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1 - - - . is or one thousandth part of unity; if three ciphers be in
l * terposed, it will express one tenth of To or one ten thou
sandth part of unity; and so on without end P. 215. The point which is employed to mark the place of units is called the decimal mark; it separates between whole numbers and decimals; that part of any number which is to the left of the decimal mark being the whole number, and that to the right the decimal. 216. We have chosen the number 1 on account of its simplicity for the illustration of this doctrine ; but the same equally holds true of all the other figures; thus, .2 expresses two tenths, or
217. Hence it appears, that a decimal fraction is expressed by the numerator only; the denominator, (being understood to consist of an unit with as many ciphers subjoined as the
decimal has places,) is always omitted; thus, .5 denotes +, .07
denotes +. .0009 expresses –*- &c.; by this simple artifice, fractions and 100 10000
whole numbers are exhibited under one and the same form; a whole number and a decimal constituting together but one number, in all respects (the decimal mark excepted) similar to a whole number; so that if a whole number and a decimal be connected, they constitute one uniform scale, the steps of which, beginning at the right hand figure, regularly increase through both, up to the left hand figure in a tenfold proportion at each step, and decrease in the same proportion at every step through both, from the left hand figure to the right.
218. Another advantage follows from the similarity of decimals to whole numbers, mamely, that the modes of operation are
P Hence it follows, that every cipher on the left hand in any decimal expression decreases the value of the decimal tenfold, and therefore such ciphers when they occur must always be put down, otherwise the value of the decimal will, for every cipher omitted, be increased tenfold, &c. more than its proper value.
It appears also that ciphers on the right hand of a decimal do not alter its
r value, for 5 = 50 = .500, &c. that is, -", 50, 500, are equal to each other; 10 100 1000
` this is evident, for each is equal to +: and the same is true in general,
alike in both ; the only peculiarity in decimals relates to the right placing of the decimal mark, which will easily be understood from the rules and explanations which follow.
219. In addition to the foregoing observations, it has been thought necessary to subjoin the following table, whereby the whole system of notation in whole numbers and decimals will be satisfactorily shewn.
It appears, by inspection of this table, that the figure on each side of the units place, and equally distant from it, is of like denomination, one in parts, and the other in wholes; but as in reading large whole numbers we make use of an abbreviated mode of expression, the same is found convenient in reading decimals. The above table is thus read; Nine hundred and sixtyeight millions, four hundred and seventy-two thousands, five hundred and thirty-one, and thirty-five millions, two hundred and seventy-four thousands, eight hundred and sirty-nine, hundred millionths: this, as far as it relates to the whole numbers, is sufficiently obvious. With respect to the decimals, this latter mode of expressing them, and that in the table, may at first sight appear to be different; we will shew that they are exactly of the same import.
Thus, 3 tenths . . . . . . . . are equal to 30 ----
6 tens of millionths . . . . . . . . . . 60
That is, the decimals as expressed in the table are together equal to 35 millions, 274 thousands, 869 hundred millionths, as we expressed their value above; which was to be shewn. 220. There is another method of reading decimals, which is very convenient for practice, as follows: The first place of decimals next to unity is called the place of primes; the next place is called the place of seconds, &c. and the figures occupying those places are called respectively primes,
seconds, thirds, fourths, &c. Thus the number 5.27 is read five, two primes, and seven seconds : and nine thirds, eight fourths, and six sevenths, is thus expressed in figures, .0098006.
221. ExAMPLes IN Decimal NoTATION AND NuMERATION. Write in words the following decimals.
Six tenths. Thirty-nine hundredths. Four hundred and fifty-six thousandths. One millionth. Five, and twenty-three hundredths. Three thousand three hundred and thirty-three ten thousandths. One, and twenty-four ten thousandths. Seven, and eight primes. Two, and two seconds. Nine thirds, eight fourths, seven fifths, and six sixths. Seven sixths, eight ninths, and mine tenths.
222. ADDITION OF DECIMALS.
Tule. Place the numbers so that the decimal marks may stand in a line under each other; then will units stand under units, tens under tens, tenths under tenths, hundredths under hundredths, &c, then, beginning at the right hand, add the mumbers together like whole numbers; and from the right hand of the sum cut off as many figures by the decimal mark as are equal to the greatest number of decimal places in any of the given numbers".
* The rule for placing the mumbers to be added is extremely obvious; for, since figures of different denominations cannot be added together, it is plain,
ExAMPLEs. 1. Add 2.34 + 35.2 + 7831 + 1.2481 + 8.0379 together.
.7831 Having placed the numbers so that like places may
1.2481 stand under like, I add up exactly as whole numbers are
- added; then I count four (equal to the greatest number
8.0379 of decimal places) from the right hand of the sum, and
Sum 47.6091 place the decimal mark to the left of the fourth figure.
2. Add 12.9 + 7.38 + 8.2 + .945 + 1.805 together. Sum 31.23. 3. Add 7.239 + 10.0046 -- 3.27 + .89 + .0073 together. - Sum 21.4109. 4. Add 942.64 + 2.301 + 71.5 + 8,457 + 3091.9 together. Sum 41 16.798. 5. Add .4937 -- .008 + .37042+.89139-1-1.290037 together. Sum 3.053547. 6. Add 3748.2 + 9.807 3 + 120.965 + 1374.7 H-48. together.
223. SUBTRACTION OF DECIMALS.
Rule. Place the less number below the greater, with the decimal marks under each other, so that units may stand under units, tenths under tenths, &c. as in Addition; then subtract as in whole numbers, and cut off from the right hand of the remainder as many figures for decimals as there are decimal places in either of the two given numbers".
that those of the same denomination must be placed under each other, as they alone are capable of being added. With respect to the operation, it is plain that 10 hundredths make 1 tenth, 10 tenths make 1 unit, 10 units 1 ten, and so on in every denomination, whether it be above or below unity; wherefore, since the same law obtains in both decimal parts and whole numbers, both must evidently be added by the same rule, namely, by simple Addition. * The observations contained in the preceding note apply equally to this rule, which is obvious from the nature of simple Subtraction. When ciphers occur in one or more of the left hand decimal places, they
From 13.745 .470349 3.617
Take 10, 193 .46712 1.71438
Diff. 3,622 .O(33929 190262
Proof 13.745 .470349 3.617
4. From 740352 take .214091. Diff. .526261.
224. MULTIPLICATION OF DECIMALS.
Rule I. Place the factors so that the right hand figure of the multiplier may stand under the right hand figure of the multiplicand, and multiply as in whole numbers. II. Count the decimal places in both factors, and from the right hand of the product mark off as many figures for decimals as there are decimals in both factors together. To prove the operation, multiply the multiplier by the multiplicand, and proceed as before; or cast out the nines, as in simple multiplication. III. When the number of decimals in both factors exceeds the number of figures in the product, prefix as many ciphers to the left of the product as will make up the number, and to the left of them place the decimal mark".
must always be put down, but ciphers occupying the right hand decimal places may be omitted; thus in ex. 2, the two ciphers at the left of the difference are put down, and in ex. 3, the two that arise at the right hand of the proof are omitted: likewise when any of the right hand places of decimals are wanting, as in ex. 2 and 3, the operation is to be performed as though there were ciphers in those vacant places. * To make the truth of this rule plain, recourse must be had to an easy example ; thus, let .5 be multiplied by .3; these numbers are equivalent to _5. and -*. as appears from the nature of decimal notation; now 5. x 3. 10 10 10 - 10