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CONTENTS AND PRICES
Of the Twelve Sections.
W. METCALFE AND SON, TRINITY STREET, CAMBRIDGE.
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ART. 1. As the local value of figures in the denary scale of notation increases tenfold reckoned from the right towards the left, the first place being the place of units, the second the place of tens, the third the place of hundreds, the fourth the place of thousands, and so on: it follows that each successive figure decreases tenfold when reckoned from the left towards the right.
If the scale be continued towards the right and reckoned from the first place of units, the second, third, fourth, &c., places will be the places of tenths, hundredths, thousandths, &c. And if a point be placed before the figure in the place of tenths, this mark will distinguish the fractional from the integral portion of the scale; and while the integral places successively increase by tens, hundreds, thousands, &c., the fractional places successively decrease by tenths, hundredths, thousandths, &c., from the unit's place.
This extension of the numerical scale reckoning tenfold decrease towards the right constitutes the perfection of the denary system of notation, and renders it complete for expressing the smallest possible fraction as well as the largest possible number.
And further, as these fractions consist only of an extension of the scale below the place of units to tenths, hundredths, thousandths, and so on, the operations of addition, subtraction, multiplication, and division can be performed with these fractions in the same manner as integers, taking care to mark the values of the results by correctly placing the point which separates the fractions from the integers.
2. DEF. A decimal fraction or decimal, may be defined to be a fraction the denominator of which is 10, 100, 1000, &c.
There is a peculiar notation asssumed for expressing these fractions. The denominators are omitted, and the numerators only are written with a point placed before that figure, which stands as many places from the right-hand figure as there are ciphers in the denominator. If the number of figures in the numerator be less than the number of ciphers in the denominator, the required number of figures must be made up by prefixing ciphers to the significant figures of the numerator, in order that each figure of the numerator may occupy its proper place in the scale."
1 In order to avoid ambiguity, the point should be placed before the upper part of the first figure of the decimal, and not before the lower part, as a point placed between numbers or symbols in the lower part has been assumed, instead of the symbol X, to denote that the numbers or symbols are multiplied together.
Thus, the decimal fraction is represented by :5, To by .05, To by .005, 900 by 876, 1876 by 5.876.
3. Prop. The ralue of any decimal is not altered in ralue by annexing one, tuo, three, 8c., ciphers to the right hand of it.
For by annexing 1, 2, 3, &c., ciphers to the right hand of a decimal, both the numerator and the suppressed denominator of the decimal fraction are multiplied by 10, 100, 1000, &c. : and if the uumerator and denominator of a fraction be both multiplied by the same number, the fraction is both multiplied and divided by that number, and therefore remains unaltered in value.
Also, if one, two, three, fc., ciphers be found at the right hand of a alecimal, they may be omitted without altering the value of the decimal.
For by omitting 1, 2, 3, &c., ciphers, both the numerator and the suppressed denominator of the decimal fraction are divided by 10, 100, 1000, &c.; and when both the numerator and denominator of a fraction are divided by the same number, its value is unaltered.
4. Prop. To find the sum or difference of two decimals.
If the decimals be reduced to a common denominator, the sum or difference may be found as the sum or difference of two ordinary fractions.
The sum or difference, however, may be found without reducing the decimal fractions to a common denominator, by arranging the numbers under each other : units under units, tenths under tenths, and so on, and then adding or subtracting as in integers, taking care to place the decimal point in the sum or difference before the place of tenths.2
1 If the equivalent notations and •5 might be named, the former “a decimal fraction,” and the latter “a decimal,” an ambiguity would sometimes be avoided in speaking of the two forms of the same thing.
A decimal may consist wholly of significant figures, as •875 is a decimal of three places, consisting of three significant figures ; or partly of significant figures, and partly of figures which are not significant, as .005 is a decimal of three places, consisting of one significant figure.
Any integer may be expressed in the form of a decimal fraction, as 25 may be put into the forms 16", 106", 1000",
259&c. Any decimal may be exhibited as the sum of as many decimals or decimal fractions as there are significant figures in the given decimal. The decimal 875 may be exhibited as the sum of three decimal fractions.
875 800 + 70 + 5 Thus: 875
1000 800 70
•8 + 07 ·005 ? Example. – To find the sum and difference of 13.1035 and 7.8105689. 13.1035000
7.8105689 20140689 sum,
5. Prop. To multiply one decimal by another, and to deduce the general rule for the place of the decimal point in the product. To multiply 3.457 by 21:34:3457
2134 Here 3.457 =
and 21:34 1000
a decimal of five places,
= 73.77238 Hence the product of two decimals is found by multiplying the decimals as integers, and pointing off from the right-hand figure of the product as many decimal places as there are in the multiplicand and the multiplier.
If, however, there are not as many figures in the product as the number of decimal places in the multiplicand and multiplier together, the required number of decimal places must be made up by prefixing to the product as many ciphers as make up the defect, and tho product will be wholly a decimal.
6. PROP. A decimal is multiplied by 10, 100, 1000, doc., by removing the decimal point in the given decimal one, two, three, doc., places towards the right.
For any given decimal expressed as a decimal fraction is multiplied by 10, 100, 1000, &c., by dividing the denominator by 10, 100, 1000, &c., and thus the decimal fraction, when expressed as a decimal, will consist of 1, 2, 3, &c., decimal places less than before.
7. Prop. To divide one decimal by another, and to deduce the general rules for the place of the decimal point in the quotient.'
Three ciphers have been annexed to the decimal part of the larger number to make the number of decimal places equal in the two numbers.
It is evident that as ten units make 1 ten, ten tens 1 hundred, and so on, so ten tenths make 1 unit, ten hundredths 1 tenth, and so on ; the same law obtains botit in the integers and the decimal parts. The processes of addition, subtraction, multiplication, and division of decimals are effected in the same manner as in integral numbers.
A decimal is said to be correct approximately for any number of places when any number of figures on the right of the decimal have been omitted ; but if the first of the figures omitted be greater than 5, the last figure on the right must be increased by 1. As an example 5•293 is nearer to 5.2929311 than 5.292, for 5.293 exceeds 5•2929311 by .0000689, and 5.292 is less than 5.2929311 by 0009311. Hence the error is less in the former case than in the latter; the former is in excess and tho latter in defect of the truth.
1 In the operations of multiplication and division, as the terms dividend, divisor, and quotient in the latter correspond to product, multiplicand, and multiplier in the former ; it is possible that the number of decimals given in a dividend may be
Case 1. When the number of decimals in the dividend is greater than the number in the divisor :Let 73.77238 be divided by 3.457. 7377238
3457 Here 73-77238 =
7377238 3457 Then 77.77238 • 3:457
100000 1000 7377238 1000
Х 100000 5457 2134
a decimal of two places, 100
= 21:34. Hence, when the number of decimals in the dividend is greater than the number in the divisor, the number in the quotient will be equal to the excess of the number in the dividend above that in the divisor.
Case 2. When the number of decimals in the dividend is equal to the number in the divisorLet 7377.238 be divided by 3.457. 7377238
3457 Here 7377.238 =
and 3.457 = 1000
7377238 3457 Then 7377.238 • 3:457
1000 1000 7377238 1000
1000 3457 7377238
= 2134 an integer. Hence, when the number of decimals in the dividend is equal to the number in the divisor, the quotient will be an integer.
Case 3. When the number of decimals in the dividend is less than the number in the divisor. Let 737723.8 be divided by :3457. 7377238
3457 Here 737723.8 =
and •3457 10
greater than, equal to, or less than the number of decimals in a divisor. Hence there will arise three cases in division of decimals to be considered.
Abbreviated methods have been devised both for the multiplication and division of decimals, whereby the number of figures employed in these operations may be somewhat lessened; and these methods were donbtless very useful in operations requiring a large number of figures. But since the invention of logarithms, these abbreviated methods are of no great practical utility, as such operations can be easily effected by the use of logarithmic tables.