Decimals.

Decimals are distinguished by a point, separating them from integers, or whole numbers. The first decimal figure signifies tenth parts, the second figure hundredth parts, the third figure thousandth parts, &c., of a unit.

I. To add and to subtract decimals.

Proceed as in whole numbers, placing the decimal points directly under each other.

45.07; 50.758; 123.0057; 74.702; 24.8 together=318.3357

II. 1. To multiply decimals.

Multiply as in whole numbers, and cut off as many figures in the product as there are decimal places in the multiplicand and multiplier together; supplying any deficiency of figures which may occur in the product by adding ciphers to the left of it.

Multiply 2.701 by 4.2; and .1801 by .068

2. To multiply decimals so as to limit the number of places in the product, which is usually called contracted multiplication.

Invert the multiplier; write down the product of the first figure of the multiplier in the usual way, and then reject from the products of the second and remaining figures of the multiplier all the places to the right of each of them respectively.

Multiply 25.236594 by 41.2435, so as to retain but three decimal places in the product.

3. In multiplying by 10, 100, 1000, 10000, &c., remove the decimal point, as the case may be, one, two, three, four, &c., places to the

right.

III. 1. To divide decimals.

Make the divisor an integer by removing the decimal point to the end of it; removing likewise the decimal point in the dividend an equal number of places towards the right hand. Then divide as in whole numbers, and the integers in the dividend will give the integral part of the quotient, and the decimal figures (if any) remaining in the dividend will give the same number of decimals in the quotient.

In all operations in division in this work, products are rejected and remainders only reserved.

2. In dividing by 10, 100, 1000, 10000, &c., remove the decimal point, as the case may be, one, two, three, four, &c., places to the

left.

IV. DECIMALS.

VULGAR FRACTIONS.

RECIPROCALS. V.

IV. To reduce vulgar fractions to decimals.

Annex ciphers to the numerator, and divide it by the denominator; the quotient will be the required value in decimals.

In vulgar fractions, the upper figures denote the numerator, and the under figures the denominator.

Therefore 3.75, and 1.7137, the equivalent decimals.

Reduce the following vulgar fractions to their equivalent decimals :--

Form a vulgar fraction, the numerator of which is unity, and the denominator the number whose reciprocal is required, and reduce it to a decimal for the reciprocal of the given number.

Division and Multiplication may be performed by means of reciprocals:

DIVISION, by multiplying the dividend by the reciprocal of the divisor for the
quotient; and

MULTIPLICATION, by dividing the multiplicand by the reciprocal of the mul-
tiplier for the product.

Divide 610000 by 625; that is, 610000 multiplied by .0016, the reciprocal of
the divisor 625; giving for the required quotient.

. 976 Multiply 976 by 625; that is, 976.0000 divided by .0016, the reciprocal of the multiplier 625; giving for the required product.

610000

VI. To reduce numbers of different denominations to their equivalent decimal values.

Form a fraction, the numerator of which is the given quantity reduced to its lowest name, and whose denominator is an integer of the highest name in the given quantity, reduced to the same terms as the numerator. The equivalent decimal of this fraction is the value required.

Reduce 15s. to the equivalent decimal of a £.

Now 15s. are already equal to the lowest name in the given sum;

And 20s. are equal to £1, which is the integer of the highest name in the given sum; Therefore is the equivalent vulgar fraction of a £;

15

20

And £15, being reduced (IV.) to a decimal, gives £.75 for the equivalent value.

209

Reduce 15s. 6d. to the equivalent decimal of a £.

Now 15s. 6d. 186d., the lowest name in the given sum;

And 240d.£1, which is the integer of the highest name in the given sum;
Therefore is the equivalent vulgar fraction of a £;

186

240

And £, being reduced (IV.) to a decimal, gives £.775 for the equivalent value. £186 240

Reduce 15s. 6d. to the equivalent decimal of a £.

Now 15s. 6d. 746 farthings, the lowest name in the given sum;

And 960 farthings-£1, which is the integer of the highest name in the given sum; Therefore is the equivalent vulgar fraction of a £;

746 960

746

And £4, being reduced (IV.) to a decimal, gives £.777083 for the equi

valent value.

Reduce d., d., id., ld.; 14d., ltd., lid., 2d.; 2§d., 2§d., 23d., 3d.; to equivalent decimals of a £.

Continue these operations until the following Table I., embracing the equivalent decimals of every intermediate sum, up to the fifth part of a £, be completely constructed.

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