Mathematics, Basic Math and Algebra
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1 NONRESIDENT TRAINING COURSE Mathematics, Basic Math and Algebra NAVEDTRA DISTRIBUTION STATEMENT A: Approved for public release; distribution is unlimited.
2 PREFACE About this course: This is a self-study course. By studying this course, you can improve your professional/military knowledge, as well as prepare for the Navywide advancement-in-rate examination. It contains subject matter supporting day-to-day occupational knowledge and skill requirements and includes text, tables, and illustrations to help you understand the information. Any errata for this course can be found at under Products. History of the course: June 1980: Original edition released. June 1985: Reprinted. Apr 2003: Administrative update released. Technical content was not reviewed or revised. NOTE In the assignment portion of this course, a question with no answer choices given is a True/False question. For True/False questions, answer 1 for True and 2 for False. Published by NAVAL EDUCATION AND TRAINING PROFESSIONAL DEVELOPMENT AND TECHNOLOGY CENTER POINTS OF CONTACT [email protected] Phone: Toll free: (877) Comm: (850) /1181/1859 DSN: /1181/1859 FAX: (850) ADDRESS COMMANDING OFFICER NETPDTC N SAUFLEY FIELD ROAD PENSACOLA FL NAVSUP Logistics Tracking Number 0504-LP DISTRIBUTION STATEMENT A: Approved for public release; distribution is unlimited.
3 CONTENTS CHAPTER PAGE 1 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 APPENDIX I Squares, cubes, square roots, cube roots, logarithms, and reciprocals of numbers 210 II Natural sines, cosines, and tangents of angles from 0 to III Mathematical symbols IV Weights and measures V Formulas INDEX... INDEX 222 ASSIGNMENT QUESTIONS follow index.
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50 MATHEMATICS, VOLUME 1 Figure 5-1. Place values including decimals. Figure 5-2. Conversion of a decimal fraction to shortened form. are zeros in the denominator of the fractional form. Figure 5-3 shows the fraction and what is meant when it is changed to the shortened form. This figure is presented to show further that each digit of a decimal fraction holds a certain position in the digit sequence and has a particular value. By the fundamental rule of fractions, it should be clear that 5 = 50 = 500. Writing the same values in the shortened way, we have 0.5 = 0.50 = In other words, the value of a decimal is not changed by annexing zeros at the right-hand end of the number. This is not Figure 5-3. Steps in the conversion of a decimal fraction to shortened form. true of whole numbers. Thus, 0.3, 0.30, and are equal but 3, 30, and 300 are not equal. Also notice that zeros directly after the decimal point do change values. Thus 0.3 is not equal to either 0.03 or Decimals such as are frequently seen. Although the 0 on the left of the decimal point is not required, it is often helpful. This is particularlytrue in an expression such as In this expression, the lower dot of the division symbol must not be crowded against the decimal point; the 0 serves as an effective spacer. If any doubt exists concerning the clarity of an expression such as.125, it should be written as
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66 MATHEMATICS. VOLUME 1 parts of the measurement as read on the scales and then to add them. For example, in figure 6-1 (B) there are two major divisions visible (0.2 inch). One minor division is showing clearly (0.025 inch). The marking on the thimble nearest the horizontal or index line of the sleeve is the second marking (0.002 inch). Adding these parts, we have Thus, the reading is inch. As explained previously, this is read verbally as "two hundred twenty-seven thousandths." A more skillful method of reading the scales is to read all digits as thousandths directly and to do any adding mentally. Thus, we read the major division on the scale as two hundred thousandths and the minor division is added on mentally. The mental process for the above setting then would be two hundred twenty-five; two hundred twenty-seven thousandths. Practice problems: Figure 6-1. (A) Parts of a micrometer; (B) micrometer scales. 1. Read each of the micrometer settings shown in figure inch since 40 is equal to The sleeve has 40 markings to the inch. Thus each space between the markings on the sleeve is also inch. Since 4 such spaces are 0.1 inch (that is, 4 x 0.025), every fourth mark is labeled in tenths of an inch for convenience in reading. Thus, 4 marks equal 0.1 inch, 8 marks equal 0.2 inch, 12 marks equal 0.3 inch, etc. To enable measurement of a partial turn, the beveled edge of the thimble is divided into 25 equal parts. Thus each marking on the thimble is 25 of a complete turn, or 25 of 40 1 of an inch. Multiplying 25 times inch, we find that each marking on the thimble represents inch. READING THE MICROMETER It is sometimes convenient when learning to read a micrometer to writedown the component Figure 6-2. Micrometer settings. 62
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68 MATHEMATICS, VOLUME 1 The foregoing example could be followed through for any distance between markings. Suppose the 0 mark fell seven tenths of the distance between ruler markings. It would take seven vernier markings, a loss of one-hundredth of an inch each time, to bring the marks in line at 7 on the vernier. The vernier principle may be used to get fine linear readings, angular readings, etc. The principle is always the same. The vernier has one more marking than the number of markings on an equal space of the conventional scale of the measuring instrument. For example, the vernier caliper (fig. 6-5) has 25 markings on the vernier for 24 on the caliper scale. The caliper is marked off to read to fortieths (0.025) of an inch, and the vernier extends the accuracy to a thousandth of an inch. showing (0.075). The thimble division nearest and below the index is the 8 (0.008). The vernier marking that matches a thimble marking is the fourth (0.0004). Adding them all together, we have, The reading is inch. With practice these readings can be made directly from the micrometer, without writing the partial readings. Figure 6-5. A vernier caliper. Vernier Micrometer By adding a vernier to the micrometer, it is possible to read accurately to one ten-thousandth of an inch. The vernier markings are on the sleeve of the micrometer and are parallel to the thimble markings. There are 10 divisions on the vernier that occupy the same space as 9 divisions on the thimble. Since a thimble space is one thousandth of an inch, a vernier space is of inch, or inch. It is inch less than a thimble space. Thus, as in the preceding explanation of verniers, it is possible to read the nearest ten-thousandth of an inch by reading the vernier digit whose marking coincides with a thimble marking. In figure 6-6 (A), the last major division showing fully on the sleeve index is 3. The third minor division is the last mark clearly Figure 6-6. Vernier micrometer settings. Practice problems: 1. Read the micrometer settings in figure 6-6. Answers: 1. (A) See the foregoing example. (B) (E) (C) (F) (D)
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238 1-64. The product, in simplified form, of the multiplication, problem; 4 (2 hours 22 minutes 32 seconds) is 1. 8 hours 88 minutes 128 seconds 2. 8 hours 90 minutes 8 seconds 3. 9 hours 28 minutes 8 seconds 4. 9 hours 30 minutes 8 seconds The product of 12 miles and 13 miles is miles 2. miles square miles The product of 2 feet 8 inches times 3 feet 4 inches may be found by 1. multiplying 2-feet times 3 feet then multiplying 8 inches times 4 inches 2. multiplying 3 feet times 2 feet 8 inches then multiplying 4 inches times 2 feet 8 inches 2 3. converting 2 feet 8 inches to 2 feet 3 and 3 feet 4 inches to feet and then multiplying 4. changing 2 feet 8 inches to 3 feet and 3 feet 4 inches to 4 feet and then multiplying If a pipe 22 feet 6 inches long is cut into 3 equal lengths, how long are the pieces? (Neglect the width of the saw cuts.) 1. 7 feet 1 inch 2. 7 feet 3 inches 3. 7 feet 4 inches 4. 7 feet 6 inches The result of dividing 23 hours 31 minutes 20 seconds by 5 is hours minutes 4 seconds 2. 4 hours 0 minutes 4224 seconds 3. 4 hours 42 minutes 16 seconds 4. 4 hours 6 minutes 4 seconds In which of the following series of operations is the order in which the operations are performed important? l (3) (9) (7) 3. 6 x 8 x x The answer to the problem is 1. not defined The order of operations is important if division or multiplication is involved with other operations, Use the rules pertaining to a series of mixed operations to calculate the value of 6 x The number 36 is a multiple of 1, 2, 3, 4, 6, 9, 18, 36, and An odd number when divided by 2 produces a remainder of The number 7 is a factor of l What is the value of x if x = (3)(0)(4)(6)? l
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267 Assignment 7 Fundamentals of Algebra; Factoring Polynomials Textbook Assignment: Chapters 9, 10 ( ) 7-1. The literal numbers a, x, and p are more general than the numbers 9, 8, and The commutative law for addition is illustrated by the equation 1. ab = ba 2. a+a=b+b 3. a+b=b+a 4. a(b + c) = ab + ac 7-3. The associative law of multiplication is illustrated by the equation 1. abc = acb = cba 2. a x (b + c) = c x (b + a) 3. a (b + c + d) = ab + ac + ad 4. a (b c) = a b c = (a b) c 7-4. If a = 2, b = -3, and c = 4, the algebraic sum of a - b - c equals If r = 1, s = 3, t = 12, and x = 15, what is the value of the expression 2rx? t - s The algebraic expression is considered to be three numbers What is the value of the algebraic expression 5x 2-2xy + (3x) 2 when x = 2 and y = -3? What name is given to the algebraic expression ax - 2bx + cx 2-4? 1. Monomial 2. Binomial 3. Trinomial 4. Polynomial 7-9. Which statement is true regarding the monomial 17xyz? is the coefficient of xyz x is the coefficient of yz xy is the coefficient of z. 4. Each of the above is a true statement In the expression xy, what is the coefficient of xy? x 3. y 4. x y What is the coefficient of x 2 y in the expression x 2 y - ab? x 2 4. y Two terms of an expression are said to be like if they contain 1. at least one factor in common 2. the same numerical coefficient 3. the same literal factors with only their exponents different 4. the same literal factors raised to the same powers The like terms in the expression 2 ac 2-2bc 2 + ac 2-2c 2 are 1. 2ac 2 and 2bc ac 2 and ac ac 2 and 2c bc 2 and 2c 2 40
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