SECTION VI.. Decimal Fractions.

DECIMAL FRACTIONS.

126. We have seen (112) (121) the necessity of reducing fractions to a common denominator, in order to compare their magnitudes. We have seen also how much more readily operations are performed upon fractions which have the same, than upon those which have different, denominators. On this account it has long been customary, in all those parts of mathematics where fractions are often required, to use none but such as either have, or can be easily reduced to others having, the same denominators. Now, of all numbers, those which can be most easily managed are such as 10, 100, 1000, &c., where 1 is followed by ciphers. These are called DECIMAL NUMBERS; and a fraction whose denominator is any one of them, is called a DECIMAL FRACTION, or more commonly, a DECIMAL.

127. A whole number may be reduced to a decimal fraction, or one decimal fraction to another, with the greatest ease. For example,

940 9400 94000 94 is ---, or ----, or ----- (106); 10 100 1000

3 30 300 3000 ---- is ----, or ----, or ----- (108). 30 100 1000 10000

The placing of a cipher on the right hand of any number is the same thing as multiplying that number by 10 (57), and this may be done as often as we please in the numerator of a fraction, provided it be done as often in the denominator (108).

128. The next question is, How can we reduce a fraction which is not decimal to another which is, without altering its value? Take, for example, the fraction ⁷/₁₆, multiply both the numerator and denominator successively by 10, 100, 1000, &c., which will give a series of fractions, each of which is equal to ⁷/₁₆ (108), viz. ⁷⁰/₁₆₀, ⁷⁰⁰/₁₆₀₀, ⁷⁰⁰⁰/₁₆₀₀₀, ⁷⁰⁰⁰⁰/₁₆₀₀₀₀, &c. The denominator of each of these fractions can be divided without remainder by 16, the quotients of which divisions form the series of decimal numbers 10, 100, 1000, 10000, &c. If, therefore, one of the numerators be divisible by 16, the fraction to which that numerator belongs has a numerator and denominator both divisible by 16. When that division has been made, which (108) does not alter the value of the fraction, we shall have a fraction whose denominator is one of the series 10, 100, 1000, &c., and which is equal in value to ⁷/₁₆. The question is then reduced to finding the first of the numbers 70, 700, 7000, 70000, &c., which can be divided by 16 without remainder.

Divide these numbers, one after the other, by 16, as follows:

16)70(4 16)700(43 16)7000(437 16)70000(4375 64 64 64 64 -- --- --- --- 6 60 60 60 48 48 48 -- --- --- 12 120 120 112 112 --- --- 8 80 80 -- 0

It appears, then, that 70000 is the first of the numerators which is divisible by 16. But it is not necessary to write down each of these divisions, since it is plain that the last contains all which came before. It will do, then, to proceed at once as if the number of ciphers were without end, to stop when the remainder is nothing, and then count the number of ciphers which have been used. In this case, since 70000 is 16 × 4375,

70000 16 × 4375 4375 ------, which is ----------, or -----, 160000 16 × 10000 10000

gives the fraction required.

Therefore, to reduce a fraction to a decimal fraction, annex ciphers to the numerator, and divide by the denominator until there is no remainder. The quotient will be the numerator of the required fraction, and the denominator will be unity, followed by as many ciphers as were used in obtaining the quotient.

EXERCISES.

Reduce to decimal fractions

½, ¼, ²/₂₅, ¹/₅₀, ³⁹²⁷/₁₂₅₀, and ⁴⁵³/₆₂₅.

Answer, ⁵/₁₀, ²⁵/₁₀₀, ⁸/₁₀₀, ²/₁₀₀, ³¹⁴¹⁶/₁₀₀₀₀, and ⁷²⁴⁸/₁₀₀₀₀.

129. It will happen in most cases that the annexing of ciphers to the numerator will never make it divisible by the denominator without remainder. For example, try to reduce ¹/₇ to a decimal fraction.

7)1000000000000000000, &c. ------------------- 142857142857142857, &c.

The quotient here is a continual repetition of the figures 1, 4, 2, 8, 5, 7, in the same order; therefore ¹/₇ cannot be reduced to a decimal fraction. But, nevertheless, if we take as a numerator any number of figures from the quotient 142857142857, &c., and as a denominator 1 followed by as many ciphers as were used in making that part of the quotient, we shall get a fraction which differs very little from ¹/₇, and which will differ still less from it if we put more figures in the numerator and more ciphers in the denominator.

Thus, 1 {is less} 1 3 {which is not} 1 --- { } --- by --- { } --- 10 { than } 7 70 { so much as } 10

14 1 2 1 --- --- --- --- 100 7 700 100

142 1 6 1 ---- --- ---- ---- 1000 7 7000 1000

1428 1 4 1 ----- --- ----- ----- 10000 7 70000 10000

14285 1 5 1 ------ --- ------ ------ 100000 7 700000 100000

142857 1 1 1 ------- --- ------- ------- 1000000 7 7000000 1000000

&c. &c. &c. &c.

In the first column is a series of decimal fractions, which come nearer and nearer to ¹/₇, as the third column shews. Therefore, though we cannot find a decimal fraction which is exactly ¹/₇, we can find one which differs from it as little as we please.

This may also be illustrated thus: It is required to reduce ¹/₇ to a decimal fraction without the error of say a millionth of a unit; multiply the numerator and denominator of ¹/₇ by a million, and then divide both by 7; we have then

1 1000000 1428571¹/₇ --- = ------- = ----------- 7 7000000 1000000

If we reject the fraction ¹/₇ in the numerator, what we reject is really the 7th part of the millionth part of a unit; or less than the millionth part of a unit. Therefore ¹⁴²⁸⁵⁷/₁₀₀₀₀₀₀ is the fraction required.

EXERCISES.

Make similar tables} 3 17 1 with } ---, ---, and ---. these fractions } 91 143 247

} 3 The recurring} --- is 329670,329670, &c. quotient of} 91

17 --- 118881,118881, &c. 143

1 --- 404858299595141700,4048582 &c. 247

130. The reason for the recurrence of the figures of the quotient in the same order is as follows: If 1000, &c. be divided by the number 247, the remainder at each step of the division is less than 247, being either 0, or one of the first 246 numbers. If, then, the remainder never become nothing, by carrying the division far enough, one remainder will occur a second time. If possible, let the first 246 remainders be all different, that is, let them be 1, 2, 3, &c., up to 246, variously distributed. As the 247th remainder cannot be so great as 247, it must be one of these which have preceded. From the step where the remainder becomes the same as a former remainder, it is evident that former figures of the quotient must be repeated in the same order.

131. You will here naturally ask, What is the use of decimal fractions, if the greater number of fractions cannot be reduced at all to decimals? The answer is this: The addition, subtraction, multiplication, and division of decimal fractions are much easier than those of common fractions; and though we cannot reduce all common fractions to decimals, yet we can find decimal fractions so near to each of them, that the error arising from using the decimal instead of the common fraction will not be perceptible. For example, if we suppose an inch to be divided into ten million of equal parts, one of those parts by itself will not be visible to the eye. Therefore, in finding a length, an error of a ten-millionth part of an inch is of no consequence, even where the finest measurement is necessary. Now, by carrying on the table in (129), we shall see that

1428571 1 1 -------- does not differ from --- by --------; 10000000 7 10000000

and if these fractions represented parts of an inch, the first might be used for the second, since the difference is not perceptible. In applying arithmetic to practice, nothing can be measured so accurately as to be represented in numbers without any error whatever, whether it be length, weight, or any other species of magnitude. It is therefore unnecessary to use any other than decimal fractions, since, by means of them, any quantity may be represented with as much correctness as by any other method.

EXERCISES.

Find decimal fractions which do not differ from the following fractions by ¹/₁₀₀₀₀₀₀₀₀.

⅓ Answer, ³³³³³³³³/₁₀₀₀₀₀₀₀₀. ⁴/₇ ⁵⁷¹⁴²⁸⁵⁷/₁₀₀₀₀₀₀₀₀. ¹¹³/₃₅₅ ³¹⁸³⁰⁹⁸⁵/₁₀₀₀₀₀₀₀₀. ³⁵⁵/₁₁₃ ³¹⁴¹⁵⁹²⁹²/₁₀₀₀₀₀₀₀₀.

132. Every decimal may be immediately reduced to a quantity consisting either of a whole number and more simple decimals, or of more simple decimals alone, having one figure only in each of the numerators. Take, for example,

147326 147326 326 ------. By (115) ------- is 147----; 1000 1000 1000

and since 326 is made up of 300, and 20, and 6; by (112) ³²⁶/₁₀₀₀₀ = ³⁰⁰/₁₀₀₀ + ²⁰/₁₀₀₀ + ⁶/₁₀₀₀. But (108) ³⁰⁰/₁₀₀₀ is ³/₁₀, and ²⁰/₁₀₀₀ is ²/₁₀₀. Therefore, ¹¹⁴⁷³²6/₁₀₀₀ is made up of 147 + ³/₁₀ + ²/₁₀₀ + 6/₁₀₀₀. Now, take any number, for example, 147326, and form a number of fractions having for their numerators this number, and for their denominators 1, 10, 100, 1000, 10000, &c., and reduce these fractions into numbers and more simple decimals, in the foregoing manner, which will give the table below.

DECOMPOSITION OF A DECIMAL FRACTION.

147326 ------ = 147326 1

147326 6 ------ = 14732 + --- 10 10

147326 2 6 ------ = 1473 + --- + --- 100 10 100

147326 3 2 6 ------ = 147 + --- + --- + ---- 1000 10 100 1000

147326 7 3 2 6 ------ = 14 + --- + --- + ---- + ----- 10000 10 100 1000 10000

147326 4 7 3 2 6 ------ = 1 + --- + --- + ---- + ----- + ------ 100000 10 100 1000 10000 100000

147326 1 4 7 3 2 6 ------- = --- + --- + ---- + ----- + ------ + ------- 1000000 10 100 1000 10000 100000 1000000

147326 1 4 7 3 2 6 -------- = --- + ---- + ----- + ------ + ------- + -------- 10000000 100 1000 10000 100000 1000000 10000000

N.B. The student should write this table himself, and then proceed to make similar tables from the following exercises.

EXERCISES.

Reduce the following fractions into a series of numbers and more simple fractions:

31415926 31415926 --------, --------, &c. 10 100

2700031 2700031 -------, --------, &c. 10 100

2073000 2073000 -------, --------, &c. 10 100

3331303 3331303 -------, -------, &c. 1000 10000

133. If, in this table, and others made in the same manner, you look at those fractions which contain a whole number, you will see that they may be made thus: Mark off, from the right hand of the numerator, as many figures as there are ciphers in the denominator by a point, or any other convenient mark.

This will give 14732·6 when the fraction is 147326 ------ 10

1473·26 147326 ------ 100

147·326 147326 ------ 1000

&c. &c.

The figures on the left of the point by themselves make the whole number which the fraction contains. Of those on its right, the first is the numerator of the fraction whose denominator is 10, the second of that whose denominator is 100, and so on. We now come to those fractions which do not contain a whole number.

134. The first of these is ¹⁴⁷³²⁶/₁₀₀₀₀₀₀ which the number of ciphers in the denominator is the same as the number of figures in the numerator. If we still follow the same rule, and mark off all the figures, by placing the point before them all, thus, ·147326, the observation in (133) still holds good; for, on looking at ¹⁴⁷³²⁶/₁₀₀₀₀₀₀ in the table, we find it is

1 4 7 3 2 6 --- + --- + ---- + ----- + ------ + ------- 10 100 1000 10000 100000 1000000

The next fraction is ¹⁴⁷³²⁶/₁₀₀₀₀₀₀₀, which we find by the table to be

1 4 7 3 2 6 --- + ---- + ----- + ------ + ------- + -------- 100 1000 10000 100000 1000000 10000000

In this, 1 is not divided by 10, but by 100; if, therefore, we put a point before the whole, the rule is not true, for the first figure on the left of the point has the denominator which, according to the rule, the second ought to have, the second that which the third ought to have, and so on. In order to keep the same rule for this case, we must contrive to make 1 the second figure on the right of the point instead of the first. This may be done by placing a cipher between it and the point, thus, ·0147326. Here the rule holds good, for by that rule this fraction is

0 1 4 7 3 2 6 --- + --- + ---- + ----- + ------ + ------- + -------- 10 100 1000 10000 100000 1000000 10000000

which is the same as the preceding line, since ⁰/₁₀ is 0, and need not be reckoned.

Similarly, when there are two ciphers more in the denominator than there are figures in the numerator, the rule will be true if we place two ciphers between the point and the numerator. The rule, therefore, stated fully, is this:

To reduce a decimal fraction to a whole number and more simple decimals, or to more simple decimals alone if it do not contain a whole number, mark off by a point as many figures from the numerator as there are ciphers in the denominator. If the numerator have not places enough for this, write as many ciphers before it as it wants places, and put the point before these ciphers. Then, if there be any figures before the point, they make the whole number which the fraction contains. The first figure after the point with the denominator 10, the second with the denominator 100, and so on, are the fractions of which the first fraction is composed.

135. Decimal fractions are not usually written at full length. It is more convenient to write the numerator only, and to cut off from the numerator as many figures as there are ciphers in the denominator, when that is possible, by a point. When there are more ciphers in the denominator than figures in the numerator, as many ciphers are placed before the numerator as will supply the deficiency, and the point is placed before the ciphers. Thus, ·7 will be used in future to denote ⁷/₁₀, ·07 for ⁷/₁₀₀, and so on. The following tables will give the whole of this notation at one view, and will shew its connexion with the decimal notation explained in the first section. You will observe that the numbers on the right of the units’ place stand for units divided by 10, 100, 1000, &c. while those on the left are units multiplied by 10, 100, 1000, &c.

The student is recommended always to write the decimal point in a line with the top of the figures or in the middle, as is done here, and never at the bottom. The reason is, that it is usual in the higher branches of mathematics to use a point placed between two numbers or letters which are multiplied together; thus, 15. 16, a. b, (a + b). (c + d) stand for the products of those numbers or letters.

1234 4 4 I. 123·4 stands for ---- or 123--- or 123 + --- 10 10 10

1234 34 3 4 12·34 ---- or 12--- or 12 + --- + --- 100 100 10 100

1234 234 2 3 4 1·234 ---- or 1---- or 1 + --- + --- + ---- 1000 1000 10 100 1000

1234 1 2 3 4 ·1234 ----- or --- + --- + ---- + ----- 10000 10 100 1000 10000

1234 1 2 3 4 ·01234 ------ or --- + ---- + ----- + ------ 100000 100 1000 10000 100000

1234 1 2 3 4 ·001234 ------- or ---- + ----- + ------ + ------- 1000000 1000 10000 100000 1000000

II. 1003 1 3 ·01003 is ------ or --- + ------ 100000 100 100000

1003 1 3 ·1003 is ----- or --- + ----- 10000 10 10000

1003 3 10·03 is ---- or 10 + --- 100 100

1003 3 100·3 is ---- or 100 + --- 10 10

III. 1 2 8 3 ·1283 = --- + --- + ---- + ----- 10 100 1000 10000

= ·1 + ·02 + ·008 + ·0003 = ·1 + ·0283 = ·12 + ·0083 = ·128 + ·0003 = ·108 + ·0203 = ·1003 + ·028 = ·1203 + ·008

{ 1 is 1000 inches { 2 is 200 { 3 is 30 { 4 is 4 IV. In 1234·56789 { 5 is ⁵/₁₀ of an inch inches the { 6 is ⁶/₁₀₀ { 7 is ⁷/₁₀₀₀ { 8 is ⁸/₁₀₀₀₀ { 9 is ⁹/₁₀₀₀₀₀

136. The ciphers on the right hand of the decimal point serve the same purpose as the ciphers in (10). They are not counted as any thing themselves, but serve to shew the place in which the accompanying numbers stand. They might be dispensed with by writing the numbers in ruled columns, as in the first section. They are distinguished from the numbers which accompany them by calling the latter significant figures. Thus, ·0003747 is a decimal of seven places with four significant figures, ·346 is a decimal of three places with three significant figures, &c.

137. The value of a decimal is not altered by putting any number of ciphers on its right. Take, for example, ·3 and ·300. The first (135) is ³/₁₀, and the second ³⁰⁰/₁₀₀₀, which is made from the first by multiplying both its numerator and denominator by 100, and (108) is the same quantity.

138. To reduce two decimals to a common denominator, put as many ciphers on the right of that which has the smaller number of places as will make the number of places in both fractions the same. Take, for example, ·54 and 4·3297. The first is ⁵⁴/₁₀₀, and the second ⁴³²⁹⁷/₁₀₀₀₀. Multiply the numerator and denominator of the first by 100 (108), which reduces it to ⁵⁴⁰⁰/₁₀₀₀₀, which has the same denominator as ⁴³²⁹⁷/₁₀₀₀₀. But ⁵⁴⁰⁰/₁₀₀₀₀ is ·5400 (135). In whole numbers, the decimal point should be placed at the end: thus, 129 should be written 129·. It is, however, usual to omit the point; but you must recollect that 129 and 129·000 are of the same value, since the first is 129 and the second ¹²⁹⁰⁰⁰/₁₀₀₀.

139. The rules which were given in the last chapter for addition, subtraction, multiplication, and division, apply to all fractions, and therefore to decimal fractions among the rest. But the way of writing decimal fractions, which is explained in this chapter, makes the application of these rules more simple. We proceed to the different cases.

Suppose it required to add 42·634, 45·2806, 2·001, and 54. By (112) these must be reduced to a common denominator, which is done (138) by writing them as follows: 42·6340, 45·2806, 2·0010, and 54·0000. These are decimal fractions, whose numerators are 426340, 452806, 20010, and 540000, and whose common denominator is 10000. By (112) their sum is

426340 + 452806 + 20010 + 540000 1439156 --------------------------------, which is ------- 10000 10000

or 143·9156. The simplest way of doing this is as follows: write the decimals down under one another, so that the decimal points may fall under one another, thus:

42·634 45·2806 2·001 54 -------- 143·9156

Add the different columns together as in common addition, and place the decimal point under the other decimal points.

EXERCISES.

What are 1527 + 64·732094 + 2·0013 + ·00001974; 2276·3 + ·107 + ·9 + 26·3172 + 56732·001; and 1·11 + 7·7 + ·0039 + ·00142 + ·8838?

Answer, 1593·73341374, 59035·6252, 9·69912.

140. Suppose it required to subtract 91·07324 from 137·321. These fractions when reduced to a common denominator are 91·07324 and 137·32100 (138). Their difference is therefore

13732100 - 9107324 4624776 ------------------, which is ------- 100000 100000

or 46·24776. This may be most simply done as follows: write the less number under the greater, so that its decimal point may fall under that of the greater, thus:

137·321 91·07324 --------- 46·24776

Subtract the lower from the upper line, and wherever there is a figure in one line and not in the other, proceed as if there were a cipher in the vacant place.

EXERCISES.

What is 12362 - 274·22107 + ·5; 9976·2073942 - ·00143976728; and 1·2 + ·03 + ·004 - ·0005?

Answer, 12088·27893, 9976·20595443272; and 1·2335.

141. The multiplication of a decimal by 10, 100, 1000, &c., is performed by merely moving the decimal point to the right. Suppose, for example, 13·2079 is to be multiplied by 100. The decimal is ¹³²⁰⁷⁹/₁₀₀₀₀, which multiplied by 100 is (117) ¹³²⁰⁷⁹/₁₀₀, or 1320·79. Again, 1·309 × 100000 is ¹³⁰⁹/₁₀₀₀ × 100000, or (116) ¹³⁰⁹⁰⁰⁰⁰⁰/₁₀₀₀ or 130900. From these and other instances we get the following rule: To multiply a decimal fraction by a decimal number (126), move the decimal point as many places to the right as there are ciphers in the decimal number. When this cannot be done, annex ciphers to the right of the decimal (137) until it can.

142. Suppose it required to multiply 17·036 by 4·27. The first of these decimals is ¹⁷⁰³⁶/₁₀₀₀, and the second ⁴²⁷/₁₀₀. By (118) the product of these fractions has for its numerator the product of 17036 and 427, and for its denominator the product of 1000 and 100; therefore this product is ⁷²⁷⁴³⁷²/₁₀₀₀₀₀, or 72·74372. This may be done more shortly by multiplying the two numbers 17036 and 427, and cutting off by the decimal point as many places as there are decimal places both in 17·036 and 4·27, because the product of two decimal numbers will contain as many ciphers as there are ciphers in both.

143. This question now arises: What if there should not be as many figures in the product as there are decimal places in the multiplier and multiplicand together? To see what must be done in this case, multiply ·172 by ·101, or ¹⁷²/₁₀₀₀ by ¹⁰¹/₁₀₀₀. The product of these two is ¹⁷³⁷²/₁₀₀₀₀₀₀, or ·017372 (135). Therefore, when the number of places in the product is not sufficient to allow the rule of the last article to be followed, as many ciphers must be placed at the beginning as will make up the deficiency.

ADDITIONAL EXAMPLES.

·001 × ·01 is ·00001 56 × ·0001 is ·0056.

EXERCISES.

Shew that

3·002 × 3·002 = 3 × 3 + 2 × 3 × ·002 + ·002 × ·002 11·5609 × 5·3191 = 8·44 × 8·44 - 3·1209 × 3·1209 8·217 × 10·001 = 8 × 10 + 8 × ·001 + 10 × ·217 + ·001 × ·217.

Fraction. Square. Cube. 82·92 6875·7264 570135·233088 ·0173 ·00029929 ·000005177717 1·43 2·0449 2·924207 ·009 ·000081 ·000000729

15·625 × 64 = 1000 1·5625 × ·64 = 1 ·015625 × ·0064 = ·0001 ·15625 × ·64 = ·1 1562·5 × ·064 = 100 15625000 × ·064 = 1000000

144. The division of a decimal by a decimal number, such as 10, 100, 1000, &c., is performed by moving the decimal point as many places to the left as there are ciphers in the decimal number. If there are not places enough in the dividend to allow of this, annex ciphers to the beginning of it until there are. For example, divide 1734·229 by 1000: the decimal fraction is ¹⁷³⁴²²⁹/₁₀₀₀, which divided by 1000 (123) is ¹⁷³⁴²²⁹/₁₀₀₀₀₀₀, or 1·734229. If, in the same way, 1·2106 be divided by 10000, the result is ·00012106.

145. Before proceeding to shorten the rule for the division of one decimal fraction by another, it will be necessary to resume what was said in (128) upon the reduction of any fraction to a decimal fraction. It was there shewn that ⁷/₁₆ is the same fraction as ⁴³⁷⁵/₁₀₀₀₀ or ·4375. As another example, convert ³/₁₂₈ into a decimal fraction. Follow the same process as in (128), thus:

128)300000000000(234375 256 ---- 440 384 ---- 560 512 ---- 480 384 ---- 960 896 ---- 640 640 --- 0

Since 7 ciphers are used, it appears that 30000000 is the first of the series 30, 300, &c., which is divisible by 128; and therefore ³/₁₂₈ or, which is the same thing (108), ³⁰⁰⁰⁰⁰⁰⁰/₁₂₈₀₀₀₀₀₀₀ is equal to ²³⁴³⁷⁵/₁₀₀₀₀₀₀₀ or ·0234375 (135).

From these examples the rule for reducing a fraction to a decimal is: Annex ciphers to the numerator; divide by the denominator, and annex a cipher to each remainder after the figures of the numerator are all used, proceeding exactly as if the numerator had an unlimited number of ciphers annexed to it, and was to be divided by the denominator. Continue this process until there is no remainder, and observe how many ciphers have been used. Place the decimal point in the quotient so as to cut off as many figures as you have used ciphers; and if there be not figures enough for this, annex ciphers to the beginning until there are places enough.

146. From what was shewn in (129), it appears that it is not every fraction which can be reduced to a decimal fraction. It was there shewn, however, that there is no fraction to which we may not find a decimal fraction as near as we please. Thus, ¹/₁₀, ¹⁴/₁₀₀, ¹⁴²/₁₀₀₀, ¹⁴²⁸/₁₀₀₀₀, ¹⁴²⁸⁵/₁₀₀₀₀₀, &c., or ·1, ·14, ·142, ·1428, ·14285, were shewn to be fractions which approach nearer and nearer to ¹/₇. To find either of these fractions, the rule is the same as that in the last article, with this exception, that, I. instead of stopping when there is no remainder, which never happens, stop at any part of the process, and make as many decimal places in the quotient as are equal in number to the number of ciphers which have been used, annexing ciphers to the beginning when this cannot be done, as before. II. Instead of obtaining a fraction which is exactly equal to the fraction from which we set out, we get a fraction which is very near to it, and may get one still nearer, by using more of the quotient. Thus, ·1428 is very near to ¹/₇, but not so near as ·142857; nor is this last, in its turn, so near as ·142857142857, &c.

147. If there should be ciphers in the numerator of a fraction, these must not be reckoned with the number of ciphers which are necessary in order to follow the rule for changing it into a decimal fraction. Take, for example, ¹⁰⁰/₁₂₅; annex ciphers to the numerator, and divide by the denominator. It appears that 1000 is divisible by 125, and that the quotient is 8. One cipher only has been annexed to the numerator, and therefore 100 divided by 125 is ·8. Had the fraction been ¹/₁₂₅, since 1000 divided by 125 gives 8, and three ciphers would have been annexed to the numerator, the fraction would have been ·008.

148. Suppose that the given fraction has ciphers at the right of its denominator; for example, ³¹/₂₅₀₀. Then annexing a cipher to the numerator is the same thing as taking one away from the denominator; for, (108) ³¹⁰/₂₅₀₀ is the same thing as ³¹/₂₅₀, and ³¹⁰/₂₅₀ as ³¹/₂₅. The rule, therefore, is in this case: Take away the ciphers from the denominator.

EXERCISES.

Reduce the following fractions to decimal fractions:

1 36 297 1 ---, ----, ----, and ---. 800 1250 64 128

Answer, ·00125, ·0288, 4·640625, and ·0078125.

Find decimals of 6 places very near to the following fractions:

27 156 22 194 2637 1 1 3 --, ---, -----, ---, ----, ----, ---, and ---. 49 33 37000 13 9907 2908 466 277

Answer, ·551020, 4·727272, ·000594, 14·923076, ·266175, ·000343, ·002145, and ·010830.

149. From (121) it appears, that if two fractions have the same denominator, the first may be divided by the second by dividing the numerator of the first by the numerator of the second. Suppose it required to divide 17·762 by 6·25. These fractions (138), when reduced to a common denominator, are 17·762 and 6·250, or ¹⁷⁷⁶²/₁₀₀₀ and ⁶²⁵⁰/₁₀₀₀. Their quotient is therefore ¹⁷⁷⁶²/₆₂₅₀, which must now be reduced to a decimal fraction by the last rule. The process at full length is as follows: Leave out the cipher in the denominator, and annex ciphers to the numerator, or, which will do as well, to the remainders, when it becomes necessary, and divide as in (145).

625)17762(284192 1250 ----- 5262 5000 ----- 2620 2500 ----- 1200 625 ----- 5750 5625 ----- 1250 1250 ---- 0

Here four ciphers have been annexed to the numerator, and one has been taken from the denominator. Make five decimal places in the quotient, which then becomes 2·84192, and this is the quotient of 17·762 divided by 6·25.

150. The rule for division of one decimal by another is as follows: Equalise the number of decimal places in the dividend and divisor, by annexing ciphers to that which has fewest places. Then, further, annex as many ciphers to the dividend as it is required to have decimal places, throw away the decimal point, and operate as in common division. Make the required number of decimal places in the quotient.

Or remove ciphers from the divisor; or make up the number of ciphers partly by removing from the divisor and annexing to the dividend, if there be not a sufficient number in the divisor.

Thus, to divide 6·7173 by ·014 to three decimal places, I first write 6·7173 and ·0140, with four places in each. Having to provide for three decimal places, I should annex three ciphers to 6·7173; but, observing that the divisor ·0140 has one cipher, I strike that one out and annex two ciphers to 6·7173. Throwing away the decimal points, then divide 6717300 by 014 or 14 in the usual way, which gives the quotient 479807 and the remainder 2. Hence 479·807 is the answer.

The common rule is: Let the quotient contain as many decimal places as there are decimal places in the dividend more than in the divisor. But this rule becomes inoperative except when there are more decimals in the dividend than in the divisor, and a number of ciphers must be annexed to the former. The rule in the text amounts to the same thing, and provides for an assigned number of decimal places. But the student is recommended to make himself familiar with the rule of the characteristic given in the Appendix, and also to accustom himself to reason out the place of the decimal point. Thus, it should be visible, that 26·119 ÷ 7·2436 has one figure before the decimal point, and that 26·119 ÷ 724·36 has one cipher after it, preceding all significant figures.

Or the following rule may be used: Expunge the decimal point of the divisor, and move that of the dividend as many places to the right as there were places in the divisor, using ciphers if necessary. Then proceed as in common division, making one decimal place in the quotient for every decimal place of the final dividend which is used. Thus 17·314 divided by 61·2 is 173·14 divided by 612, and the decimal point must precede the first figure of the quotient. But 17·314 divided by 6617·5 is 173·14 by 66175; and since three decimal places of 173·14000 ... must be used before a quotient figure can be found, that quotient figure is the third decimal place, or the quotient is ·002.....

EXAMPLES.

3·1 ·00062 ----- = 1240, ------ = ·00096875 ·0025 ·64

EXERCISES.

15·006 × 15·006 - ·004 × ·004 Shew that ----------------------------- = 15·002, 15·01

and that

·01 × ·01 × ·01 + 2·9 × 2·9 × 2·9 --------------------------------- = 2·9 × 2·9 - 2·9 × ·01 + ·01 × ·01 2·91

1 1 365 What are -------, ---------, and ------, as far as 6 places 3·14159 2·7182818 ·18349

of decimals?--Answer, ·318310, ·367879, and 1989·209221.

Calculate 10 terms of each of the following series, as far as 5 places of decimals.

1 1 1 1 1 + --- + ----- + --------- + ------------- + &c. = 1·71824. 2 2 × 3 2 × 3 × 4 2 × 3 × 4 × 5

1 1 1 1 1 + --- + --- + --- + --- + &c. = 2·92895. 2 3 4 5

80 81 82 83 84 ---- + ---- + ---- + ---- + ---- + &c. = 9·88286. 81 82 83 84 85

151. We now enter upon methods by which unnecessary trouble is saved in the computation of decimal quantities. And first, suppose a number of miles has been measured, and found to be 17·846217 miles. If you were asked how many miles there are in this distance, and a rough answer were required which should give miles only, and not parts of miles, you would probably say 17. But this, though the number of whole miles contained in the distance, is not the nearest number of miles; for, since the distance is more than 17 miles and 8 tenths, and therefore more than 17 miles and a half, it is nearer the truth to say, it is 18 miles. This, though too great, is not so much too great as the other was too little, and the error is not so great as half a mile. Again, if the same were required within a tenth of a mile, the correct answer is 17·8; for though this is too little by ·046217, yet it is not so much too little as 17·9 is too great; and the error is less than half a tenth, or ¹/₂₀. Again, the same distance, within a hundredth of a mile, is more correctly 17·85 than 17·84, since the last is too little by ·006217, which is greater than the half of ·01; and therefore 17·84 + ·01 is nearer the truth than 17·84. Hence this general rule: When a certain number of the decimals given is sufficiently accurate for the purpose, strike off the rest from the right hand, observing, if the first figure struck off be equal to or greater than 5, to increase the last remaining figure by 1.

The following are examples of a decimal abbreviated by one place at a time.

3·14159, 3·1416, 3·142, 3·14, 3·1, 3·0

2·7182818, 2·718282, 2·71828, 2·7183, 2·718, 2·72, 2·7, 3·0

1·9919, 1·992, 1·99, 2·00, 2·0

152. In multiplication and division it is useless to retain more places of decimals in the result than were certainly correct in the multiplier, &c., which gave that result. Suppose, for example, that 9·98 and 8·96 are distances in inches which have been measured correctly to two places of decimals, that is, within half a hundredth of an inch each way. The real value of that which we call 9·98 may be any where between 9·975 and 9·985, and that of 8·96 may be any where between 8·955 and 8·965. The product, therefore, of the numbers which represent the correct distances will lie between 9·975 × 8·955 and 9·985 × 8·965, that is, taking three decimal places in the products, between 89·326 and 89·516. The product of the actual numbers given is 89·4208. It appears, then, that in this case no more than the whole number 89 can be depended upon in the product, or, at most, the first place of decimals. The reason is, that the error made in measuring 8·96, though only in the third place of decimals, is in the multiplication increased at least 9·975, or nearly 10 times; and therefore affects the second place. The following simple rule will enable us to judge how far a product is to be depended upon. Let a be the multiplier, and b the multiplicand; if these be true only to the first decimal place, the product is within (a + b)/20 of the truth; if to two decimal places, within (a + b)/200; if to three, within (a + b)/2000; and so on. Thus, in the above example, we have 9·98 and 8·96, which are true to two decimal places: their sum divided by 200 is ·0947, and their product is 89·4208, which is therefore within ·0947 of the truth. If, in fact, we increase and diminish 89·4208 by ·0947, we get 89·5155 and 89·3261, which are very nearly the limits found within which the product must lie. We see, then, that we cannot in this case depend upon the first place of decimals, as (151) an error of ·05 cannot exist if this place be correct; and here is a possible error of ·09 and upwards. It is hardly necessary to say, that if the numbers given be exact, their product is exact also, and that this article applies where the numbers given are correct only to a certain number of decimal places. The rule is: Take half the sum of the multiplier and multiplicand, remove the decimal point as many places to the left as there are correct places of decimals in either the multiplier or multiplicand; the result is the quantity within which the product can be depended upon. In division, the rule is: Proceed as in the last rule, putting the dividend and divisor in place of the multiplier and multiplicand, and divide by the square of the divisor; the quotient will be the quantity within which the division of the first dividend and divisor may be depended upon. Thus, if 17·324 be divided by 53·809, both being correct to the third place, their half sum will be 35·566, which, by the last rule, is made ·035566, and is to be divided by the square of 53·809, or, which will do as well for our purpose, the square of 50, or 2500. The result is something less than ·00002, so that the quotient of 17·324 and 53·809 can be depended on to four places of decimals.

These are not quite correct, but sufficiently so for every practical purpose.

153. It is required to multiply two decimal fractions together, so as to retain in the product only a given number of decimal places, and dispense with the trouble of finding the rest. First, it is evident that we may write the figures of any multiplier in a contrary order (for example, 4321 instead of 1234), provided that in the operation we move each line one place to the right instead of to the left, as in the following example:

2221 2221 1234 4321 ---- ---- 8884 2221 6663 4442 4442 6663 2221 8884 ------- ------- 2740714 2740714

Suppose now we wish to multiply 348·8414 by 51·30742, reserving only four decimal places in the product. If we reverse the multiplier, and proceed in the manner just pointed out, we have the following:

3488414 2470315 | ---------+ 17442070 | 3488414| 1046524|2 24418|898 1395|3656 69|76828 ----------+------ 17898·1522|23188

Cut off, by a vertical line, the first four places of decimals, and the columns which produced them. It is plain that in forming our abbreviated rule, we have to consider only, I. all that is on the left of the vertical line; II. all that is carried from the first column on the right of the line. On looking at the first column to the left of the line, we see 4, 4, 8, 5, 9, of which the first 4 comes from 4 × 1′, the second 4 from 1 × 3′, the 8 from 8 × 7′, the 5 from 8 × 4′, and the 9 from 4 × 2′. If, then, we arrange the multiplicand and the reversed multiplier thus,

3488414 2470315

each figure of the multiplier is placed under the first figure of the multiplicand which is used with it in forming the first four places of decimals. And here observe, that the units’ figure in the multiplier 51·30742, viz. 1, comes under 4, the fourth decimal place in the multiplicand. If there had been no carrying from the right of the vertical line, the rule would have been: Reverse the multiplier, and place it under the multiplicand, so that the figure which was the units’ figure in the multiplier may stand under the last place of decimals in the multiplicand which is to be preserved; place ciphers over those figures of the multiplier which have none of the multiplicand above them, if there be any: proceed to multiply in the usual way, but begin each figure of the multiplier with the figure of the multiplicand which comes above it, taking no account of those on the right: place the first figures of all the lines under one another. To correct this rule, so as to allow for what is carried from the right of the vertical line, observe that this consists of two parts, 1st, what is carried directly in the formation of the different lines, and 2dly, what is carried from the addition of the first column on the right. The first of these may be taken into account by beginning each figure of the multiplier with the one which comes on its right in the multiplicand, and carrying the tens to the next figure as usual, but without writing down the units. But both may be allowed for at once, with sufficient correctness, on the principle of (151), by carrying 1 from 5 up to 15, 2 from 15 up to 25, &c.; that is, by carrying the nearest ten. Thus, for 37, 4 would be carried, 37 being nearer to 40 than to 30. This will not always give the last place quite correctly, but the error may be avoided by setting out so as to keep one more place of decimals in the product than is absolutely required to be correct. The rule, then, is as follows:

The 1′ here means that the 1 is in the multiplier.

154. To multiply two decimals together, retaining only n decimal places.

I. Reverse the multiplier, strike out the decimal points, and place the multiplier under the multiplicand, so that what was its units’ figure shall fall under the nᵗʰ decimal place of the multiplicand, placing ciphers, if necessary, so that every place of the multiplier shall have a figure or cipher above it.

II. Proceed to multiply as usual, beginning each figure of the multiplier with the one which is in the place to its right in the multiplicand: do not set down this first figure, but carry its nearest ten to the next, and proceed.

III. Place the first figures of all the lines under one another; add as usual; and mark off n places from the right for decimals.

It is required to multiply 136·4072 by 1·30609, retaining 7 decimal places.

1364072000 906031 ---------- 1364072000 409221600 8184432 122766 ----------- 178·1600798

In the following examples the first two lines are the multiplicand and multiplier; and the number of decimals to be retained will be seen from the results.

·4471618 33·166248 3·4641016 3·7719214 1·4142136 1732·508 ========= ========== ============ 37719214 033166248 346410160 8161744 63124141 8052371 -------- ---------- ------------ 15087686 3316625 346410160 1508768 1326650 242487112 264034 33166 10392305 3772 13266 692820 2263 663 173205 38 33 2771 30 10 ------------ --------- 2 6001·58373 1·6866591 -------- 46·90415

Exercises may be got from article (143).

155. With regard to division, take any two numbers, for example, 16·80437921 and 3·142, and divide the first by the second, as far as any required number of decimal places, for example, five. This gives the following:

3·142)16·80437921(5·34830 15·710 ------- 1·0943 | 9426 | -----| 15177| (A) 12568| ---- -----|- 2609 2609|9 2514 2513|6 ---- ----|-- 95 96|32 94 94|26 -- --|--- 1 2|061

Now cut off by a vertical line, as in (153), all the figures which come on the right of the first figure 2, in the last remainder 2061. As in multiplication, we may obtain all that is on the left of the vertical line by an abbreviated method, as represented at (A). After what has been said on multiplication, it is useless to go further into the detail; the following rule will be sufficient: To divide one decimal by another, retaining only n places: Proceed one step in the ordinary division, and determine, by (150), in what place is the quotient so obtained; proceed in the ordinary way, until the number of figures remaining to be found in the quotient is less than the number of figures in the divisor: if this should be already the case, proceed no further in the ordinary way. Instead of annexing a figure or cipher to the remainder, cut off a figure from the divisor, and proceed one step with this curtailed divisor as usual, remembering, however, in multiplying this divisor, to carry the nearest ten, as in (154), from the figure which was struck off; repeat this, striking off another figure of the divisor, and so on, until no figures are left. Since we know from the beginning in what place the first figure of the quotient is, and also how many decimals are required, we can tell from the beginning how many figures there will be in the whole quotient. If the divisor contain more figures than the quotient, it will be unnecessary to use them: and they may be rejected, the rest being corrected as in (151): if there be ciphers at the beginning of the divisor, if it be, for example,

·3178 ·003178, since this is -----, 100

divide by ·3178 in the usual way, and afterwards multiply the quotient by 100, or remove the decimal point two places to the right. If, therefore, six decimals be required, eight places must be taken in dividing by ·3178, for an obvious reason. In finding the last figure of the quotient, the nearest should be taken, as in the second of the subjoined examples.

Places required, 2 8 Divisor, ·41432 3·1415927 Dividend, 673·1489 2·71828180 41432 2·51327416 -------- ---------- 258828 20500764 248592 18849556 ------- -------- 10237 1651208 8286 1570796 ----- ------- 1951 80412 1657 62832 ----- ----- 294 17580 290 15708 --- ----- 4 1872 4 1571 - ---- 0 301 283 --- 18 19 -- Quotient, 1624·71 ·86525596

This is written 7 instead of 6, because the figure which is abandoned in the dividend is 9 (151).

Examples may be obtained from (143) and (150).