-6 ten thousandths 8 ten thousandths. 8 is written under the column of ten thousandths, and 1 carried to the next figure for the 1 thousandth borrowed: 1 thousandth+3 thousandths=4 thousandths; and 5 thousandths4 thousandths=1 thousandth.

The remainder, 1, is written under the column of thousandths, and the subtraction continued.

3 hundredths-4 hundredths; the subtraction is not possible: 1 tenth,= 10 hundredths, is borrowed; 10 hundredths+3 hundredths=13 hundredths; and 13 hundredths-4 hundredths=9 hundredths. The 9 is written in its proper place, and 1 tenth carried for the 1 tenth or 10 hundredths borrowed.

1 tenth+0=1 tenth, which, taken from the 1 tenth of the minuend, gives O to be written in the place of tenths of the remainder.

The remainder is, therefore, 0918 or 918 ten thousandths.

2d Example. Let it be required from 176-384 to take 98-75864. The arrangement of the numbers, both with and without the reduction of the decimals to the same denominator, is exhibited below. The remainder, also, is given, but not the detail of the calculation, this seeming unnecessary.

231. Rule for the subtraction of decimal fractions:

Arrange the figures of the minuend and subtrahend, the latter under the former, according to their relative values. Then subtract as in whole numbers, and place the decimal point of the remainder in the vertical column containing the decimal points of the minuend and subtrahend.

232. Exercises in the subtraction of decimal fractions.
1st. Find the difference between 274.5 and 4.0076?

Ans. 240.9862.

233. Rules for the addition and subtraction of decimals may be easily deduced from the rules for the addition and subtraction of vulgar fractions. For any given decimals may be reduced to a common denominator by Article 221; and, considered as fractions having a common denominator, their sum is equal to the sum, and their difference to the difference, of the numerators, placed over the common denominator.

In decimal fractions the value of the denominator is indicated by the position of the decimal point. This value being the same for the numbers which are added or subtracted, and for their sum or difference, the position of the decimal point in the sum or difference must be the same as it is in the numbers added or subtracted.

Whence, generally, to find the sum or difference of any decimal fractions, reduce the given decimals to a common denominator, add or subtract them as whole numbers, and point off as many decimal figures in the sum or difference as there are decimal figures in each of the numbers after reduction to the same denominator.

MULTIPLICATION OF DECIMAL FRACTIONS.

234. The decimal fractions whose product is required, can be expressed as vulgar fractions (Art. 220), and their product obtained by multiplying together the numerators of the factors for the numerator of the product, and the denominators for its denominator (Art. 194).

Now, the numerator of a decimal, expressed as a vulgar fraction, is the given number without the decimal point; that is, it is the given number considered as an integer; and the denominator is 1 with as many zeros annexed as there are decimal figures in the given number (Art. 217 and 220).

Whence the numerator of the product of two decimal factors is equal to the product of the given figures, considered as integers; and the denominator of their product is equal to the product of 1, followed by as many zeros as there are decimal figures in the multiplicand into 1 followed by as many zeros as there are decimal figures in the multiplier; that is, to 1 followed by as many zeros as there are decimal figures in both multiplicand and multiplier.

In decimal fractions this denominator is indicated by the decimal point, which must be placed so as to cut off, from the right of the numerator, as many figures for decimals as there are zeros in the denominator (Art. 220). Consequently the product of two given decimal fractions is found by, 1st, multiplying together the given factors as whole numbers; And, 2d, pointing off from this product as many figures for decimals as there are decimal figures in both the multiplicand and multiplier.

The product of three decimal fractions is obtained by forming the product of any two of the factors, and then multiplying this product by the third factor; and in the same manner is found the product of four or any greater number of decimal fractions.

235. To apply these deductions, let it be required to multiply 35 407 by

12:54:

Performed in this manner the calculation serves to illustrate the preceding observations on the multiplication of decimal, considered as vulgar, fractions. In accordance, however, with the conclusion which has been deduced, 35407 ought to be multiplied by 1254, and 3+2 or 5 decimal figures to be pointed off from the product thus,

35.407

12.54

141628

177035

70814

35407

444-00378

The product of 3054 by 23 is 70242. To give this product its proper relative value: the multiplicand contains 5 decimal figures, and the multiplier 3; the product must consequently contain 8 decimal figures; and therefore the last figure, 2, is the 8th from the decimal point. But the product of the factors, taken as whole numbers, contains only 5 figures. In order, therefore, that the last figure may hold the 8th place from the decimal point, it is necessary to write three zeros to the left of the significant figures of the product, which thus becomes 00070242.

The calculation by the factors, expressed fractionally, is as follows,

A conclusion to be drawn from this example (or any other in which the absolute product of the given factors contains fewer figures than the factors contain decimals) is, that when the number of figures in the product of two decimal fractions is less than the number of decimal figures in the multiplicand and multiplier, zeros must be prefixed to the product, till the number of decimal figures contained in it is made equal to the number contained in both factors.

237. Let any decimal, expressed as a vulgar fraction, be multiplied by 10; the multiplication can be effected by dividing the denominator of the multiplicand by 10 (Art. 149); the denominator of the product consequently contains 1 zero fewer than that of the multiplicand, and, therefore, the product 1 decimal figure fewer than the multiplicand.

Similarly, the multiplier being 100, the product contains two decimal figures fewer than the multiplicand; being 1000, the product contains 3 decimal figures fewer than the multiplicand.

Whence, to multiply a decimal by any power of 10, remove the decimal point of the multiplicand 1 figure to the right for each zero contained in the multiplier; the result is the product required.

Example 1. Multiply 0.00587 by 1000.

The multiplier contains 3 zeros; therefore to obtain the product, the decimal point of the multiplicand must be removed three figures to the right. The product consequently is 005-87, or, omitting the zeros, which on the left of a whole number have no value, 5.87.

Example 2. Multiply 4.76 by 10000.

The multiplier contains 4 zeros; therefore the product is obtained by removing the decimal point of the multiplicand 4 figures to the right. The latter containing but two decimal figures, this transference of the decimal point to the right can only be made by annexing 2 zeros to the multiplicand.

Therefore 47600 is the product required.

The multiplication of 4.76 by 10000 may be represented also thus,

476 10000 476 100

100

1

x=47600, the same result as before.

238. Rule for the multiplication of decimal fractions:

Multiply the given factors together as whole numbers, and point off from the product as many figures for decimals as there are decimal figures in both the multiplicand and multiplier.

When the number of figures in the product is less than the number of decimal figures in the multiplicand and multiplier, zeros, equal in number to the deficient figures, must be written to the left of the number which expresses the absolute product of the factors.

To multiply a decimal by any power of 10, remove the decimal point of the multiplicand one figure to the right for each zero contained in the multiplier.

239. Exercises in the multiplication of decimal fractions:

..Ans. 286-283415.

....Ans. 2·662915.

..Ans. 0.29700236. ......Ans. 0·065341. ..Ans. 0·004320797. ....Ans. 0·375. .Ans. 602·301. .Ans. 2·8413165. ..Ans. 0·00065341.

Ans. 6.7579854. ..Ans. 0.072891. ..Ans. 33·08884. .......Ans. 1·819. ..Ans. 3415. .......Ans. 760. ..Ans. 5884000. ..Ans. 10363-28475. ..Ans. 0·00001234. ....Ans. 0.01. .Ans. 0·0001. ...Ans. 0.74. .....Ans. 7400. ..Ans. '05. ......Ans. 0.1.

..Ans. 0·00000001.

DIVISION OF DECIMAL FRACTIONS.

Ans. 456. ..Ans. 12600. ...Ans. 0·729.

240. If the dividend and divisor are brought to the same denominator, and then expressed as vulgar fractions, the denominators are equal; the quotient of the decimal is also equal to that of the vulgar fractions.

Now, in division of vulgar fractions, the quotient is obtained by multiplying the dividend by the reciprocal of the divisor; but, since the numerator of the reciprocal of the divisor is equal to the denominator of the dividend, these factors may be cancelled in the numerator and denominator of the quotient, which is thus reduced to a fractional expression, having for numerator the numerator of the dividend, and for denominator the numerator of the divisor.

These numbers are, respectively, the given dividend and divisor, reduced to the same denominator, and then considered as whole numbers.

Whence, to divide one decimal fraction by another, bring the decimals to the same denominator (or make the number of decimal figures in the dividend and divisor equal), and divide as in whole numbers.

3.4703=
34703 10000 34703
X
10000 10000 10000 270 270

84703

270

+

270 10000

143 347031284 (Art. 156.)

270

The fractional part of this result being a vulgar, and not a decimal, fraction, it becomes necessary to investigate a means of converting the former into the latter form of expression.

Now 14 is 43 of one simple unit.

80

270

80

800

Since one simple unit is equal to 10 tenths, 143 of one simple unit= 1430 tenths 5 tenths, or 143 of one unit-5 tenths+ of a tenth. Again, 1 tenth 10 hundredths, tenths- hundredths, and 998 hundredths=238 hundredths 2 hundredths+25% of a hundredth. hundredths=9178 thousandths.

Similarly

.*. 143 of 1 simple unit=58 tenths.

270

=5 tenths+269 hundredths.

=5 tenths+2 hundredths+9178 thousandths.
5 tenths+2 hundredths+9 thousandths+6,80% ten
thousandths+ &c. &c.

Whence 343 0.5296+*,
And 34703128·5296+.

270

From the preceding discussion it appears that to convert a proper vulgar into a decimal fraction, it is necessary to annex zeros to the numerator, and to divide by the denominator.

The number of decimal figures in the result is equal to the number of zeros annexed to the numerator of the vulgar fraction; for the quotient of the first division is tenths; that of the second, hundredths, &c. If any partial division give no significant figure to the quotient, a zero must be written to occupy the place, in order that their proper relative values may be preserved to the succeeding figures.

The reduction of 143 to a decimal may be represented thus,

270)143-00000(*52962+

1350

800

540

2600

2430

1700

1620

800

540

260

The remainders, 80, 260, 170, being reproduced, it is evident that the process cannot terminate in a last remainder equal to zero. The approximation to an exact result may, however, be always carried to any degree of accuracy required. In the present instance one unit of the last quotient figure is the part of 1; and as the value of all the succeeding figures of the quotient, how far soever carried, cannot amount to one unit of this order, (for then the last figure would be, not 2, but 3,) it follows that this quotient falls short of the exact quotient by less than the 100000th part of unity.

If carried one place farther, the approximation would be to within less than the millionth part of unity.

* + written after a decimal fraction is employed to indicate that the expression is incomplete, and greater than that which is put down.