0504LP0267940.pdf

[Virtual Presenter] DISTRIBUTION STATEMENT A: Approved for public release; distribution is unlimited. NONRESIDENT TRAINING COURSE June 1985 Mathematics, Basic Math and Algebra NAVEDTRA 14139.

[Audio] DISTRIBUTION STATEMENT A: Approved for public release; distribution is unlimited. 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..

[Audio] i 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. COURSE OVERVIEW: This course provides a review of basic arithmetic and continues through some of the early stages of algebra. Emphasis is placed on decimals, percentages and measurements, exponents, radicals and logarithms. Exercises are provided in factoring polynomials, linear equations, ratio, proportion and variation, complex numbers, and quadratic equations. The final assignment affords the student an opportunity to demonstrate what he or she has learned concerning plane figures, geometric construction and solid figures, and slightly touches on numerical trigonometry. 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. 1980 Edition Reprinted 1985 Published by NAVAL EDUCATION AND TRAINING PROFESSIONAL DEVELOPMENT AND TECHNOLOGY CENTER NAVSUP Logistics Tracking Number 0504-LP-026-7940.

[Audio] ii 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.".

Chapter 2. 3. 4. 5. 6. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. Appendix 1. n. CONTENTS Number systems and sets . . . Positive integers . Signed numbers . . . . . . . . . . Common fractions . Decimals . . . Percentage and measurement Exponents and radicals . Logarithms and the slide rule Fundamentals Of algebra . . . . . . Factoring polynomials . Linear equations in one variable Linear equations in two variables . Ratio, proportion, and variation . Dependence, functions, and formulas . Complex numbers . Quadratic equations in one variable Plane figures . Geometric constructions and solid figures . Numerical trigonometry Squares, cubes, square roots, cube roots, logarithms, and reciprocals Of numbers . . . . Natural sines, cosines, and tangents of angles from 00 to 900 Mathematical symbols Weights measures . . . . . . . . Formulas . . . , . iii Page 1 19 28 45 55 65 80 98 111 120 130 141 151 158 167 181 190 199 210 213 219 220 221 222.

[Audio] iv 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: Advantages to Internet grading are: • 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: http://courses.cnet.navy.mil Grading by Mail: When you submit answer sheets by mail, send all of your assignments at one time. Do NOT submit individual answer sheets for grading. Mail all of your assignments in an envelope, which you either provide yourself or obtain from your nearest Educational Services Officer (ESO). Submit answer sheets to: COMMANDING OFFICER NETPDTC N331 6490 SAUFLEY FIELD ROAD PENSACOLA FL 32559-5000 Answer Sheets: All courses include one "scannable" answer sheet for each assignment. These answer sheets are preprinted with your SSN, name, assignment number, and course number. Explanations for completing the answer sheets are on the answer sheet. Do not use answer sheet reproductions: Use only the original answer sheets that we provide—reproductions will not work with our scanning equipment and cannot be processed. Follow the instructions for marking your answers on the answer sheet. Be sure that blocks 1, 2, and 3 are filled in correctly. This information is necessary for your course to be properly processed and for you to receive credit for your work. COMPLETION TIME Courses must be completed within 12 months from the date of enrollment. This includes time required to resubmit failed assignments..

[Audio] 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 assignments only once. Internet students will 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. ERRATA Errata are used to correct minor errors or delete obsolete information in a course. Errata may also be used to provide instructions to the student. If a course has an errata, it will be included as the first page(s) after the front cover. Errata for all courses can be accessed and viewed/downloaded at: http://www.advancement.cnet.navy.mil 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. For subject matter questions: E-mail: n3222.products@cnet.navy.mil Phone: Comm: (850) 452-1001, Ext. 1520 or 1518 DSN: 922-1001, Ext. 1520 or 1518 FAX: (850) 452-1694 (Do not fax answer sheets.) Address: COMMANDING OFFICER NETPDTC N3222 6490 SAUFLEY FIELD ROAD PENSACOLA FL 32509-5237 For enrollment, shipping, grading, or completion letter questions E-mail: fleetservices@cnet.navy.mil Phone: Toll Free: 877-264-8583 Comm: (850) 452-1511/1181/1859 DSN: 922-1511/1181/1859 FAX: (850) 452-1370 (Do not fax answer sheets.) Address: COMMANDING OFFICER NETPDTC N331 6490 SAUFLEY FIELD ROAD PENSACOLA FL 32559-5000 NAVAL RESERVE RETIREMENT CREDIT If you are a member of the Naval Reserve, you may earn retirement points for successfully completing this course, if authorized under current directives governing retirement of Naval Reserve personnel. For Naval Reserve retirement, this course is evaluated at 22 points, which will be credited in units as shown below: Unit 1 – 12 points upon satisfactory completion of Assignments 1 through 6 Unit 2 – 10 points upon satisfactory completion of Assignments 7 through 11 (Refer to Administrative Procedures for Naval Reservists on Inactive Duty, BUPERSINST 1001.39, for more information about retirement points.).

[Audio] vii Student Comments Course Title: Mathematics, Basic Math and Algebra NAVEDTRA: 14139 Date: We need some information about you: Rate/Rank and Name: SSN: Command/Unit Street Address: City: State/FPO: Zip 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.

CHAPTER 1 NUMBER SYSTEMS AND SETS Mathematics is a tool. Some use of mathematics is found in every rating in the Navy, from the simple arithmetic of counting for inventory mxrposes to the complicated equa- dons encountered in commiter and engineering work. Storekeepers need mathematical compu- tauon in their bookkeeping. Damage Control- men need mathematics to commxte stress, cen- ters of gravity, maximum permissible roll. Electronics principles are frequently stated by means mathematical formulas. Navigation and engineering also use mathematics to a great extent. As maritime warfare becomes more and more .complex, mathematics achieves ever increasing importance as an essential tool. From the pant of view of the individual there are many incentives for learning the subject. Mathematics better equips him to do his pres- ent job. It will help him in attaining promotiona and the corresponding increases. Statisti- cany it has been found that one of the best indi- cators a man's potential success as a naval officer is his of mathematics. This training course begins with the basic facts of arithmetic and cmtinues through some ot the early stages of algebra. An attempt ie made throughout to give an understanding of why the rules of mathematics are true. Thig is done because it is felt that rules are easier to learn remember if the ideas that Led to their development are understood. Many us have areas in our maülematice that are hazy, barely or troublesome. Thug, while it may at firat seem beneath your dignity to read chapters on funda- mental arithmetic, these concepts may be just the spots where your difficulties lie. These chapters attempt to treat the subject on an adult level that will be interesting informative. COUNTING Counting is such a basic natural process that we rarely stop to think about it. The proc- ess iS based on the idea of ONE-TO-ONE COR- RESPONDENCE, Which is easily demonstrated by using the fingers. When children cmmt on their fingers, they are placing each finger in one-to-one correspondence with me of the ob- jects being counted. Having outgrown finger counting, we use numerals. NUMERALS Numerals are number symbols. One of the simplest numeral systems iS the Roman nu- meral system, in which tally marks are used to represent the objects being counted. Roman numerals appear to be a refinement of the tally method still in use today. By this one makes short vertical until a total ot is reached; when the fifth tally is counted, a diagonal mark 18 drawn through the first fmar marks. Grouping by fives in this way ig remi- niscent of the Roman numeral system, in which the multiples of five are represented by symbols. A number may have many "names." For aample, the number 6 may be indicated by any of the following symbols: 9 - 3, 12/2, 5 + 1, or 2 x 3. The important thing to remember is that a number ie idea; various symbols used to a number are merely different waye of expregeing ere same idea. WHOLE NUMBERS The numbers which are u•ed for counting in number system are someäme• called ral numbers. They are the poeiUve Whole num- bera, or to more preci•e mathematical term, positive NTEGERS. The Arabic nu- merals from O throudl 9 are caned digit', an integer may have any number digits. For example, 5, 32, and 7,049 are all integer'. me number of digits in an integer indicates it' rank; that is, whether it ig "in the hundreds 'f "in thousands," etc. The idea Of ranÄng numbers in terms of tens, thoueands, etc., is based on the PLACE VALUE concept. PLACE - VALUE a system such as the Roman nu- meral system is adequate for recording the 1.

MATHEMATICS results Of counting, it is too cumbersome for purposes Of calculation. Before arithmetic could develop as we know it today, the following two important concepts were needed as addi- tions to the counting process: 1. The idea of 0 as a number. 2. Positional notation (place value). Positional notation is a form Of coding in which the value of each digit of a number de- pends upon its position in relation to the other digits Of the number. The convention used in our number system is that each digit has a higher place value than those digits to the right of it. The place value which corresponds to a given position in a number is determined by the BASE Of the number system. The base which is most commonly used is ten, and the system with ten as a base is called the decimal system (decem is the Latin word for ten). Any number is as— sumed to be a base-ten number, unless some other base is One exception to ulis rule occurs when the subject of an entire dis- cussion is some base oüler than ten. For ex- ample, in the discussion of binary (base two) numbers later in this chapter, all numbers are assumed to be binary numbers unless some other base is indicated. DECIMAL SYSTEM In the decimal System, each digit position in a number has ten times the value of the position adjacent to it cm the right. For example, in the number 11, the 1 on the left is said to be in the 'tens place," and its value is 10 times as great as that of the 1 on the right. The 1 on the right is said to be in the "units place," with the un- derstanding that the term "unit" in our system refers to the numeral 1. Thus the number 11 is actually a coded symbol which means ' 'one ten plus one unit." Since ten plus one is eleven, the symbol 11 represents the number eleven. Figure 1-1 shows the names of several digit positions in the decimal System. If we apply this nomenclature to the digits Of the integer 235, then this number symbol means ' 'two hun- dreds plus three tens plus five units." This number may be expressed in mathematical symbols as follows: 2x lox 10+3 x 10+5 xl Notice that this bears out our earlier Statement: each digit position has 10 times the value Of the adjacent to it on the right. 2 VOLUME 1 1999 NITS ENS Figure of digit positions. The integer 4,372 is a number symbol whose meaning is "four thousands plus three hundreds plus seven tens plus two units." Expressed in mathematical symbols, this number is as fol- lows: 4 x 1000 + 3 x 100 + 7 x 10 +2 x 1 This presentation may be broken down further, in order to show that each digit position as 10 times the place value Of the position on its right, as follows: 4 x 100 + 3x 10 x 10 + 7 x +2 xl The comma which appears in a number Sym— bol Such as 4,372 is used for "pointing Off" the digits into groups of three beginning at the right-hand side. The first group of three digits on the right is the units group; the second group is the thousands group; the third group is the millions group; etc. Some Of these groups are shown in table 1-1. Table values and grouping. Billions group Millions Thousands group group Units group By reference to table 1-1, we can verify that 5,432,786 iS read as follows: five million, four.

Chapter I—NUMBER SYSTEMS AND SETS hundred thirty-two thousand, Seven hundred eighty-six. Notice that the wor•d "and" is not necessary when reading numbers of this kind. 1. 2. 3. 4. 2. 4. Practice problems: Write the number Symbol for seven thousand two hundred eighty-one. Write the meaning, in words, of the symbol 23,469. If a number iS in the millions, it must have at least how many digits ? If a number has 10 digits, to what number group (thousands , millions , belong ? Answers : 1. 7281 thousand, four nine. Billions BNARY SYSTEM etc.) does it hundred sixty— The binary number system is constructed in the same manner as the decimal system. How- ever, since the base in this system is two, only two digit symbols are needed for writing num- bers. These two digits are 1 and O. In order to understand why only two digit symbols are needed in the binary system, we may make Some observations about the decimal system and then generalize from these. One of the most striking observations about number systems which utilize the concept of place value is that there is no single-digit sym- bol for the base. For example, in the decimal system the symbol for ten, the base, is 10. This symbol is compounded from two digit symbols, and its meaning may be interpreted as "one base plus no units." Notice the implication of this where other bases are concerned: Every system uses the same symbol for the base, namely 10. Furthermore, the symbol 10 is not called 'ten" except in the decimal system. Suppose Qiat number system were con- structed with five as a base. Then the only digit symbols needed would be O, 1, 2, 3, and 4. No single-digit symbol for five is needed, since symbol 10 in a base-five system with place value means "one five plus no units." gen- eral, in a number system using base N, the largest number for which a single-digit symbol iS needed is N minus 1. Therefore, when the base is two the only digit symbols needed are 1 and O. An example Of a binary number is the sym- bol 101. We can discover the meaning of this symbol by relating it to the decimal System. Figure 1-2 shows that the place value of each digit position in the binary system is two times the place value of the position adjacent to it on the right. Compare this with figure 1-1, in which the base is ten rather than two. 010 UNITS Figure 1-2.—Digit positions in the binary system, Placing the digits of the number 101 in their respective blocks on figure 1-2, we find that 101 means "one four plus notwos plus one unit." Thus 101 is the binary equivalent Of decimal 5. we wish to convert a decimal number, such as 7, to its binary equivalent, we must break it into parts which are multiples of 2. Since 7 iS equal to 4 plus 2 plus 1, we say that it "con- tains" One 4, One 2, and One unit. Therefore the binary symbol for decimal 7 is 111. The most common use Of the binary number system is in electronic digital commxters. All data fed to a typical electronic digital comB1ter is converted to binary form and the computer performs its calculations using binary arith- metic rather than decimal arithmetic, One of the reasons for this is the fact that electrical electronic equipment utilizes many switch- ing circuits in which there are only two operat- ing conditions. Either the circuit is ' 'on" or it is "off," and a two-digit number system is ideally suited for symbolizing such a situation. Details concerning binary arithmetic are be- yond the scope of this volume, but are available in Mathematics, Volume 3, NavPers 10073, and in Basic Electonics, NavPers 10087-A. 1. 2. 1. 2. Practice problems: Write the decimal equivalents of the binary Write the binary equivalents of the decimal numbers 12, 7, 14, and 3. Answers: 13, 10, 9, and 15 1100, 111, 1110, and 11 3.

Chapter I—NUMBER SYSTEMS AND SETS increased to a larger number simply by adding planes (flat surfaces). A mathematical plane 1 to it. is determined by three points which do not lie One way to represent the setof natural num- on the same line. It is also determinedby two bers symbolically would be as follows: intersecting lines. Line Segments and Rays The three dots, called ellipsis, indicate that the When we draw a 'tline," label its end points pattern established by the numbers shown con- A and B, and call it 'line AB," we really mean tinues without limit. In other words, the next SEGMENT AB. A line segment is a sub- number in the set is understood to be 7, the set of the set of points comprising a line. next after that is 8, etc. When a line is considered to have a starting PONTS AND LNES point int no stopping point (that is, it extends without limit in one direction), it is called a In addition to the many sets which can be RAY. A ray is not a line segment, because it formed with number symbols, we frequently does not terminate at both ends; it may be ap- find it necessary in mathematics to work with propriate to refer to a ray as a 'half-line." sets composed of points or lines. A point is an idea, rather than a tangible ob- ject, just as a number is. The mark which is THE NUMBER LNE made on a piece of paper is merely a symbol representing åe point. strict mathematical terms, a point hu nodimensiong (physical size) As in the case of a line segment, a ray is a at all. Thus a pencil dot is only a rough picture subset Of the set of points comprising a line. of a point, useful for indicating the location of All three—lines, line segments, and rays—are the point certainly not to be confused with subsets of the set of points comprising a plane. the idea. Now suppose that a large numbér 01 points are placed side by side to form a "string." Among the many devices used for represent- Picturing this arrangement by drawing dots on ing a set of numbers, one of the most useful is paper, we would have a "dotted line." If more the number line. To illustrate the construction dots were placed between the dots already in of a number line, let us place the elements of the string, with the number of dots increasing the set of natural numbers in one-to-one cor- until we could not see between them, we would respondence with points on a line. Since the have a rough picture of a line. Once again, it natural numbers are equally spaced, we select is important to emNtasize that the picture is points such that the distances between them are only a symbol which represents an ideal line. equal. The starting point is labeled O, the next The ideal line would have length but no width or point is labeled 1, the next 2, etc., using the thickless. natural numbers in normal counting order. (See fig. 1-3.) Such an arrangement is often referred The foregoing discussion leads to the con- clusion that a line is actually a set of points. to as a scale, a familiar example being the The number of elements in the set is infinite, scale on a thermometer. since the line extends in both directions without limit. Thus far in our discussion, we have not men- The idea of arranging points together to tioned any numbers other than integers. The form a line may be extended to the formation of number line is an ideal device for picturing ule 2 3 Figure 1 4 -3.—A number line. 5 5 6 7.

MATHEMATICS, VOLUME 1 we had added 1 to the digits 5, 4, and 2 in the tens column Of the original problem. There- fore, the thought process in addition is as fol- lows: Add the 7, 5, and 2 in the units column, getting a sum Of 14. Write down the 4 in the units column of the answer and carry the I to the tens column. Mentally add the i along with the other digits in the tens column, getting a sum of 12. Write down the 2 in the tens column Of the answer and carry the 1 to the hundreds column. Mentally add the 1 along with the other digits in the hundreds column, getting a sum of 12. Write down the 2 in the hundreds column of the answer and carry the 1 to the thousands column. If there were other digits in the thou- sands column to which the 1 could be added, the process would continue as before. Since there are no digits in the thousands column Of the original problem, this final 1 is not added to anything, but is simply written in the thousands place in the answer. The borrow process is the reverse of carry- ing and is used in subtraction. Borrowing is not necessary in such problems as 46 - 5 and 58 - 53. In the first problem, the thought proc- ess may be "5 from 6 is 1 and bring down the 4 to get the difference, 41." In the second prob- lem, the thought process is "3 from 8 is 5" and "5 from 5 is zero," and the answer is 5. More explicitly, the subtraction process in these ex- amples is as follows: 40 +6 5 41 50 +8 50 +3 0+5=5 This illustrates that we are subtracting units from units and tens from tens. Now consider Üle following problem where borrowing is involved: 43 8 If the student uses the borrowing method, he may think "8 from 13 is 5 and bring down 3 to get the difference, 35." In this case what actu- ally was done is as follows: 30 + 13 8 30+5 = 35 A 10 has been borrowed from the tens column and combined with the 3 in the units column to make a number large enough for subtraction of the 8. Notice that borrowing to increase the value of the digit in the units column reduces the value of the digit in the tens column by 1. Sometimes itis necessary to borrow in more than one column. For example, suppose that we wish to subtract 2,345 from 5,234. Grouping the minuend and subtrahend in units, tens, hun- dreds, etc., we have the following: 5,000 + 200 + 30+4 2,000 + 300 + 40 + 5 Borrowing a 10 from the 30 in the tens column, we regroup as follows: 5,000 + 200 + 20 + 14 2,000 + 300 +40+ 5 The units column is now ready for subtrac- tion. By borrowing from the hundreds column, we can regroup so that subtraction is possible in the tens column, as follows: 5,000 + 100 + 120 + 14 2,000 300+ 40 + .5 the final regrouping, we borrow from Ole thousands column to make subtraction possible in the hundreds column, with the following result: 4,000 + 1,100 + 120 + 14 2,000+ 300+ 40+ 5 2,000+ 800+ 80+ 9=2,889 In actual practice, the borrowing and re- grouping are done mentally. The numbers are written in the normal manner, as follows: 5,234 -2,345 2,889 The following thought process is used: Borrow from the tens column, making the 4 become 14. Subtracting in the units column, 5 from 14 is 9. In the tens column, we now have a 2 in the min- uend as a result Of the first borrowing opera- tion. Some students find it helpful at first to cancel any digits that are reduced as a result of borrowing, jottingdown the digit of next lower 8.

has the subtractim, one column at a time. We borrow from the hundreds column to change the 2 that we now have in the tens column into 12. Sub- tracting in the tens column, 4 from 12 is 8. Proceeding in the same way for the hundreds column, 3 from 11 is 8. Finally, in the column, 2 from 4 is 2. Practice problems. problems 1 through 4, add the indicated numbers. In problems 5 through 8, subtract the lower number from the upper. 1. Au 23, 468, 7, and 9,045. 2. 129 958 787 436 5. 709 594 Chapter 2— POSITIVE NTEGERS value just above the canceled digit. This been done in the following example: 4 12 -2,345 2,889 After canceling the 3, we proceed with 1 in. , we s. 9,497 6,364 4,2e9 9 785 6. 8,700 5 008 a. 2,310 i. 9,543 5. 115 e. 3,692 Denominate Numbers 4. 67 ,856 22,851 44,238 97 158 7. 7,928 5 349 s. 29,915 7. 2,519 8. 75,168 28 089 4. 232,101 8. 47,079 A similar problem would be to 20 de- grees 44 minutes 6 seconds to 13 degrees 22 minutes 5 seconds. Thig is illustrated as fol- lows: 20 deg 44 min 6 sec 13 deg 22 min 5 sec 33 deg 66 min 11 sec This answer is regrmaped as 34 deg 6 min 11 sec. Numbers must be expressed in units of the same Hnd, in order to be combined. For in- stance, the sum of 6 kilowatts plus 1 watt is not 7 Hlowatt8 nor is it 7 watts. The sum can only be indicated (raÜier than performing the opera- tion) unless some meU10d is used to write these numbers in units of the same value. Subtraction of denominate numbers algo Ln- volves the regrouping idea. we wish to sub- tract 16 deg 8 min 2 gec from 28 deg 4 min 3 sec. for example, we would have the following arrangement: 28 deg 4 min 3 sec -16 deg 8 min 2 sec order to subtract 8 min from 4 min we re— group u follows: 27 deg 64 min S •ec -16 deg 8 min 2 •ec ii deg 58 miff i Practice problems. In problem: i, 2, and 3 add. In problem: 4, 5, and 6 subtract the Iowa number• from the upper. Numbers that have • unat ot meuure u•o- ciated them, •uch u yard, Elow•tt, pænd, pint, etc., are called DENOMNATE NUMBERS. The word '<enominzte" means the numbers have been aven name; they are not just ab- strut •ymbol•. To add denominate numbers, units ot the game Knd. SimplUy the re- gult, possible. The following example illus- trate• the addition of ft 8 in. to 4 ft 5 Ln.: ett 8 in. 10 ft 13 In. Since IS in. iS åe quivalent the an•wer ii ft 1 in. 2. 3. 1. 6yd2ft lin. Ift 9in. 2 yd 10 in. 9 hr 47 min 51 3 hr 36 min 23 sec Shr IS min 23 sec 10 wks 5 days 22 wks 3 days 3 wks 4 days 1 hrs 10 hrs 12 hrs 4. IS hr 25 min 10 see 6 hr 30 min 33 S. 123 deg 47 des 9 min 14 sec 6. 20 wks 2days 10hrs 1 wks 6 days IS hrs 9.

MATHEMATICS, 2. 3. 4. 5. 6. 26, 37 - - Final an Answers: 18 hr 39 min 37 sec 36 wks 6 days 5 hr 8 hr 34 min 35 sec 77 deg 50 min 46 sec 12 wks 2 days 19 hr VOLUME 1 Practice umns from example: problems. Add the following col- the top down, as in the precedi.ng 7 3 6 4 1 Answers, 2, 12, 22, 6 7 8 1 8 3. 88 36 59 82 28 57 4. 57 32 64 97 79 44 Mental Calculation Mental regrouping can be used to avoid the necessity of writing down some of the steps, or rewriting in columns, when groups one- digit or two-digit numbers are to be added or subtracted. of the most common devices for rapid addition is recognition of groups digits whose sum is 10. For example, in the following prob- lem two "ten groups" have been marked with braces: 7 64} 5 1 10 9 To add this column as grouped, you would say to yourself, "7, 17, 22, 32." The thought should be just the successive totals as shown above and not such cumbersome steps as "7 + 10, 17, +5, 22, + 10, 32." When successive digits appear in a column and their sum is less than 10, it ig often con- venient to think Of them, too, as a sum rather than separately. Thus, if adding a column in which the sum two successive digits iS 10 or 1. 2. 3. 4. showing successive mental steps: 23 - - Final answer, 23 Final answer, 34 10, 17, 26, 34-- Units column: 14. O, carry 4. Tens column: 12, ewer, 350. 9, Units column: 3, carry 3. 8, Tens column: swer, 373. 23, 20, 20, 17, 33, 40 - - Write down 30, 35 - - Final an- 29, 33 - - Write down SUBTRACTION .—ln an example such as 73 - 46, the conventional approach is to place 46 under 73 and subtract units from units and tens from tens, and write only the difference without the intermediate steps. To do this, the best method is to begin at the left. Thug, in the example 73 - 46, we take 40 from 73 and then take 6 from the result. This is mentally, however, and the thought would be "73, 33, 27," or "33, 27." the example 84 - 21 the thot%ht is "64, 63" and in the example 64 - 39 the thought is "34, 25." Practice problems. Mentally subtract and write only the difference: 1. 47 - 24 69 - 38 87 - 58 Answers, less, group them as follows: 3 1 81} 9 4 10 The thought process here might be, by the grouping, "5, 14, 24." 2. 3. 1. 2. 3. 4. 5. 6. 27, 39, 37, 16, 42, 20, 4. 86- 73 5. 82 - 41 6. 30- 12 showing successive mental etepa: 23 - - Final answer, 23 31 - - Final answer, 31 Final answer, 29 29 - - 13 - - Fi.nal answer, 13 41 - - Final answer, 41 18 — - Final anewer, 18 MULTIPLICATION AND DIVEION as shown 10 Multiplication may be indicated by a multi- plication sign (x) between two numbers, a dot.

between two numbers, or parentheses around one or both of the numbers to be multiplied. The following examples illustrate these methods: 48 6-8=48 6(8) 48 (6)(8) 48 Notice that when a dot is used to indicate multiplication, it is distinguished from a deci- mal point or a period by being placed above the line of writing, as in example 2, whereas a period or decimal point appears on the line. Notice also that when parentheses are used to indicate multiplication, the numbers to be mul- tiplied are spaced closer together than they are when the dot or x is used. In each of the four examples just given, 6 is the MULTIPLIER and 8 is the MULTIPLICAND. Both the 6 and the 8 are FACTORS, and the more modern texts refer to them this way. The "answer" in a multiplication problem is the PRODUCT; in the examples just given, the product is 48. Division usually is indicated either by a division sign (+) or by placing one number over another number with a line between the num- bers, as in the following examples: 1. 8-4=2 8 The number 8 iS the DIVIDEND, 4 is the DIVI- SOR, and 2 is the QUOTIENT. MULTIPLICATION METHODS The multiplication of whole numbers may be thought as a short process adding equal numbers. For example, 6(5) and 6 x 5 are read as six 5's. Of course we could write 5 six times and add, but ii we learn that the result is 30 we can save time. Although the concept of adding equal numbers ig quite adequate in explaining multiplication of whole numbers, Lt iS only a special case a more general definition, which Will be later in multiplication involv- ing fractions. Grouping Ikt us examine the process involved in mul- tiplying 6 times 27 to get the product 162. We first arrange the factors in thefonowing manner: Chapter 2—POSITIVE NTEGERS 6(27) = 162 27 162 The thought process is as follows: 1. 6 times 7 is 42. Write down the 2 and carry the 4. 2. 6 times 2 is 12. Add the 4 that was car- ried over from step 1 and write result, 16, beside the 2 that was written in step 1. 3. The final answer is 162. Table 2-1 shows that the factors were grouped in units, tens, etc. The multiplication was done in three steps: Six times 7 units is 42 units (or 4 tens and 2 units) and six times 2 tens is 12 tens (or 1 hundred and 2 tens). Then the tens were added and the product was written as 162. Table 2-1. —Multiplying by a one-digit number. 1 1 2 4 2 6 7 6 2 2 In preparing numbers for multiplication as in table 2-1, it is important to place the digits of the factors in the proper columns; that is, units must be placed in the units column, tens in tens column, and hundreds in hundreds col- umn. Notice that it is not necessary to write the zero in the case Of 12 tens (120) Since the 1 and 2 are written in the proper columns. practice, the addition is done mentally, and just the product is written without the intervening steps. Multiplying a number with more Elan two digits by a one-digit number, as shown in table 2-2, involves no new ideas. Three times 6 tmits iS 18 units (1 ten and 8 units), 3 times 0 tens is O, and 3 times 4 hundreds is 12 hundreds (1 11.

MATHEMATICS Table 2-2.—Multiplying a three- digit number by a one-digit number. 3(406) = 1,218 1 4 2 2 o 1 6 3 8 thousand and 2 hundreds). Notice that it is not necessa.ry to write the O's resulting from the step "3 times O tens is O." The two terminal O's of the number 1,200 are also omitted, since the 1 and the 2 are placed Ln their correct col- umns by the position of the 4. Partial hoducts In the example, 6(8) • 48, notice that multiplying could be done another way to the correct product ag follows: +5) -ex3+6x5 VOLUME 1 43 x27 301 = 3 hundreds, O tens, 1 unit 86 = 8 hundreds, 6 tens 1,161 As long as the partial products are written in the correct columns, we can multiply begin- ning from either the left or the right the multiplier. Thus, multiplying from the left, we have 43 x27 86 301 1,161 Multiplication by a number having more places involves no new ideas. End Zeros The placement of partial products must be kept when multiplying in problems volving end zeros, ag in the following example: the get That Is, we can break 8 into 3 and 5, multiply each of üle•e by the other tactor, ud add the partial product'. This idea is employed in multiplying by L two-digit number. Consider the following example: 43 1,161 Breaking the 27 Into 20 + 7, we have 7 unit' timee 43 plug 2 teng times 43, U follows: 43(20 + 7) + Since 7 units times 4S is SOI units, and 2 tens times 43 is 86 tens, we have the following: 27 1,080 We have O units time• 27 plug 4 tens times 27, foLow•: 27 108 1, The zero in the units place plays an importmt part in the the product. End zeros are dten caned "place holders" their only function in the problem to hold the digit positions which they occupy, thus helping to place the other digits Ln the problem correctly. The end zero in the foregoing problem can be accounted for very nicely, while at the game time placülgt.he other digits correctly, by means a shortcut. This consists the 40 one place to the right and then simply bringing 12.

Chapter 2—POSITIVE NTEGERS down the O, without using it as a multiplier at an. The problem would appear as follows: 27 x40 1,080 H the problem involves a multiplier with more than one end O, the multiplier is offset as many places to the right as there are end O's. For example, consider the following multipli- cation in which the multiplier, 300, has two end O' s: 220 x300 66,000 Notice that there are as many place-holding zeros at the end in the product as there are place-holding zeros in the multiplier and the multiplicand combmed. Placement Decimal Points any whole number in the decimal system, there is understood to be a terminating mark, called a decimal point, at the right-hand end Of number. Although the decimal point is sel- dom shown except in numbers involving decimal tractions (covered in chapter 5 this course), its location must be k10wn. The placement the decimal point is automatically taken care when the end O's are correctly placed. Practice problems. Multiply in each of the following problems: 1. 287 x 8 2. 67 x 49 3. 940 x 20 Answers: 1. 2 296 2. 3,283 3. 18 800 4. 807 x 28 5. 694 x 80 6. 9,241 x 7,800 4. 22,596 5. 55,520 6. can be considered as DIVISION METHODS Just as multiplication repeated addition, division can be considered as repeated subtraction. For example, if we wish to divide 12 by 4 we may subtract 4 from 12 in successive steps and tally the number of times that the subtraction is performed, as follows: 12 8 4 As indicated by the asterisks used as tally marks, 4 has been subtracted 3 times. This result Ls sometimes described by saymg that "4 is contained in 12 three times." Since successive subtraction is too cumber- some for rapid, concise calculation, methods which treat division as the inverse of multipli- cation are more useful. &wwledge of the mul- tiplication tables should lead us to an answer for a problem such as 12 + 4 immediately, since we that 3 x 4 is 12. However, a such as 84 4 is not 80 easy to solve by direct reference to the multiplication table. way to divide 84 by 4 is to note that 84 is the same as 80 plus 4. Thus 84 + 4 is the same as 80 4 plus 4 + 4. symbols, this can be indicated as follows: 20 + 1 (When this tme division symbol is used, the quotient 18 written above tie vinculum as shown.) Thus, 84 divided by 4 is 21. From the foregoing example, it can be seen that the regrouping is useful in division aa well as in multiplication. However, the mechanical procedure used in division does not include writing down the regrouped form of the divi- dend. After becoming familiar With the proc- ess, we find that the division can be performed directly, one digit at a time, with the regrouping taking place mentally. The following example illustrates this: 14 4 16 16 The thought process is as follows: "4 iS con- tained in 5 once" (write 1 in tens place over the 5); "one times 4 is 4" (write 4 in tens place 13.

MATHEMATICS under 5, take the difference, and bring down 6); and "4 is contained in 16 four times" (write 4 in units place over the 6). After a little prac- tice, many people can do the work shown under the dividend mentally and write only the quo- tient, if the divisor has only 1 digit. The divisor is sometimes too large to be contained Ln the first digit of the dividend. The following example illustrates a problem of this 36 21 42 42 Since 2 is not large enough to contain 7, we divide into the number formed by the first two digits, 25. Seven is contained 3 times in 25; we write 3 abovethe 5 the dividend. Multiplying, 3 times 7 is 21; We write 21 below the first two digits af the dividend. Subtracting, 25 minus 21 is 4; we write down the 4 and bring down the 2 in the units place of the dividend. We have now formed a new dividend, 42. Seven is contained 6 times in 42; we write 6 above the 2 of the dividend. Multiplying as before, 6 times 7 is 42; we write this product below the dividend 42. Subtracting, we have nothing left and the divi- Sion is complete. Estimation Wtten there are two or more digits Ln the divisor, it is not always easy to determine the first digit of the quotient. An estimate must be made, and the resulting trial quotient may be too large or too small. For example, if 1,862 is to be divided by 38, we might estimate that 38 is contained 5 times in 186 and thefirst digit of our trial divisor would be 5. However, mul- tiplication reveals that the product of 5 and 38 is larger than 186. Thus we would change the 5 in our quotient to 4, and the problem would then appear as follows: 49 152 342 342 14 VOLUME 1 the other hand, suppose that we had esti- mated that 38 is contained in 186 only 3 times. We would then have the following: 3 114 72 Now, before we make any further moves in the division process, it should be obvious that some- thing is wrong. our new dividend is large to contain the divisor before bringing down a digit from the original dividend, then the trial quotient should have been larger. In other words, our estimate is too small. Proficiency in estimating trial quotients is gained through practice and familiarity with number combinations. For example, after a little experience we realize that a close esti- mate can be made in the foregoing problem by thinking of 38 as "almost 40." It is easy to see that 40 is contained 4 times in 186, since 4 times 40 is 160. Also, since 5 times 40 is 200, we are reasonably Certain that 5 is too large for Our trial divisor. Uneven Division some division problems such as 7 + 3, there is no other whole number that, when mul- tiplied by the divisor, will give the dividend. We use the distributive idea to show how divi- Sion is done in such a case. For example, 7 3 could be written as follows: 1 3 Thus, we see that the quotient also carries one unit that is to be divided by 3. It should now be clear that 3/Sä = 3/ßܯöT, and that this can be further reduced as follows: 30 6 12 3 In elementary arithmetic the part of the divi- dend that cannot be divided evenly by ule divisor is often called a REMAINDER and is placed next to the quotient with the prefix R. Thus, in the foregoing example where the quotient was 12 S, the quotient could be written 12 R 1. This.

MATHEMATICS, step 20 - 10 - 3 - 67 1. 61 3. 92 — 25 s = 55 VOLUME 1 multiplications may be performed in any order. Thus, in 20 4 gal (left Over) Sep 2: Convert the 4 gal left over add to the 1 qt. Step 3: or to 16 qt and 4+2+7+5= 18 100 - 4 x 2 x 7 x 5 280 5 q 15 2 qt Qep 4: Convert the 2 add to the i pt. 5: 5 (left over) qt left over pt 4 pt and Therefore, 24 gai I qt 1 pt divided by S is 4 gal 3 qt 1 pt. Practice problems. problems 1 through 4, divide ag indicated. problems S through 8, multiply or divide u indicated. 1. 549 9 2. 470/63 2. 7 R 29 4. 1,169 5. 6. 7. 5. 6. 7. 8. 4 hr 26 min 16 sec 3(4 gal S qt 1 pt) 67 deg 43 min 12 sec 2 22 hr 11 min 20 gec 14 gal 2 qt 1 pt 33 deg 51 min 36 gec 12 1b 11 4/5 oz the numbers may be combined in any order de- sired. For example, they may be grouped easily to give 6 + 12 18 97 30 67 40 x 7 280 A series divisions be taken in the order written. 100 + 10 2. 10 2-5 In a series of mixed operations, perform multi- plications and divisions in order from left to right, then perform additions and subtractions in order from left to right. For example 100 = 25 x s 125 60 Now consider ORDER OF OPERATIONS When a series involving addi- tion, subtraction, multiplication, or division is indicated, ule order in which the operations are performed ig important only division ie in- volved or if the operations are mixed. A se- ries individual additions, subtractions, or 16 60 - 25 + = 115 - IOS = 10 5 + 15 -60-5+ - 100 10 —5+ 15 — 100 +4 x 10 - 100 +40.

Chapter 2 Evaluate each —POSITIVE INTEGERS Practice problems. following expressions: 2. 12 the 1. 2. 3. 4. 5. 18 -2 x 5+4 90 + 2 9 75 5 x 3 +5 7+1-8 x 4+ Answers: 16 MULTIPLES AND FACTORS 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 theee 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 algo ot i. number that i' multiple 2 is an EVEN NUMBER. The even number. begin With 2 and progress by 2" follow•: 2, 4, e, 8, 10, 12, .. Aty number that not a multiple 2 is an ODD NUMBER. The odd number. begin With 1 and progress by 2", u follows: Any number that can b' divided into a given number without a remainder i' a FACTOR of the given number. The given number a mul- tiple any number that i' one its tactors. For example, 2, 3, 4, 6, 8, and 12 are tactor• 24. The following four equalities •how vari- ou combmatlon• the tactors 24: 24 -24 • 1 24 • 12 • 2 24.8. 3 24-6.4 the number 24 factored completely possible, it usumes the torm 24-2-2. 2-3 ZERO AS A FACTOR any number is multiplied by zero, the product is zero. For example, 5 times zero equals zero and may be written 5(0) = O. The zero factor law tells us that, the product two or more factors is zero, at least one Ole factors must be zero. PRIME FACTORS A number that has factors other than itseu and 1 is a COMPOSrrE NUMBER. For exam- pie, 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 some- times advantageous to separate a composite number into prime factors, it is helpful to be able to recognize a few prime numbers quickly. The following series shows an the prime num- bers up to 60: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 3i, 37, 41, 43, 47, 53, 59. Notice that 2 is the only even prime number. Ail other even numbers are divisible by 2. Notice also that 51, tor example, doen not pear in the series, Since it is a composite num- ber equal to 3 x 17. a factor a number prime, it called a PRmdE FACTOR. To separate a number into prime factors, begin by taking out the smallest factor. M the number i' even, take out the 2's tir•t, then try 3 a factor, etc. Thu, we have the following example: 540 2 • 270 •2 • 2 • 135 • 2-2. 3 • 45 . 2-2.s.s.15 .2-2-3•3-3-5 Since 1 i' understood tactor every num-, b", we do not wute •pace recording it u on' of the tactors in presentation thia kind. A convenient way keeping track the prime tactors in the •hort divi'ion process u follows: 3Lüå 3Liå- 51.5— 17.

a number is odd, its factors will be odd numbers. To separate an odd number into prime factors, take out the 3's first, if there are any. Then try 5 as a factor, etc. As an example, 5, 775 MATHEMATICS, VOLUME 1 18, 51 1,925 • 5 • 385 5 •5 77 •5 •5-7. 11 Practice problems: 1. Which of the following are prime numbers and which are composite numbers ? 25, 7, 18, 29, 51 2. What prime numbers are factors Of 36? 3. Which of the following are multiples of 3 ? 45, 53, 51, 39, 47 4. Find the prime factors 27. Answers: 1. Prime: 7, 29 Composite: 25, 2. 36 3. 45, 51, 39 4. 27 =3-3-3 •3 Tests for Divisibility It is often useful to be able to tell by inspec- tion whether a number is exactly divisible by one or more of the digits from 2 through 9. An expression which is frequently used, although it is sometimes misleading, is "evenly divisible." This expression has nothing to do with the con- cept of even and odd numbers, and it probably should be avoided in favor of the more descrip- tive expression, "exactly divisible." For the re- mainder of this discussion, the word "divisible" has the same meaning as "exactly divisible. " Several tests for divisibility are listed in the following paragraphs: 1. A number is divisible by 2 if its right- hand digit is even. 2. A number is divisible by 3 if the sum of its digits is divisible by 3. For example, the digits Of the number 6,561 add to produce the sum 18. Since 18 is divisible by 3, We know that 6,561 is divisible by 3. 3. A number is divisible by 4 if the number formed by the two right-hand digits is divisible by 4. For example, the two right-hand digits Of the number 3,524 form the number 24. Since 24 is divisible by 4, We know that 3,524 is di- visible by 4. 4. A number is divisible by 5 if its right- hand digit is O or 5. 5. A number is divisible by 6 if it is even and the sum of its digits is divisible by 3. For example, the sum Of the digits Of 64,236 is 21, which is divisible by 3. Since 64,236 is also an even number, we know that it is divisible by 6. 6. No short method has been found for de- termining whether a number is divisible by 7. 7. A number is divisible by 8 if the number formed by the three right-hand digits is divisi- ble by 8. For example, the three right-hand digits Of the number 54,272 form the number 272, which is divisible by 8. Therefore, we know that 54,272 is divisible by 8. 8. A number is divisible by 9 if the sum of its digits is divisible by 9. For example, the sum Of the digits Of 546,372 is 27, which is di- visible by 9. Therefore we know that 546,372 is divisible by 9. Practice problems. Check each Of the fol- lowing numbers for divisibility by all of the digits except 7: 1. 242,431,231,320 2. 844,624,221,840 3. 988,446,662,640 4. 207,634,542,480 Answers: All of these numbers are divisible by 2, 3, 4, 5, 6, 8, 9. 18.

CHAPTER 3 SIGNED NUMBERS The positive numbers with which we have worked in previous chapters are not sufficient for every situation which may arise. For ex- ample, a negative number results in the opera- tion of subtraction when the subtrahend is larger than the minuend. NEGATIVE NUMBERS When the subtrahend happens to be larger than the minuend, this fact is indicated by plac- ing a minus sign in front of the difference, as in the following: 12 - 20 = -8 The difference, -8, is said to be NEGATIVE. A number preceded by a minus sign is a NEGA- TIVE NUMBER. The number -8 iS read "minus eight." Such a number might arise when we speak of temperature changes. If •the tempera- ture was 12 degrees yesterday and dropped 20 degrees to&y, the reading today would be 12 - 20, or -8 degrees. Numbers that show either a plus or minus Sign are called SIGNED An un- signed number is understood to be positive and is treated as though there were a plus sign preceding it. If it is desired to emphasize the fact that a number is positive, a plus sign is placed in front of the number, as in +5, which is read ' 'plus five." Therefore, either +5 or 5 indi- cates that the number 5 is positive. If a num- ber is negative, a minus sign must appear in front of it, as in -9. In dealing with signed numbers it should be emphasized that the plus and minus signs have two separate and distinct functions. They may indicate whether a number is positive or nega- tive, or they may indicate the operation of ad- dition or subtraction. When operating entirely with positive num- bers, it is not necessary to be concerned with ülis distinction since plus or minus signs indi- cate only addition or subtraction. However , when negative numbers are also involved in a 19 computation, it is important to distinguish be- tween a sign of operation and the sign of a number. DIRECTION OF MEASUREMENT Signed numbers provide a convenient way of indicating opposite directions with a minimum of words. For example, an altitude of 20 ft above sea level could be designated as +20 ft. The same distance below sea level would then be designated as -20 ft. One of the most com- mon devices utilizing signed numbers to indicate direction of measurement is the thermometer. Thermometer The Celsius (centigrade) thermometer shown in figure 3-1 illustrates the use of positive and negative numbers to indicate direction of travel above and below O. The O mark is the change- over point, at which the signs of the scale num- bers change from - to 4. When the thermometer is heated by the sur- rounding air or by a hot liquid in which it is placed, the mercury expands and travels up the tube. After the mercury O, the mark at which it comes to rest is read as a positive temperature. If the thermometer is allowed to cool, the mercury contracts. After passing O in its downward movement, any mark at which it comes to rest is read as a negative temperature. Rectangular Coordinate System As a matter of convenience, mathematicians have agreed to follow certain Conventions as to the use of Signed numbers in directional meas- urement. For example, in figure 3-2, a direc- tion to the right along the horizontal line is positive, while the opposite direction (toward the left) is negative. On the vertical line, di- rection upward is positive, while direction downward is negative. A distance of -3 units along the horizontal line indicates a measure- ment of 3 units to the left of starting point O. A distance of -3 units on the vertical line indicates.

MATHEMATICS, VOLUME 1 WATER BEGIP•S BOIL ING STEAM BEGINS CONDENSING ICE BEGINS MELTING WATER BEGINS FREEZING MER URY 00 o 130•c 120 110 100 100 DEGREES 30 +10 0 -10 o o WATER 0 Figure 3-1. —Celsius (centigrade) temperature scale. a measurement of 3 units below the surting point. The two lines of the rectangular coordinate system which through the O position are the vertical axis and horizontal axis. Other vertical and horizontal lines may be included, forming a grid. When such a grid is used for the location ot points and lines, the resulting "picture" containing points and lines is called a GRAPH. Figure 3-2. —Rectangular coordinate system. The Number Line Sometimes it is important to know the rela- tive greatness (magnitude) of positive and nega- tive numbers. TO determine whether a ular number is greater or less than another number, think of an the numbers both positive and negative as being arranged along a hori- zontal line. (See fig. 3-3.) Figure 3-3. —Number line showing both positive and negative numbers. Place zero at the middle oi the line. Let the positive numbers extend from zero toward the right. Let the neßtive numbere extend from zero toward the left. With this arrangement, positive and negative numbers are go located that they progress from smaller to larger num- bers as we move from left to right along the line. Any number that lies to the right of a given number Is greater than the given number. A number that lies to the left of a given number is less than the given number. This arrange- ment shows that any negative number is smaller than any positive number. The symbol for "greater than" is >. The symbol for "less than" is It is easy to dis- tinguish between these symbols because the symbol used always opens toward the larger number. For example, "7 is greater than 4" can be written 7 > 4 and "-5 is less than -1" can be written -5 < -1. 20.

Chapter 3—SIGNED NUMBERS Absolute Value The ABSOLUTE VALUE of a number is its numerical value when the sign is dropped. The absolute value Of either +5 or -5 is 5. Thus, two numbers that differ only in 'Sign have the same absolute value. The symbol for absolute value consists of two vertical bars placed one on each side of the number, as in I -5 1 = 5. Consider also the following: - 201= 16 = 1-71 7 The expression -71 is read "absolute value of minus seven." When positive and negative numbers are used to indicate direction of measurement, we are concerned only with absolute value, if we wish to know only the distance covered. For example, in figure 3-2, if an object moves to the left from Ole starting point to the point in- dicated by -2, the actual distance covered is 2 units. We are Concerned only with the fact that 1-21 = 2, if our only interest is in the distance and not the direction. OPERATING WITH SIGNED NUMBERS The number line can be used to demonstrate addition of signed numbers. Two cases must be considered; namely, adding numbers with like signs and adding numbers with unlike signs. ADDING WITH LIKE SIGNS AS an example Of addition with like signs, suppose that we use the number line (fig. 3-4) to add 2 + 3. Since these are signed numbers, we indicate this addition as (+2) + (+3). This emphasizes that, among the three + signs shown, two are number signs and one is a sign of 0 234 S Figure 3-4.—Using the number line to add. operation. Line a (fig. 3-4) above the number line shows this addition. Find 2 on the number line. To add 3 to it, go three units more in a positive direction and get 5. To add two negative numbers on the number line, such as -2 and -3, find -2 on the number line and then go three units more in the nega- tive direction to get -5, as in b (fig. 3-4) above the number line. Observation of the results of the foregoing operations on the number line leads us to the following conclusion, which may be stated as a law: To add numbers with like signs, add the absolute values and prefix the common sign. ADDING WITH UNLIKE SIGNS To add a positive and a negative number, such as (-4) + (+5), find +5 on the number line and go four units in a direction, as in line c above the number line in figure 3-4. Notice that this addition could be performed in the other direction. That is, we could start at -4 and move 5 units in the positive direction. (See line d, fig. 3-4.) The results of our operations with mixed signs on the number line lead to the following conclusion, which may be stated as a law: To add numbers with unlike signs, find the differ- ence between their absolute values and prefix the Sign Of the numerically greater number. The following examples show the addition of the numbers 3 and 5 with the four possible com- binations of signs: 3 5 8 -3 -5 -8 3 -5 -2 -3 5 2 In the first example, 3 and 5 have like signs and the common sign is understood to be posi- tive. The sum of the absolute values is 8 andno sign is prefixed to this sum, thus signifying that the sign of the 8 is understood to be positive. In the second example, the 3 and 5 againhave like signs, but their common sign is The sum of the absolute values is 8, and this time the common sign is prefixed to the sum. The answer is thus -8. In the third example, the 3 and 5 have unlike signs. The difference between their absolute values is 2, and the sign of the Larger addend is negative. Therefore, the answer is -2. In the fourth example, the 3 and 5 again have unlike signs. The difference of the absolute 21.

1. 2. 4. MATHEMATICS, values is still 2, but this time the sign of the larger addend is positive. Therefore, the sign prefixed to the 2 is positive (understood) and the final answer is simply 2. These four examples could be written in a different form, emphasizing the distinction be- tween the sign of a number and an operational sign, as follows: Practice problems. Add as indicated: 3. -11 4. -35 VOLUME 1 minuend is numerically greater than the Sub- trahend, as in the following examples: 8 5 3 8 -5 13 -8 5 -13 -8 -5 -3 -10 +5- Add -9, -16, and 25 Add -22 and -13 Answers: SUBTRACTION Subtraction is the inverse of addition. When subtraction is performed, we "take away" the subtrahend. This means that whatever the value of the subtrahend, its effect is to be reversed when subtraction is indicated. In addition, the sum of 5 and -2 is 3. In subtraction, however, to take away the effect of the -2, the quantity +2 must be added. Thus the difference between +5 and -2 is +7. Keeping this idea in mind, we may now pro- ceed to examine the various combinations of subtraction involving signed numbers. Let us first consider the four possibilities where the We may show how each Of these results is obtained by use oi the number line, as shown in figure 3-5. In the first example, we find +8 on the num- ber line, then subtract 5 by making a movement that reverses its sign. Thus, we move to the left 5 units. The result (difference) is +3. (See line a, fig. 3-5.) In the second example, we find +8 on the number line, then subtract (-5) by making a movement that will reverse its sign. Thus we move to the right 5 units. The result in this case is +13. (See line b, fig. 3-5.) In the third example, we find -8 on the num- ber line, then subtract 5 by making a movement that reverses its sign. Thus we move to the left 5 units. The result is -13. (See line c, fig. 3-5.) In the fourth example, we find —8 on the number line, then reverse the sign of -5 by moving 5 units to the right. The result is -3. (See line d, fig. 3-5.) Next, let us consider the four possibilities that arise when the subtrahend is numerically greater than the minuend, as in the following examples: 5 8 -3 5 -8 13 -5 8 -13 -5 -8 3 In the first example, we find +5 on the num- ber line, then subtract 8 by making a movement 2 34 S 67 B 9 10 e 13 -13-12-11*0-9 -e -7 -6-5 4 -3-2 -l o Figure 3-5.—Subtraction by use Of the number line. 22.

that reverses its sign. Thus we move to the left 8 units. The result is -3. (See line e, fig. 3-5.) In the second example, we find +5 on the number line, then subtract -8 by making a move- ment to the right that reverses its sign. The result is 13. (See line f, fig. 3-5.) In the third example, we find -5 on the num- ber line, then reverse the sign of 8 by a move- ment to the left. The result is -13. (See line g, fig. 3-5.) In the fourth example, we find -5 on the num- ber line, then reverse the sign -8 by a move- ment to the right. The result is 3. (See line h, fig. 3-5.) Careful study Of the preceding examples leads to the following conclusion, which is stated as a law for subtraction of signed num- bers: In any subtraction problem, mentally change the sign of the subtrahend and proceed as in addition. Practice problems. In problems 1 through 4, subtract the lower number from the upper. 5 through 8, subtract as indicated. Chapter 3—SIGNED NUMBERS 2. -12 8. -11 In 7 16 1. 5. 6. 7. 1. 17 -10 14 - 7 -(- Answers: 27 3. -9 4. 8 When the multiplier is negative, as in -3(7), we are to take away 7 three times. Thus, -3(7) is equal to - (7) - (7) which is equal to -21. For example, if 7 shells were expended in one firing, 7 the next, and 7 the next, there would be a loss of 21 shells in all. Thus, the rule is as follows: The product of two numbers with unlike signs is negative. The law Of signs for unlike Signs is some— times stated as follows: Minus times plus is minus; plus times minus is minus. Thus a problem such as 3(-4) can be reduced to the following two steps: 1. Multiply the signs and write down the sign of the answer before working with the numbers themselves. 2. Multiply the numbers as if they were un— signed numbers. Using the suggested procedure, the sign of the answer for 3(-4) is found to be minus. The product 3 and 4 is 12, and the final answer is -12. When there are more than two numbers to be multiplied, the signs are taken in pairs until the final sign is determined. Like Signs When both factors are positive, as in 4(5), the sign Of the product is positive. We are to add +5 four times, as follows: 4(5) 5_+5 +5+5= 20 When both factors are negative, as in -4(-5), 7. -13 10 the sign of the product is positive. take away -5 four times. -4(-5) - ( 5) = 20 We are to 2. -20 MULTIPLICATION To explain the rules for multiplication Of signed numbers, we recall that multiplication of whole numbers may be thought as short- ened addition. Two types of multiplication problems must be examined; the first type in- volves numbers with unlike signs, and the sec- ond involves numbers with like signs. Unlike Signs Consider the example 3(-4), in which the multiplicand is negative. This means we are to add -4 three times; that is, 3(-4) is equal to (-4) + (-4) + (-4), which is equal to -12. For example, ii we have three 4-dollar debts, we owe 12 dollars in all. Remember that taking away a negative 5 is the same as adding a positive 5. For example, suppose someone owes a man 20 dollars and pays him back (or diminishes the debt) 5 dollars at a time. He takes away a debt of 20 dollars by giving him four positive 5-doUar bills, or a total of 20 positive dollars in all. The rule developed by the foregoing example is as follows: The product of two numbers with like signs is positive. Knowing that the product of two positive num- bers or two negative numbers is positive, we can conclude that the product of any even num- ber negative numbers is positive. Similarly, 23.

MATHEMATICS, 3(-4) = -12 the product Of any Odd number Of negative num- bers is negative. The laws of signs may be combined as fol- lows: Minus times plus is minus; plus times minus is minus; minus times minus is plus; plus times plus is plus. Use of this combined rule may be illustrated as follows: 4(-2) • ( 5) (6) • (-3) = -720 Taking the signs in pairs, the understood plus on the 4 times the minus on the 2 produces a minus. This minus times the minus on the 5 produces a plus. This plus times the under- stood plus on the 6 produces a plus. This plus times the minus on the 3 produces a minus, so we know that the final answer is negative. The product of the numbers, disregarding their signs, is 720; therefore, the final answer is -720. 3. 24 VOLUME 1 Therefore, -4 Thus the rule for division with like signs is: The quotient of two numbers with like signs is positive. The following examples show the application of the rules for dividing signed numbers: 12 = 4 -3 pr±lems. Practice problems. Multiply as indicated: Practice indicated: I. 15 + -5 Answers: 3 12 = -4 Multiply and divide as 3. 4. -81/9 1. 5(-8) ? -7(3) (2) 2. 3. -2(3) (-4) (5) 6) 4. Answers: 1. -40 2. -42 DIVISION 4. -720 Because division is the inverse of multipli- cation, we can quickly develop the rules for division of signed numbers by comparison with the corresponding multiplication rules, as in the following examples: 1. Division involving two numbers with un- like signs is related to multiplication with un- like signs, as follows: Therefore, 3(-4) = -12 12 3 Thus, the rule for division with unlike signs is: The quotient of two numbers with unlike signs is negative. 2. Division involving two numbers with like signs is related to multiplication with like signs, as follows: 24 SPECIAL CASES Two special cases arise frequently in which the laws signs may be used to advantage. The first such usage is in simplifying subtrac- tion; the second is in changing the signs of the numerator and denominator when division is indicated in the form Of a fraction. Subtraction The rules for subtraction may be simplified by use the laws of signs, if each expression to be subtracted is considered as being multi- plied by a negative sign. For example, 4 -(-5) is the same as 4 + 5, since minus times minus is plus. This result also establishes a basis for the rule governing removal of parentheses. The parentheses rule, as usually stated, is: Parentheses preceded by a minus sign may be removed, the signs of all terms within the parentheses are changed. This is illustrated as follows: 12 -(3 - 2 +4) = 12 - 3+2-4.

The reason for the changes Of sign is clear when the negative sign preceding the parenthe- ses is considered to be a multiplier for the whole parenthetical expression. Division in Fractional Form Division is often indicated by writing the dividend as the numerator, and the divisor as the denominator, of a fraction. In algebra, every fraction is consideredto have three signs. The numerator has a sign, the denominator has a si.gn, and the fraction itself, taken as a whole, has a sign. In many cases, one or more of Olese signs will be positive, and thus will not be shown. For example, in the following fraction the sign the numerator and the sign the denominator are both positive (understood) and the sign the fraction itseM is negative: 5 Fractions with more than one negative eign are always reducible to a simpler form with at most one negative sign. For example, the sign the numerator and the sign the denomina- tor may be both negative. We note that minus divided by minus gives ule same result as plus divided by plus. Therefore, we may Change to the legs complicated form having plus signs (understood) for both numerator and denomina- tor, as follows: -15 +15 15 Since -15 divided by -5 3, and 15 divided by 5 is also 3, we conclude that the change Sign does not alter the final answer. The game reasoning may be applied In the following ex- ample, in which the Sign the fraction itgell ig Chapter 3—SIGNED NUMBERS = -15 a fraction has a negative sign in one of the three sign positions, this sign may be moved to another position. Such an adjustment is an ad- vantage in some types of complicated sions involving fractions. type of sign change follow: 15 Examples of this 15 -5 negative: -15 15 When the traction Ltseu haa a negative sign, as in this example, the fraction may be enclosed in parentheses temporarily, for the purpose ot working with the numerator and denominator only. Then the Bign of the fraction is applied separately to the result, as follows: All this can be done mentally. first expression the foregoing ex- ample, the sign of the numerator is positive (understood) and the sign of the fraction is neg- ative. Changing both these signs, we obtain the second expression. To obtain the third ex- pression from the second,• we change the sign the numerator and the sign of the denomina- tor. Observe that the sign changes in each case involve a pair of Signs. This leads to the law of signs for •fractions: Any two of the three signs a fraction may be changed without al- tering the value of the fraction. AXIObS AND LAWS An axiom is a sell-evident truth. It is a truth that is 80 universally accepted that it does not require prod. För example, the statement that ' 'a straight line ig the shortest distance between two points" ig an axiom from plane geometry. tends to accept the truth axiom Without proof, because anything which is axiomatic is, by its very nature, obviously true. the other hand, a law (in the mathematical sense) ig result of defining certain quanti- ties and relationships and then developing logi- cal conclusions from the definitions. AXIObC OF EQUALITY The four uiom• of equality with which we are concerned in arithmetic and algebra are stated u followe: 1. the same qua.ntity is added to each two equal quantities, the resulting quantities are equal. This is sometimes stated u follows: equale are added to equals, the results are equal. For example, by adding the same quan- tity (3) to both Bides of the following equatim, we obtain two sums which are equal: -2 +3 -3 +1+3 1=1 25.

MATHEMATICS, 2. the same quantity is subtracted from each of two equal quantities, the resulting quan- tities are equal. This is sometimes stated as follows: equals are subtracted from equals, the results are equal. For example, by sub- tracting 2 from both sides of the following equa- tion we obtain results which are equal: 5=2+3 -2 5-2=2+3 3=3 3. two equal quantities are multiplied by the same quantity, the resulting products are equal. This is sometimes stated as follows: equals are multiplied by equals, the products are equal. For example, both sides of the fol- lowing equation are multiplied by -3 and equal results are obtained: 5=2+3 15 = -15 4. two equal quantities are divided by the same quantity, the resulting quotients are equal. This is sometimes stated as follows:. equals are divided by equals, the results are equal. For example, both sides of the following equa- tionare divided by 3, and the resulting quotients are equal: 15 12 + 3 - 15 3 4+1=5 These axioms are especially useful when letters are used to represent numbers. we know that 5x = -30, for instance, then dividing both 5x and -30 by 5 leads to the conclusion that x = -6. LAWS FOR COMBINING NUMBERS Numbers are combined in accordance with the following basic laws: 1. The associative laws of addition and mul- tiplication. 2. The commutative laws of addition and multiplication. 3. The distributive law. 26 VOLUME 1 Associative Law of Addition The word "associative" suggests association or grouping. This law states that the sum of three or more addends is the same regardless of the manner in which they are grouped. For example, 6 + 3 + 1 is the same as 6 + (3 + 1) or This law can be applied to subtraction by changing Signs in such a way that all negative signs are treated as number signs rather than operational signs. That is, some Of the ad- dends can be negative numbers. For example, 6 - 4 - 2 can be rewritten as 6 + (-4) + (-2). By the associative law, this is the same as However, 6 — 4 - 2 is not the same as 6 — (4 - 2); the terms must be expressed as addends before applying the associative law addition. Associative Law of Multiplicatim This law states that the product of three or more factors is the same regardless of manner in which they are grouped. For ex- ample, 6 • 3 • 2 is the same as (6 • 3) • 2 or 6 • (3 • 2). Negative signs req.lire no special treatment in the application of this law. (-2) is the same For exam le, 6 • . as [6 • or 6 .(-2)]. Commutative Law of Addition The word ' 'commute" means to change, sub- stitute or move from place to place. The com- mutative law of addition states that the sum of two or more addends is the same regardless of the order in which they are arranged. For ex- ample, 4 +3 + 2 is the same as 4 + 2 + 3 or 2+4 +3. This law can be applied to subtraction by changing signs so that all negative signs be- come number signs and all signs ci operation are positive. For example, 5 - 3 - 2 is changed to 5 + (-3) + (—2), which is the same as 5 + (-2) + or +5 + Commutative Law of Multiplication This law states that tie product of two or more factors is the same regardless Of the order in which the factors are arranged. For example, 3 • 4 • 5 is the same as 5 • 3 • 4 or.

CHAPTER 4 COMMON FRACTIONS The emphasis in previous Chapters Of this course has been on integers (whole numbers). In this chapter, we turn our attention to num- bers which are not integers. The simplest type of number other than an integer is a COMMON FRACTION. Common fractions and integers together comprise a set of numbers called the RATIONAL NUMBERS; this set iS a subset of the set of real numbers. The number line may be used to show the relationship between integers and fractions. For example, if the interval between O and 1 is marked off to form three equal spaces (thirds), then each space formed is one-third of the total interval. If we move along the number Line from O toward 1, we will have covered two of the three "thirds" when we reach the second mark. Thus the position of the second mark represents the number 2/3. (See fig. 4-1.) -2 2 Figure 4-l.—ültegers and fractions on the number line. The numerals 2 and 3 in the fraction 2/3 are named so that we may distinguish between them; 2 is the NUMERATOR and 3 iS the DENOMINA- TOR. general, the numeral above the di- viding line in a fraction iB the numerator and the numeral below the line is the denominator. The numerator and denominator are ule TERMS of the fraction. The word "numerator" is re- lated to the word "enumerate." To enumerate means to "tell how many"; thus the numerator tells us how many fractional parts we tLve in the indicated fraction. To denominate means to "give a name" or "tell what kind"; thus the de- nominator tells us what kind Of parts we have (halves, thirds, fourths, etc.). Attempts to define the word "fraction" in mathematics usually result in a statement sim- ilar to the following: A fraction is an indicated division. Any division maybe indicated by plac- ing the dividend over the divisor and drawing a line between them. By this definition, any num- ber which Can be written as the ratio Of two in- tegers (one integer over the other) Can be con- sidered as a fraction. This leads to a further definition: Any number which can be expressed as the ratio of two integers is a RATIONAL number. Notice that every integer is a rational number, because we can write any integer as the numerator of a fraction having 1 as its de- nominator. For example, 5 is the same as 5/1. It should be obvious from the definition every common fraction is algo a rational number. TYPES OF FRACTIONS Fractions are often classified as pmper or improper. A proper fraction is one in which the numerator is numerically smaller than the de- nominator. An improper fraction has a nu— merator which is larger than its denominator. MMED NUMBERS When the denominator of an improper frac- tion is divided into its numerator, a remainder is produced along with the quotient, Ulless the numerator happens to be an exact multiple of the denominator. For example, 7/5 is equal to 1 plus a remainder of 2. This remainder may be shown as a dividend with 5 as its divisor, as follows: 7 5+2 2 5 The e'qression 1 + 2/5 iS a MDCD NUM- BER. Mixed numbers are usually written with- out showing the plus sign; that is, 1 + 2/5 is the same as 1— or 1 2/5. When a mixed num- ber is written as 1 2/5, care must be taken to insure that there is a space between the 1 and the 2; otherwise, 1 2/5 might be taken to mean 12/5. 28.

MATHEMATICS, VOLU>IE 1 Answers: For example, if I in the form is multiplied S, the prcduct will still have a value of Ent will be in a different form, as follows: 2 3 2-3 6 Figure 4-4 shows that of line a is equal to 6 — of line b where line a equals line b. Line a 10 is marked off in fifüis and line b is marked off in ten ths. It can be seen that and measure distances of equal length. a 0 4 6 23 4 s 67 e Figure 4-4. —Equivalent fractions. The markings on a ruler show equivalent fractions. The major division of an inch divides it into two equal parts. One of these represents The next smaller markings divide the inch into four equal parts. It will be noted that two of these parts represent the same distance as 1 • that is, — equals Also, the next smaller markings break the inch into 8equal parts. How many Of these parts are equivalent to inch? The answer is found by noting that equals i. Practice problems. Using the divisions on a ruler for reference, complete the following exercise: 1 7 1 3 1 4 3. 12 A review of the foregoing exercise will re- veal that in each case the right-hand fraction could be formed by multiplying both the nu- merator a.nd the denominator of the left-hand fraction by the same number. In each case the number may be determined by dividing the de— nominator Of the right—rund fraction by the de- nominator of the left-hand fraction. Thus in problem 1, both terms of — were multiplied by 2. Ln problem 3, both terms were multiplied by 4. It is seen that multiplying both terms of a frac- tion by the same number does not change the value of the fraction. Since equals the reverse must also be true; that is — must be equal to This can likewise be verified on a ruler. We have al- 1 12 i' 16 equals 7, ready seen that — is the same as — and — equals We see that dividing both terms of a fraction by the same number does not change the value Of the fraction. FUNDAMENTAL RULE OF FRACTIONS The foregoing results are combined to form the fundamental rule of fractions, which is stated as follows: Multiplying or dividing both terms Of a fraction by the same number does not change the value Of the fraction. This is one Of the most important rules used in dealing with fractions. The following examples show how the mental rule is used: 1. Change 1/4 to twelfths. This problem is set up as follows: 1 The first step is to determine how many 4's are contained in 12. The answer is 3, so we klow that the multiplier for both terms of the fraction is 3, as follows: 3 30.

The truth of this can be verified another way: If I equals •s, then 2 equals Thus, 26 These examples lead to the following con- clusion, which is• stated as a rule: To change an improper fraction to a mixed number, divide the numerator by the denominator and write the fractional part of the quotient in lowest terms. Practice problems. Change the following MATHEMATICS, VOLUME 1 2. 5.1 Thus , EXAMPLE: SOLUTION: Thus, 1 5 5 fractions to mixed numbers: 1. 2. 1. 31/20 33/9 Answers: 11 1— 20 3. 65/20 4. 45/8 Write 5— as an improper fraction. 5 = 45 1(9) 45 2 47 2 47 9 3. 4. 1 3 OPERATING WITH NUMBERS In comßltation, mixed numbers are often un- wieldy. As it is possible to change any im- proper fraction to a mixed number, it is like- wise possible to change any mixed number to an improper fraction. The problem can be reduced to the finding of an equivalent fraction and a simple addition. EXAMPLE: Change 2— to improper fraction. SOLUTION : Step 1: Write 2— as a whole number plus a 1 fraction, 2 + Step 2: Change 2 to an equivalent fraction with a denominator Of 5, as follows: In each of these examples, notice that the multiplier used in Step 2 is the same number as the denominator of the fractional part of the original mixed number. This leads to the fol- lowing conclusion, which is stated as a rule: To change a mixed number to an improper frac- tion, multiply the whole-number by the • denominator of the fractional part and add the numerator to this product. The result is the numerator of the improper fraction; its denom- inator is the same as the denominator of the fractional part of the original mixed number. Practice problems. Change the following mixed numbers to improper fractions: 1. 1 S 11 20 Answers: 6 20 4. 10 7 43 10 2 25 10 1 10 11 Step 3: Add -s + s = v NEGATIVE FRACTIONS A fraction preceded a minus sign is nega- tive. Any negative fraction is equivalent to a positive fraction multiplied by -1. For example, 32.

Cha*er 4 —COMMON FRACTIONS The number - — is "minus two-fifths." We that the quotient of two numbers With unlike signs is negative. Therefore, -2 2 5 This indicates that a negative fraction is equiv- alent to a fraction With either a negative nu- merator or a negative denominator. The fraction — is read "two over minus -2 five." The fraction is read "minug two over five." A minus sign in a fraction can be moved about at will. It can be placed before the nu- merator, before the denominator, or before the of likeness applies also to fractions. We can add eighths to eighths, fourths to fourths, but not eighths to fourths. To add— inch to inch we simply add the numerators and retain the denominator unchanged. The denomination is fifths; as with denominate numbers, we 1 3 fifth to 2 fifths to get 3 fifths, or LIKE AND UNLIKE FRACTIONS We have shown that like fractions are by simply the numerators and keeping the denominator. Thus, 3 2 3+2 5 8 2 5 7 fraction itself. Thus, -2 2 2 Similarly we can subtract like fractions by subtracting the numerators. Moving the minus sign from numerator to denominator, or vice versa, is eqGivalent to multiplying the terms of the fraction by -1. This i8 shown in the following examples: 7-2 8 5 -21-1) 2 2(-1) A fraction may be regarded as having three sigrs associated with it—the sign of the numer- ator, the sign of the denominator, and the preceding the fraction. Any two of these Signs may be changed wiülout changing the value of the fraction. Thug, -3 3 3 -3 The following will show that like fractions may be divided by dividing the nu- merator of the dividend by numerator of the divisor. 3 SOLUTION: We may state the problem u a question: "How many times does — appear in 2, or how many times may be taken from 2?" OPERATIONS WITH FRACTIONS It will be recalled from the discussion of denominate numbers that numbers must be the same denomination to be added. We can add pounds to pmmds, pints to pints, but not ounces to pints. If we think of fractions loosely as de- nominate numbers, it will be seen that the rule 3/8 - 1/8 = 2/8 2/8 - 1/8 = 1/8 1/8 - 1/8 0/8 = o (1) (2) (3) We see that 1/8 can be subtracted from 3/8 three times. Therefore, 3/8 + 1/8 3 33.

MATHEMATICS When the denominators of fractions are un- equal, the fractions are said to be unlike. Ad- dition, subtraction, or division cannot be per- formed directly on unlike fractions. The proper application the fundamental rule, however, can change their form so that they become like fractions; then ail the rules for like fractions apply. LOWEST COMMON DENOMINATOR To change unlike fractions to like fractions, it iS necessary to find a COMMON DENOPÆNA- TOR and it is usually advantageous to find the LOWEST COMMON DENOMINATOR (L c D). This is nothing more than the least common multiple of the denominators. Least Common Multiple a number is a multiple of two or more different numbers, it is called a COMMON MULTIPLE. Thus, 24 iS a common multiple of 6 and 2. There are many common multiples of these numbers. The numbers 36, 48, and 54, to name a few, are also common multiples 6 and 2. The smallest Of the common multiples Of a set numbers iS caned the LEAST COMMON MULTIPLE. It is abbreviated LCM. The least common multiple of 6 and 2 is 6. To find the least common multiple of a set of numbers, tirst separate each of the numbers into prime factors. Suppose that we wish to find the LCM Of 14, 24, and 30. Separating these numbers into VOLUME 1 Greatest Common Divisor The largest number that can be divided into each of two or more given numbers without a remainder is caned the GREATEST COMMON DIVEOR0f the given numbers. It is abbreviated GCD. It is also sometimes called the HIGHEST COMMON FACTOR. In finding the GCD of a set numbers, se- parate the numbers into prime factors just as for LCM. The GCD is the product of only those factors that appear in ail Of the numbers. Notice in the example the previous section that 2 the greatest common divisor of 14, 24, and 30. Find the GCDof 650, 900, and 700. The pro- cedure is as follows: 650 = 2 • 52 • 13 900 22 • 32 700 22 52 GCD = 2 52 = 50 Notice that 2 and 52 are factors of each num- ber. The greatest common divisor is 2 x 25 50. USING THE LCD Consider the example 1 1 The numbers 2 and 3 are both prime; so the prime factors we have 24=23 30 = 2 3 LCD iS 6. Therefore Thus, the follows: 1 3 addition Of 3 6 — and 1 The LCM will contain each of the various prime factors shown. Each prime factor is used the greatest number of times that it occurs in any oneoi the numbers. Notice that 3, 5, and 7 each occur only once in any one number. the other hand, 2 occurs three times in one number. — is performed as 5 1 In the example 10 iS the LCD. 1 1 3 2 We get the following result: LCM = 23 •3 •5 •7 Thus, 840 is the least common multiple 24, and 30. 3 of 14, 34.

Chapter 4— COMMON FRACTIONS Therefore, 1 Practice problems. Change the fractions in each of the following groups to like fractions with least common denominators: Answers: iä' 12 ADDITION 1 1 6 5 It has been Shown that in tions we the numerators. äö' äö adding like frac- In adding unlike fractions, the fractions must first be changed So that they have common denominators. We apply these same rules in mixed numbers. It will be remembered that a mixed number is an indicated sum. Thus, 2 i is really 2 + Add- ing can be done in any order. The following examples will show the application of these rules: EXAMPLE: TMs could have been written as follows: EXAMPLE: 10 — Here we change — to the mixed number 1 Then 10 = 10 +— = 11 — EXAMPLE: We first change the fractions so that they are like and have the least common denominator and then proceed as before. 11 12 EXAMPLE: 11 Since equals I as follows: 11 the final answer is found 35.

MATHEMATICS, VOLUME 1 11 Practice problems. Sums to simplest terms: 31 3 Add, and reduce 1 3 2 5 2. 4. 5. uue 1 8 55 (B) Answers: 25 2. 21 13 4. 2 31 5. The following example demonstrates a prac- tical application of addition of fractions: EXAMPLE: Find the total length the piece of metal shown in figure 4-5 (A). SOLUTION: First indicate the sum as follows: 9 9 Changing to like fractions and adding numerators , 12 12 9 56 9 14 8 1 The total length is 3 inches. Practice problem. Find the distance from the center of the first hole to the center of the last hole in the metal plate shown in figure Figure 4-5.—Adding fractims to obtain EXAMPLE: total length or spacing. 2 Subtract 1 from 5 2 5 1 4-5 (B). Answer: 2 inches The rule likeness applies in the sub- traction fractions as well as in addition. Some examples will show that cases likely to arise may be solved by use of ideas previously developed. We see numbers are subtracted from whole numbers; fractions from fractions. 4 EXAMPLE: Subtract from 4 1 Changing to like fractions with an LCD, we have 32 5 27 36.

COMMON FRACTIONS EXAMPLE: Subtract 11 Chapter 4— 2 11 from 3 8 11 49 15 64 49 64 64 Regrouping 3 —i we have 8 12 3 8 12 20 12 9 Practice problems. Subtract the lower num- ber from the upper number and reduce the di.fference to simplest terms: 4. 1 Answers: 1. 11 1 5 22 S 3. 25 7 5. 23 5 T 5. 12 The following problem demonstrates sub- traction of fractions in a practical situation. EXAMPLE: What the length the dimen- Bion marked X on the machine bolt shown in figure 4-6 (A) ? SOLUTION: Total the the hiown parts. 1 1 64 64 - 64 Subtract this gum from the overan length. The answer is 1 inch. (B) 292" Figure 4-6.—Finding unknown dimensions by subtracting fractions. Practice problem. Find ule length the dimension marked Y on the machine bolt in figure 4-6 (B). Answer: 2 inches MULTIPLICATION The fact that multiplication by a fraction not increase the value of the product may con- fuse those who remember the definition of mul- tiplication presented earlier for whole numbers. was stated that 4(5) means 5 ig taken as an addend 4 times. How is it then that -k4) 2, a number legs than 4 ? our idea multiplication must be broadened. Consider the following products: 4(4) = 16 3(4) 12 2(4) 8 37.

products their factors and then dividing like factors, or canceling. Thus, Dividing the factor 3 in the numerator by 3 in the denominator gives the following simplified Chapter 4—COMMON FRACTIONS D = Ax-a = T - 4 21 x 3 X 3 5 2 1 2 1 result: 1 1 Practice the following 2 products, using the general where possible: rule and canceling 5 1. x 12 Answers: 1 15 4 This method is most advantageous when done before any other computation. Consider the example, 1 The product in factored form is Rather than doing the multiplying and then reducing the result it is simpler to cancel like factors first, as follows: 1 4 1 2 1 2 1 1 1 1 1 1 1 1 1 The following problem illustrates the mul- tiplication fractions in a practical situation. EXAMPLE: Find distance between the cen- ter lines of the first and fKth rivets connecting the two metal plates shown in figure 4-7 (A). SOLUTION: The distance between two adjacent rivets, centerline to centerline, is 4 1/2 times the diameter one of them. I space = 4 L x _ 45 There are 4 such spaces between the fk•st and Bere we mentally factor 6 to the form 3 x 2, and 4 to the form 2 x 2. Cancellation is a valuable tool in shortening Qerationg with fractions. The general rule may be applied to mixed numbers by simply changing them to improper fractions. Thus, fifth rivets. Therefore, is found as follows: 1 45 4 the total distance, D, 45 39.

The distance iS 11 inches Practice problem. Find the the centers the two rivets MATHEMATICS, VOLUME i RIVET SPACING: 4k DIAMETERS (A) * : DIAMETER (B) 4 RIVET SPACING: DILUETERS Figure Of multiplication Of fractions in determining rivet spacing. -3 4+3 distance between shown in figure 12 4 1 the dividend and divisor are both fractions, as in 1/3 divided by 1/4, we proceed as follows: 4-7 (B). Answer: DIVISION 13 4 — inches 16 3 • 4 12 4 = 12 12 + 12 There are two methods commonly used for performing division with fractions. Gle is the common denominator method and the Other is the reciprocal method. Common Denominator Method The common denominator method is an adap- tation the method of like fractions. The rule is as follows: Change the dividend and divisor to like fractions and divide the numerator the dividend by the numerator of the divisor. This method can be demonstrated with whole numbers, first changing them to fractions with 1 as the denominator. For example, 12 4 can be written as follows: 12 12 • 4 = + 12 4 1 Reciprocal Method The word ' 'reciprocal" denotes an Inter- changeable relationship. is used in mathe- matics to describe a specific relationship be- tween two numbers. We say that two numbers are reciprocals of each other if their product 4 is one. In the example 4 x — = 1, the fractions and L are reciprocals. Notice the interchange- ability. 4 is the reciprocal of — and — iS the re- ciprocal of 4. 40.