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After the above it is easy to see that we have only to remove the decimal point one place to the right or left, in order to increase or diminish respectively the value of a decimal tenfold. And if we allow the term decimal to include numbers which are greater than unity, as 35.721, we may extend the principle thus::
A decimal may be divided by 10, by removing the dot one place to the left; by 100 or 102 by removing the dot two places to the left; by 1000 or 103 by removing the dot three places to the left, and so on; and further, a decimal may be multiplied by 10, 100, 1000, &c., that is, by 10, 102, 103, &c., by removing the dot 1, 2, 3, &c., places respectively to the right. Thus, 6872-3476 divided by 10, 100, 1000 respectively, becomes 687-23476, 68-723476, 6-8723476; and multiplied by the same becomes 68723-476, 687234-76, 6872347-6 respectively.
Addition and Subtraction.
2. If the student has understood the preceding article, he will at once perceive that, provided we keep the units' figure under the units' figure in every case, there is no difference between the addition and subtraction of decimals, and the addition and subtraction of ordinary integers. All he has to take care of is that the decimal points are kept under each other.
Ex. 1.-Add together 325-02, 647, 5.6073, 00214, 290, and 4-7001. Proceeding as in ordinary addition:
Ex. 2. Take 6.291 from 18.3064. Proceeding as in
Ex. 3.-Find the difference between 15.02 and 6732.
NOTE. In an example of this kind, where the number of decimal figures in the lower line exceeds the number in the upper, it is advisable to mentally supply ciphers to make up the deficiency in the upper line. This may be done, as we have seen, without altering the value of the upper line.
3. Suppose we have to multiply 2.935 by 6.34, and let us suppose the dot in each case removed to the extreme right. Then (Art. 1), we have multiplied the number 2.935 by 1000. and the number 6.34 by 100, and we have obtained the numbers 2935 and 634 respectively. As these numbers are integers, we may omit the dot, and write them 2935 and 634. Now 2935 x 634 = 1860790, but as we increased our original numbers one thousand and one hundredfold respectively, it is evident that our product is increased 1000 × 100, or one hundred thousandfold. Dividing, therefore, the above result, 1860790 by 100000, or what is the same thing (Art. 1), writing it 1860790 and removing the dot 5 places. to the left, we get for our product of the numbers 2.935 and 6.34 the result, 18.60790. We may remark that the number of decimal figures in the product, namely, 5, is the sum of the numbers of decimal figures in the two given numbers.
We have, therefore, the following rule for multiplication:Multiply the given numbers exactly as integers, regardless of the decimal points, and after the operation is finished, point off as many decimal figures in the product as there are together in the multiplier and multiplicand.
Ex. 1.-Multiply 6.35 by 1703.
Now, the number of decimal figures in the multiplier and multiplicand together, is (4 + 2), or 6, and therefore we mark off 6 decimal figures in our product. This gives us 1.081405.
Ex. 2.-Multiply 0063, by 017.
And pointing off (4 + 3), or 7 decimal figures, we obtain for our product 0001071.
4. Suppose we have to divide 76875 by 6.25. We will proceed as in the case of multiplication, by imagining the decimal points in each number removed to the extreme right. The numbers will then be 76875, and 625, or, omitting the dot, as they are now integers, they will be 76875, and 625. Proceed now as in ordinary division (which operation it is unnecessary to explain), and we get for our quotient 123.
Now we must remember that we have increased our dividend 100,000 or 105-fold, and that, consequently, our quotient will require to be diminished 105-fold. This is done (Art. 1) by removing the dot of the number 123 five places to the left. But, before doing that, we know that the divisor has been increased 100 or 102-fold, and on this account our quotient must be increased 102-fold. This is done (Art. 1) by removing the dot two places to the right. Hence, to get the true quotient of 76875 by 6.25, we must remove the dot from the extreme right of the number 123, five minus two, or three places to the left. Now three is the excess of the number of decimal figures in the dividend over the number in the divisor. We hence arrive at the following rule :
Proceed as in ordinary division, and when all the figures of the dividend have been brought down, and the remainder, if any, obtained, cut off as many decimal figures in the
quotient, as the number of decimal figures in the dividend exceeds the number in the divisor.
NOTE.-When the number of decimal figures in the dividend is less than the number in the divisor, affix a sufficient number of ciphers to make the number of decimals in the dividend equal to the number in the divisor. After finishing the operation of ordinary division, there will be no decimal figures to cut off in the quotient. If there be a remainder, and the division carried on further, by affixing ciphers to the successive remainders, all the quotient figures thus obtained will be decimals.
Ex. 1.-Divide 117.85088 by 6.272.
We see that there are five decimal figures in the dividend, and three in the divisor, and so we cut off (5-3) or 2 in the quotient. The answer is therefore 18.79.
Ex. 2.-Divide 527.2 by 0008.
Here it will be necessary to affix three ciphers to the dividend, and the operation will stand thus
As there is no remainder, and the number of decimal figures in the dividend is equal to that in the divisor, we have none to cut off. The answer is therefore 659000.
Ex. 3.-Divide 463-7 by 2.769 to four places of decimals.
Here we must affix two ciphers to the dividend, and the operation, as far as the ordinary remainder of long division, stands thus:—
The quotient up to this point is integral; but, as we have a remainder, we must continue the operation of division, first placing a dot at the right of the figures in the quotient, and affixing a cipher to the present and each successive remainder, until we have the requisite number of decimals in the quotient. By thus proceeding, it is easy to see we arrive at an answer-167.4611.
1. Increase the numbers 4.523, 29, 02367, 07 respectively 10, 100, 1,000, 10,000-fold.
2. Divide by inspection the numbers 0.05, 1111, 4.0020, 45 respectively by 100, 10,000, 1,000, 10.
3. Express in words 3467, 34-67, 0003467, 3.467; and compare the values of the last three with the first.
4. Add together
(1.) 6-732, 14.9, 0064, 14.27006.
5. By how much does 5 exceed 4.2763, and 16.021 exceed 12.70009 ?
7. What number added to four thousandths will give three Lundredths, and what number subtracted from 8,000 units will give 291 units 29 hundredths?