SUBTRACTION OF DECIMAL FRACTIONS.

109. The rule for the Subtraction of Decimal Fractions is also the same as the rule for the Subtraction of whole numbers. If the number of decimal figures in the minuend and subtrahend is not the same, zeros may be annexed to either, so as to make the number of decimal figures the same in both. This, though a convenient preparation of the numbers for subtraction, is not indispensable.

The Rule for Subtraction of Decimal Fractions is,- Place the decimal point of the subtrahend under the decimal point of the minuend, and the figures of the former under the figures of the latter, according to their relative values. Then subtract as in whole numbers, and place the decimal point of the remainder in the vertical column of the other decimal points. Ex. From 587.654 take 98.78654.

110. It is required to find the difference between,

MULTIPLICATION OF DECIMAL FRACTIONS.

111. By substituting denominators for decimal points in both factors, they are reduced to the form of vulgar fractions. (Arts. 105. and 106.) The product of two such factors is obtained by multiplying together the numerators of the factors for the

numerator of the product and the denominators for its denominator. (Art. 92. c.) The numerator of the product is therefore the product of the proposed factors, taken as whole numbers, and its denominator is 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, 1 followed by as many zeros as there are decimal figures in the multiplicand and multiplier.

But in decimal fractions, the denominator is replaced by the decimal point; the place of the point being determined by taking one decimal figure for each zero in the denominator. (Art. 105.) Hence, the number of zeros in the denominator of the product being equal to the sum of the numbers of zeros in the multiplicand and multiplier, it follows that the number of decimal figures in the product of two decimal factors is equal to the number of decimal figures in both multiplicand and multiplier. Ex. 1st. Let it be required to multiply 7.54 by 3.82.

2nd. Let it be required to multiply 024 by 07.

The calculation of these examples is set down in both forms, viz., by vulgar fractions and by decimals; the former for the sake of illustrating the preceding remarks, and the latter as the form to be followed in practice.

In Example 1., the multiplicand and multiplier contain two decimal figures each; therefore the product contains four decimal figures. In Example 2., the multiplicand contains three decimal figures and the multiplier two. Whence the product must contain five decimal figures; and therefore 8, the last figure of the product, is the fifth from the decimal point. But the product of the factors, taken as whole numbers, contains

only three figures. In order that the last figure may fill the fifth place from the decimal point it is therefore necessary to prefix two zeros to the significant figures of the product, which thus becomes 00168. This remark belongs properly to the calculation conducted in the decimal form. The other form shows at once that the zeros after the decimal point are necessary, and the reason why they are so.

In the same manner it is found that the product of any number of factors is obtained by multiplying the factors together, as whole numbers, and pointing off from the product a number of decimal figures equal to the number of decimal figures contained in all the factors.

a. General Rule for the Multiplication of Decimal Fractions. Find the product of the factors, considered as whole numbers; and point off from the right of the product as many figures for decimals as there are decimal figures in the factors.

If the number of figures in the product is less than the number of decimal figures in the factors, one zero for each deficient figure must be prefixed to the number which expresses the absolute product of the factors.

112. Multiply together,

113. If one factor is 10 or some power of 10, the product of the multiplicand by this factor is obtained by merely changing the position of the decimal point.

...

...

For if the multiplicand be expressed as a vulgar fraction, the product of this fraction by 10, 100, ... is obtained by dividing the denominator by 10, 100, (Art. 70. b.); the denominator of the product consequently contains one zero, two zeros, less than the multiplicand, and therefore the product, one zero, two zeros, ... less than the multiplicand; in other words, the product is obtained by moving the decimal point as many figures to the right as there are zeros in the multiplier.

For example 01576 × 1000=15.76.

31.439 × 10000=314390.

a. If the multiplier is any other number than 1 followed by zeros, it may be decomposed into the product of the significant figures by 10, 100, 1000, ... Then the product of the multiplicand by 10, 100, &c., is obtained by moving the decimal point, and the product of this result by the significant figures of the multiplier, by the general rule for the multiplication of decimal fractions.

Ex. Let it be required to multiply 0854 by 7000.

Hence, to multiply a decimal fraction by 10 or any power of 10, remove the decimal point of the multiplicand one figure to the right for each zero contained in the multiplier; and to multiply by any other number followed by zeros, first remove the decimal point of the multiplicand one figure to the right for each zero in the multiplier, and then multiply by the significant figure or figures of the multiplier.

114. Examples. Multiply together,

DIVISION OF DECIMAL FRACTIONS.

115. The dividend and divisor may be reduced to the same denominator (Art. 105.), and expressed as vulgar fractions. Then the quotient of the decimal fractions is equal to that of the vulgar fractions.

But the quotient of the vulgar fractions is obtained by multiplying the dividend by the reciprocal of the divisor; or, since the denominator of the dividend is equal to the numerator of the reciprocal of the divisor, the quotient is 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 dividend and divisor reduced to the same denominator, and then taken as whole numbers.

As an example, let it be required to divide 8.447 by 3.22.

8-447-47.3-22=3228;

... 8·447+3·22=8447÷1030=1447 × 1000=8147=23007.

The fractional part of this result being a vulgar fraction, it is necessary to reduce it to the form of a decimal; that is, to a fraction which has 10, 100, 1000, or some power of 10 for its denominator.

This may be accomplished by dividing 10, raised to a suitable power (which may be denoted by 10m), by 3220, and multiplying the terms of the fraction 3993 by the quotient; thus,

2007

3220

3220)1000000000(310559+

9660

3400

3220

18000

16100

19000

16100

29000

28980

20

... $220 × 310559=1000000000, very nearly.