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

    NONRESIDENTTRAINING

    COURSE

    Mathematics, Basic Math and

    AlgebraNAVEDTRA 14139

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    MATHEMATICS, VOLUME 1

    Figure 5-1.Place values including decimals.

    Figure 5-2.Conversionof a decimal fractionto shortened form.

    are zeros in the denominator of the fractionalform.

    Figure 5-3 shows the fraction 24358 and100000

    what is meant when it is changed to the short-ened form. This figure is presented to showfurther that each digit of a decimal fractionholds a certain position in the digit sequenceand has a particular value.

    By the fundamental rule of fractions, it

    should be clear that 550 500

    = = . Writing10 100 1000

    the same values in the shortened way, we have0.5 = 0.50 = 0.500. In other words, the value ofa decimal is not changed by annexing zeros atthe right-hand end of the number. This is not

    46

    Figure 5-3.Steps in the conversion of adecimal fraction to shortened form.

    true of whole numbers. Thus, 0.3, 0.30, and0.300 are equal but 3, 30, and 300 are not equal.

    Also notice that zeros directly after the deci-mal point do change values. Thus 0.3 is notequal to either 0.03 or 0.003.

    Decimals such as 0.125 are frequently seen. Although the 0 on the left of the decimal pointis not required, it is often helpful. This is par-

    ticularlytrue in an expression such as 32 0.1.In this expression, the lower dot of the divisionsymbol must not be crowded against the decimalpoint; the 0 serves as an effective spacer. Ifany doubt exists concerning the clarity of anexpression such as .125, it should be written as0.125.

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    MATHEMATICS. VOLUME 1

    0.2000.0250.002

    0.227

    Figure 6-1.(A) Parts of a micrometer;(B) micrometer scales.

    of an inch. Multiplying times 0.025 inch, we

    find that each marking on the thimble repre-sents 0.001 inch.

    0.025 inch since1

    is equal to 0.025. The40

    sleeve has 40 markings to the inch. Thus eachspace between the markings on the sleeve isalso 0.025 inch. Since 4 such spaces are 0.1inch (that is, 4 x 0.025), every fourth mark islabeled in tenths of an inch for convenience inreading. Thus, 4 marks equal 0.1 inch, 8 marksequal 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 into25 equal parts. Thus each marking on the

    thimble is1

    of a complete turn, or1

    of1

    25 25 401

    25

    READING THE MICROMETER

    It is sometimes convenient when learning toread a micrometer to writedown the component

    62

    parts of the measurement as read on the scalesand then to add them. For example, in figure6-1 (B) there are two major divisions visible(0.2 inch). One minor division is showingclearly (0.025 inch). The marking on the thimblenearest the horizontal or index line of the sleeve

    is the second marking (0.002 inch). Addingthese parts, we have

    Thus, the reading is 0.227 inch. As explainedpreviously, this is read verbally as "two hun-dred twenty-seven thousandths." A more skill-ful method of reading the scales is to read alldigits as thousandths directly and to do anyadding mentally. Thus, we read the major divi-sion on the scale as two hundred thousandthsand the minor division is added on mentally.The mental process for the above setting thenwould be two hundred twenty-five; two hundredtwenty-seven thousandths.

    Practice problems:

    1. Read each of the micrometer settings shownin figure 6-2.

    Figure 6-2.Micrometer settings.

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    MATHEMATICS, VOLUME 1

    The foregoing example could be followedthrough for any distance between markings.Suppose the 0 mark fell seven tenths of the dis-tance between ruler markings. It would takeseven vernier markings, a loss of one-hundredthof an inch each time, to bring the marks in lineat 7 on the vernier.

    The vernier principle may be used to getfine linear readings, angular readings, etc.The principle is always the same. The vernierhas one more marking than the number of mark-ings on an equal space of the conventional scaleof the measuring instrument. For example, thevernier caliper (fig. 6-5) has 25 markings onthe vernier for 24 on the caliper scale. Thecaliper is marked off to read to fortieths (0.025)of an inch, and the vernier extends the accuracyto a thousandth of an inch.

    0.30000.07500.00800.0004

    0.3834

    Figure 6-5.A vernier caliper.

    Vernier Micrometer

    By adding a vernier to the micrometer, it is

    possible to read accurately to one ten-thousandthof an inch. The vernier markings are on thesleeve of the micrometer and are parallel tothe thimble markings. There are 10 divisionson the vernier that occupy the same space as 9divisions on the thimble. Since a thimble spaceis one thousandth of an inch, a vernier space is1

    of9

    inch, or9

    inch. It is1

    inch10 1000 10000 10000

    less than a thimble space. Thus, as in the pre-ceding explanation of verniers, it is possible toread the nearest ten-thousandth of an inch byreading the vernier digit whose marking coin-cides with a thimble marking.

    In figure 6-6 (A), the last major divisionshowing fully on the sleeve index is 3. Thethird minor division is the last mark clearly

    64

    showing (0.075). The thimble division nearestand below the index is the 8 (0.008). The ver-nier marking that matches a thimble markingis the fourth (0.0004). Adding them all together,we have,

    The reading is 0.3834 inch. With practice thesereadings can be made directly from the microm-eter, without writing the partial readings.

    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) 0.1539 (E) 0.4690

    (C) 0.2507 (F) 0.0552

    (D) 0.2500

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    Assignment 7

    Fundamentals of Algebra; Factoring Polynomials

    Textbook Assignment: Chapters 9, 10 (111-117)

    7-1. The literal numbers a, x, and p are moregeneral than the numbers 9, 8, and 7.

    7-2. The commutative law for addition is il-lustrated by the equation1 . ab = ba2. a + a = b + b3. a + b = b + a4. a(b + c) = ab + ac

    7-3. The associat ive law of mult ipl icat ion isi l lustrated by the equation1. abc = acb = cba2. a x (b + c) = c x (b + a)3. a (b + c + d) = ab + ac + ad4. a (bc) = abc = (ab)c

    7-4. If a = 2, b = -3, and c = 4, the algebraicsum of a - b - c equals

    1. -52. 13 . 34. 9

    7-5. If r = 1, s = 3, t = 12, and x = 15, whatis the value of the expression

    2rx?t -

    s

    1 . - 3 3. 22. - 2 4. 3

    7-6. The algebraic expression

    is considered to be three numbers.

    7-7. What is the value of the algebraic ex-pression

    5x2 - 2xy + (3x)

    2

    when x = 2 and y = -3?1 . 442. 683. 1244. 14 8

    7-8. What name is given to the algebraic ex-pression ax - 2 bx + cx

    2- 4?

    1. Monomial2 . B in om ia l3 . T ri n om i al4 . Po lynom ial

    7-9. Which statement is true regarding themonomial 17xyz?1. 17 i s t he coe f fi c i en t o f xyz .2. 17x is the coef f ic ient of yz.3. 17xy i s the coe ff i c i en t of z .4. Each of the above is a true statement.

    7-10. In the expression xy, what is the co-e f f i c ien t o f xy?1. 12. x

    3. y

    4 . x y

    7-11. What is the coef f ic ient of x2y in theexpression x

    2y - ab?

    1. 12. 23. x

    2

    4. y

    7- 12 . Two terms of an expression are said to bel ike i f they con ta in1. at least one factor in common2. the same numerical coefficient3. the same l i tera l factors wi th only

    their exponents d i f ferent4. the same l i teral factors raised to the

    same powers

    7-1 3. The like terms in the expression

    2 ac2

    - 2bc2

    + ac2

    - 2c2

    are

    1 . 2ac2

    and 2bc2

    2 . 2ac2

    and ac2

    3. 2ac2

    and 2c2

    4. 2bc2 and 2c2

    40

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