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Thus, to express 10367 in its natural form, we first write 83. Then, as there are four ciphers in the denominator, beginning on 3, we count towards the left, one, two, three, four, placing a cipher, as we say three, and another as we say four; and, prefixing the comma, we have ,0083 for the required decimal.
228. In the scale of Natural Numbers, (222,) the value of any figure is determined by its distance from the comma. Therefore, in any
finite number, to remove the comma one, two, three, &c: places to the right, is the same as to advance each figure one, two, three, &c. places towards the left; that is, (36) it is the same as to multiply the number by 10, 100, 1000, &c. Also, to remove the comma one, two, three, &c. places to the left, is to advance each figure one, two, three, &c. places towards the right; that is, (223,) to divide the number by 10, 100, 1000, &c.
Wherefore, to multiply a decimal number by 10, 100, 1000, &c., we remove the comma one, two, three, &c. places towards the right. Also, to divide the number by 10, 100, 1000, &c., we remove the comma one, two, three, &c. places towards the left. Thus :
,7854 X 10 = 7,854; ,7854 X 1000=785,4. If, in either direction, we do not find a sufficient number of places, we supply the deficiency with ciphers. Thus : 785,4 X 100=78540; 1,6 100 = ,016; ,7854 - 1000 =,0007854; ,7854 x 1000000 =785400
Numeration of Decimals. 229. The denominator of a decimal figure is (223) a unit followed by as many ciphers as there are places from the comma to the figure inclusive ; and is also (225) the alt of a whole number containing the same number of places. We have, therefore, two methods by which to determine the order of any decimal figure ; namely, we can point off the places of figures from the figure, considered as units, to the comma. Then, having read the figure, we pronounce the name of the alt or unit of the next higher grade with the usual denominating termination, ths. Or, beginning at the comma, we can pronounce on the places successively, tenths, hundredths, thousandths, tenthousandths, hundred-thousands, millionths, ten-millionths, hundred-millionths, billionths, &c., till we reach the figure.
230. In the scale of Natural Numbers, a number of consecu
tive figures, taken at pleasure, may (48) be read as a number of units of the order of the right-hand place of the part taken. To read a decimal fraction, therefore, we find, by either of the above methods, the name of the order of its right-hand place, which name we give to the whole, after reading it as a whole number. For example, to read the fraction ,5083020461079, having separated it into periods, as a whole number, we find that its highest order is units of trillions, and hence, its alt (87) ten trillions; consequently, the fraction is a number of tentrillionths. We therefore read, five trillions, eighty-three billions, twenty millions, four hundred and sixty-one thousand and seventy-nine ten-trillionths.
If all the figures of the above fraction, except the two last, were ciphers, we should still separate the places of figures into periods, find the denomination, as above, and read seventy-nine ten-trillionths. Express in words the following
5. 91,002009 In reading a mixed decimal number, we insert the word and between the integral and decimal parts. The present example is read ninety-one, and two thousand and nine millionths.
10. ,81302000625031171811132569 231. From the above, we easily derive the method of writing the figures of a decimal fraction from its expression in words. Omitting the last word, write the number enounced as a whole number. Ascertain the number of ciphers in the alt, which is the last word divested of ths. Count from right to left, upon the figures of the number written, (and, if necessary, (227,) supply ciphers,) as many places for decimals, as there are ciphers in the alt
. Then place a comma on the left, and the decimal is complete. Thus, to write ninety-five millionths, we first write 95 as a whole number. Then, as there are six ciphers in the alt, which is 1000000, we begin on 5 and count, towards the left, six places for decimals, supplying, of course, four ciphers; and have, 000095 for the required decimal.
We write two hundred and nineteen, and three hundred and twenty-seven billionths, thus :
219,000000327 first the integral part 219, with a comma on the right; then, as there are 9 ciphers in a billion, and only 3 figures in the number 327, we write 6 ciphers on the right of the comma, and on the right of these the number 327. Express in figures the following
Examples. 1. Nine units, and twenty-seven thousandths.
2. Six hundred and three units, and ninety-five ten-thousandths.
3. Eighteen units, and twelve thousand six hundred and four millionths.
4. Ten thousand and twenty-five ten-millionths.
6. Forty millions six hundred units, and ninety-seven thousand three hundred and fifty-eight hundred-millionths.
7. Sixty-four billions, forty-two millions, eleven thousand and seventeen hundred-billionths.
8. Write nine units, and twenty-seven thousandths; say how many thousandths this number contains, and how many times it would be lessened by changing it to millionths.
9. How many ten-thousandths are there in five hundred and four units, and sixty-seven ten-thousandths ? and how many times would this number be lessened by changing it to hundred-millionths ?
10. How many millionths are there in seventy thousand units, and thirty-six thousand and twenty-five millionths? and how many times would this number be lessened, by changing it to hundred-billionths ?
ADDITION OF DECIMALS.
232. "As the unit of any order, whether on the right or left of the comma, contains ten units of the next order on the right of it, we add numbers containing decimals, or decimals
alone, in the same manner as whole numbers ; that is to say, we add units of the same kind together, and, for every ten in the sum of any order, we carry one to the next order on the left.
We also place the numbers under each other, so that the commas may be under each other; that is, so that units of the same order may be under each other, and place a comma in the sum on the left of the tenths, before we add the units.
For example, to add 54, 75,2, 95,56, and ,273, we place the numbers thus :
and having added the tenths, we place a comma on the left under the other commas. We prove by adding both ways, as in whole numbers.
Examples. 1. 376,3 + 5,674 +,23+ 150 = 532,204 2. ,36 + ,536 + 789,3 + 1,16 = 791,356 3. 373 + 25,25 +,789 + 236,1 +5,4 = 640,539 4. ,234 +,5+,567 + ,462+,0005= 1,7635 5. 532,204 + 791,356 + 640,539 +1,7635 = 1965,8625 The scholar may prove this example by adding the numbers which compose the four preceding examples. .
Subtraction of Decimals. 233. Place the numbers so that units of the same order may be under each other : subtract as in whole numbers, and place the comma in the remainder on the left of the tenths, or under the other commas, as in addition, (198.)
When one of the given numbers has more decimal figures than the other, we may (226) render the number the same in both. Or, without placing ciphers on the right, we may subtract as though they were so placed.
For example, to subtract 78,358 from 80,3, we place the numbers thus :
80,300 or thus : 80,3
1,942 Subtract as usual, and, placing the comma on the left of the tenths, we have 1,942 for the remainder. This and the suoceeding examples may be proved both by addition and subtraction.
5. 1-,99951807324=,00048192676 6. What is the difference between the sum of the answers to the five preceding examples, and 100000 ?
Answer: Seventy-six thousand, seven hundred and sixtysix units, and seventeen billions, eight hundred and fifty-one millions, eight hundred and seven thousand, three hundred and twenty-four hundred-billionths.
Multiplication of Decimals. 234. The denominator of a decimal number (225) is a unit, followed by as many ciphers as there are decimal places in the number. Therefore, (206,) the denominator of the product of two such numbers is a unit, followed by as many ciphers as there are decimal places in both numbers : that is, (227,) there are as many decimal places in the product as in both factors. Hence, the following rule: Place the numbers and multiply them as whole numbers; and, in the product, point off for decimals as many places as there are decimal places in both factors.
When the number of figures in the product is not sufficient, we complete it by placing ciphers on the left, and then prefix the comma.
To multiply 7,54 by 7,1, without regard to the comma, we place the numbers thus :