number, may be written as a common
`fraction. The number 20, for example, may be
`written as 20/1. This or any other fraction
`that amounts to more than 1 is an IMPROPER
`fraction. For example, 8/3 is an improper
`fraction, The accepted practice is to reduce an
`improper fraction to a mixed fraction (a whole
`number plus a proper fraction). Perform the
`indicated division and write the fractional part of
`the quotient in its lowest term. In this case,
`8/3 would be 2 2/3. A fraction that amounts to
`less than 1 is a PROPER fraction, such as the
`fraction 1/4.
`To refresh your memory, we are including the
`following rules in the solution of fractions:
`
`1. If you multiply or divide both the
`numerator and denominator of a fraction by the
`same number, the value does not change. The
`resulting fraction is called an EQUIVALENT
`fraction.
`2. You can add or subtract fractions only if
`the denominators are alike.
`3. To multiply fractions, simply find the prod-
`ucts of the numerators and the products of the
`denominators. The resulting fractional product
`must be reduced to the lowest term possible.
`
`1-2
`
`

`

`4. TO divide a fraction by a fraction, invert
`the divisor and proceed as in multiplication.
`5. The method of CANCELING can be used
`to advantage before multiplying fractions (using
`the principle of rule No. 1) to avoid operations
`with larger numbers.
`
`A decimal fraction is a fraction whose
`denominator is 10 or some power of 10, such as
`100, 1,000, and so on. For example,
`
`are decimal fractions. Accordingly, they could
`be written as 0.7, 0.23 and 0.087 respec-
`tively. Decimal fractions have certain char-
`acteristics that make them easier to use in
`computations than other fractions. Chapter 5
`of NAVEDTRA 10069-D1 deals entirely with
`decimal fractions. A thorough understanding
`of decimals will be useful to the Engineering
`Aid in making various engineering compu-
`tations. Figure 1-1 shows decimal equivalents
`of fractions commonly used by Builders,
`Steelworkers, Utilitiesmen, and other trades.
`
`Figure 1-1. Decimal equivalents.
`
`1-3
`
`

`

`Figure 1-2.-2-percent grade.
`
`In connection with the study of decimal
`fractions, businessmen as early as the fifteenth
`century made use of certain decimal fractions so
`much that they gave them the special designation
`PERCENT. The word percent is derived from
`Latin. It was originally per centum, which means
`“by the hundredths.” In banking, interest rates
`are always expressed in percent; statisticians use
`percent; in fact, people in almost all walks of life
`use percent to indicate increases or decreases in
`production, population, cost of living, and so on.
`The Engineering Aid uses percent to express
`change in grade (slope), as shown in figure 1-2.
`Percent is also used in earthwork computations,
`progress reports, and other graphical representa-
`tions. Study chapter 6 of NAVEDTRA 1-0069-D1
`for a clear understanding of percentage.
`
`POWERS, ROOTS, EXPONENTS,
`AND RADICALS
`
`Any number is a higher power of a given root.
`To raise a number to a power means to multiply,
`using the number as a factor as many times as the
`power indicates. A particular power is indicated
`by a small numeral called the EXPONENT;
`for example, the small 2 on 32 is an exponent
`indicating the power.
`
`indicate cube, fifth, and seventh roots, respec-
`tively. The square root of a number may be either
`+ or – . The square root of 36 may be written
` =
` since 36 could have been the
`thus:
`product of ( + 6)( + 6) or ( – 6)( – 6). However, in
`practice, it is more convenient to disregard
`the double sign ( ± ). This example is what we
`call the root of a perfect square. Sometimes
`it is easier to extract part of a root only
`after separation of the factors of the number, such
`
` =
` =
` As you can see, we
`were able to extract only the square root
`of 9, and 3 remains in the radical because
`it is an irrational factor. This simplification
`of the radical makes the solution easier because
`you will be dealing with perfect squares and
`smaller numbers.
`
`Examples:
`
`Radicals are multiplied or divided directly.
`
`Examples:
`
`Like fractions, radicals can be added or sub-
`tracted only if they are similar.
`
`Examples:
`
`Many formulas require the power or roots of
`a number. When an exponent occurs, it must
`always be written unless its value is 1.
`A particular ROOT is indicated by the radical
`sign
` together with a small number called the
`INDEX of the root. The number under the radical
`sign is called the RADICAND. When the radical
`sign is used alone, it is generally understood to
`
` and
`mean a square root, and
`
`When you encounter a fraction under the
`radical, you have to RATIONALIZE the
`denominator before performing the indicated
`operation. If you multiply the numerator and
`denominator by the same number, you can
`
`1-4
`
`

`

`extract the denominator, as indicated by the
`following example:
`
`Now perform the indicated subtraction and
`bring down the next group to the right, thus:
`
`The same is true in the division of radicals;
`for example,
`
`Next, double the portion of the answer already
`found (4, which doubled is 8), and set the result
`down as the first digit of a new divisor, thus:
`
`Any radical expression has a decimal
`equivalent, which may be exact if the radicand
`is a rational number. If the radicand is not
`rational, the root may be expressed as a decimal
`approximation, but it can never be exact. A
`procedure similar to long division may be used
`for calculating square root. Cube root and higher
`roots may be calculated by methods based on
`logarithms and higher mathematics. Tables of
`powers and roots have been calculated for use in
`those scientific fields in which it is frequently
`necessary to work with roots. Such tables may be
`found in appendix I of Mathematics, Vol. 1,
`NAVEDTRA 10069-D 1, and in Surveying Tables
`and Graphs, Army TM 5-236. This method is,
`however, slowly being phased out and being
`replaced by the use of hand-held scientific
`calculators.
`
`Arithmetic Extraction of Square Roots
`
`If you do not have an electronic calculator,
`you may extract square roots arithmetically as
`follows:
`
`Suppose you want to extract the square root
`of 2,034.01. First, divide the number into
`two-digit groups, working away from the decimal
`point. Thus set off, the number appears as
`follows:
`
`Next, find the largest number whose square
`can be contained in the first group, This is the
`number 4, whose square is 16. The 4 is the first
`digit of your answer. Place the 4 above the 20,
`and place its square (16) under the first group,
`thus:
`
`1-5
`
`The second digit of the new divisor is
`obtained by a trial-and-error method. Divide the
`single digit 8 into the first two digits of the
`remainder 434 (that is, into 43) until you obtain
`the largest number that you can (1) add as another
`digit to the divisor and (2) use as a multiplier
`which, when multiplied by the increased divisor,
`will produce the largest result containable in the
`remainder 434. In this case, the first number you
`try is 43 + 8, or 5. Write this 5 after the 8 and
`you get 85. Multiply 85 by 5 and you get 425,
`which is containable in 434.
`The second digit of your answer is therefore
`5. Place the 5 above 34. Your computation will
`now look like this:
`
`Proceed as before to perform the indicated
`subtraction and bring down the next group, thus:
`
`Again double the portion of the answer
`already found, and set the result (45 x 2, or 90)
`down as the first two digits of a new divisor thus:
`
`

`

`Proceed as before to determine the largest
`number that can be added as a digit to the divisor
`90 and used as a multiplier which, when multiplied
`by the increased divisor, will produce a result
`containable in the remainder, 901. This number
`is obviously 1. The increased divisor is 901, and
`this figure, multiplied by the 1, gives a result
`exactly equal to the remainder 901.
`The figure 1 is therefore the third and final
`digit in the answer, The square root of 2,034.01
`is therefore 45.1
`Your completed computation appears thus:
`
`Fractional and Negative Exponents
`
`In some formulas, like the velocity (V) of
`liquids in pipes, which you will encounter later
`in Engineering Aid 1 & C, it is more convenient
`to use FRACTIONAL EXPONENTS instead of
`radicals.
`
`It is readily observed that the index of the root
`in the above examples is the denominator of the
`fractional exponent. When an exponent occurs in
`this exponent becomes the
`the radicand,
`numerator of the fractional exponent. Roots of
`numbers not found in tables may be easily
`computed by proper treatment of the radical used.
`
`Examples:
`
`1-6
`
`Very small or very large numbers used
`in science are expressed in the form 5.832 x 10-4
`or 8.143 x 106 to simplify computation. To write
`out any of these numbers in full, just move
`the decimal point to either left or right, the
`number of places equal to the exponent,
`supplying a sufficient number of zeros depending
`upon the sign of the exponent, as shown below:
`
`RECIPROCALS
`
`The reciprocal of a number is 1 divided by the
`number. The reciprocal of 2, for example, is 1/2,
`and the reciprocal of 2/3 is 1 divided by 2/3,
`which amounts to 1 x 3/2, or 3/2. The reciprocal
`of a whole number, then, equals 1 over the
`number, while the reciprocal of a fraction equals
`the fraction inverted.
`In problems containing the power of 10,
`generally, it is more convenient to use reciprocals
`rather than write out lengthy decimals or whole
`numbers.
`
`Example:
`
`

`

`Reciprocal is also used in problems involving
`trigonometric functions of angles, as you will see
`later in this chapter, in the solutions of problems
`containing identities.
`
`RATIO AND PROPORTION
`
`Almost every computation you will make as
`an EA that involves determining an unknown
`value from given or measured values will involve
`the solution of a proportional equation. A
`thorough understanding of ratio and proportion
`will greatly help you in the solution of both
`surveying and drafting problems.
`The results of observation or measurement
`often must be compared to some standard value
`in order to have any meaning. For example, if the
`magnifying power of your telescope is 20
`diameters and you see a telescope in the market
`that says 50 diameter magnifying power, then one
`can see that the latter has a greate