that we can make the period to consist of as many figures as we please; for example, if, in the expression,0678, we leave out one or more figures, it is plain that, as these do not repeat, the value will be changed. Also, if we make the period to consist of more than four figures, unless the number is a multiple of four, the figures will not follow each other in the same order, and hence the order of their units will be changed. Therefore, we can only increase the number of figures in a period consisting of different figures, by making the number of figures in the new period a multiple of the number in the given one.

263. When the points which designate the period, which we call the extremes, are, in several decimal numbers, equidistant, that is, when the period in each begins at the same place, and has the same number of figures, these numbers are similar, seeing that their fractional equivalents must (254) have a common denominator. When the extremes are not equidistant, the numbers are dissimilar, because their fractional equivalents have different denominators.

If we have several dissimilar repeating decimals, which we wish to render similar, it is evident that the number of figures in some of the periods must be increased; but this (262) requires that the number of figures in the new period shall be a multiple of the number in the given one: if, therefore, we wish to find a common period for all the given numbers, which we must do before they can be similar, it is plain that the num ber of figures in this new period must be a multiple of the number in each: consequently, we take the least common multiple of the number in each, as the most convenient.

Again, if some of the decimal figures are mixed, we must be careful to extend the definite figures, (254;) namely, those between the comma and the first figure of the repeating period, so that the common period may commence at the place next, on the right, to that of the lowest definite figure in any of the given numbers. Thus, if we have

,6;,06;,639;,09527; and,47625,

which we would render similar, we count the number of figures in each period, and find respectively 1, 1, 2, 2, 3, the least common multiple of which is 6: wherefore, the common period must consist of 6 figures, and begin at the fourth place, the greatest number of definite figures being 3; that is to say,

,095. The numbers, when rendered similar, will, therefore, stand thus:

666666666

,066666666

,639393939

,095272727

,476256256

Addition of Repeating Decimals.

264. As we cannot add units of different orders, to add re- ́. peating decimals, we first render them similar. The period in each is then the numerator of a fraction, and the common denominator (261) as many nines as there are places in the common period. Wherefore, the numbers being placed under each other, we find the sum of the periods, as in whole numbers; and, if the sum does not contain more figures than the common period,--that is, if the sum of the left-hand column of the numbers within the period does not exceed 9- -we write it underneath; and, having marked the extremes of the total sum, we continue the addition as usual. But, if the sum of the left-hand column exceeds 9, we not only carry the lefthand figure of this sum to the next column on the left, but we also add this same figure to the right-hand figure of the sum total of the periods. The reason of this is, that this left-hand figure has as many places on the right of it, as there are nines in the denominator of the period; and, if we divide its value by the common denominator, which shows the order of its units, we shall have the same figure (258) for both quotient and remainder. But the quotient is evidently a number of units of the next higher decimal order, and the remainder, being a number of units of the order expressed by the common denominator, has its true value in being added to the unit figure of the period.

For example, if we would add the numbers given in the preceding article, having prepared and placed them as directed. we proceed thus:

,666666666

,066666666

,639393939

,095272727

,476256256

1,944256254

the addition being performed, from beginning to end, as in whole numbers, with the sole exception, that having found the sum 22 of the left-hand column of the common period, we write its left-hand figure 2 under the unit figure 4 of the period, to remind us that it must be added to that figure. The sum of the given numbers is, therefore,

1,944256256 or 1,944256

The scholar may prove the work by finding (254) the fractional equivalents, and adding them, as usual in vulgar fractions, thus:

63

§ + b + § 3 3 + b22+13378=21385138=1,944256

990

99900

10989000

If definite or finite decimals be found amongst the numbers to be added, it is plain that we must extend the definite figures of the mixed and repeating decimals, so that the common period may begin at the next place on the right of the lowest order of definite figures in any of the given numbers.

Examples.

1. 5,237+65,3+16,789+750,037+,9286—838,3263999 2. ,0075+32,7+113,003652+2,987+,2359 =

149,0128935269

3. 25,8+9,541+2,007+,6258+97,97-136,043290472

Subtraction of Repeating Decimals.

265. As we cannot subtract units of different orders, we render the numbers similar, as in addition, and place them so that the commas may be under each other. Then, if the number which constitutes the period in the lower number, be not greater than that of the period above it, as we have merely to diminish the upper period by a number of units of the same kind, the subtraction is performed throughout as in integers, except the marking of the period in the remainder.

But, if the number in the lower period be the greater, we consider that, if to the upper period we add a unit of the next higher decimal order, this unit is equal to the common denominator, and in adding this, which is a number of nines equal to the number of figures in the period, to the upper period, —that is to say, the alt minus 1,-the unit figure of this period will be one unit less, the other figures of the period will

not be altered, and there will be a unit projected one place towards the left, which is ten with regard to the left-hand figure of this upper period. Now, in subtracting in the ordinary way, as the left-hand figure of the lower period, with what may be to carry, is supposed greater than the one above it, we shall have to add ten to this upper figure, and carry one to the next lower figure on the left: now, this carrying of one to the next lower figure on the left, is just what we should have to do in consequence of having added a unit of this order to the upper number. Wherefore, we suppose the unit figure of the upper period to be one unit less; the operation is, in every other respect, the same as in whole numbers, except the marking of the extremes in the remainder.

For example, to subtract 3,2357 from 8,0235, we render the numbers similar, and place them thus:

8,02352352

3,23575757

4,78776594

Then, as the number in the lower period is greater than that in the upper, instead of saying 7 from 12, we say, 7 from 11, four. In every other respect, we subtract as in whole numbers, except that in the remainder we mark the extremes of the period.

To prove the correctness of the above result, as well as of those in the following examples, the scholar may find the fractional equivalents, subtract as in vulgar fractions, and then reduce the fractional remainder to decimals. In dividing, he may proceed as in Art. 243, and, omitting the ciphers on the right of the divisor, continue the operation till the quotient figures begin to repeat.

2. 23,005-7,35492 = 15,65012558

3. 42,0036,97684 = 41,02678

4. 378,20583-67,37598310,82984169

Multiplication of Repeating Decimals.

When the repeating decimal is found in the multiplicand only, and the multiplier is a single figure.

266. To multiply a number, is to add it to itself a certain number of times; we may, therefore, suppose the period of the repeating decimal written under itself a certain number of times, and the addition performed as in Art. 264; that is, if the sum or product of the left-hand order of the period does not exceed 9, the whole work is performed as in whole numbers, except the marking of the extremes. But if the sum er product of the highest order of the period exceeds 9, we not only carry its tens to the next place on the left, but add them as units to the right-hand figure of the period in the product. If the period consists of a single figure, as this is a number of ninths, we divide its product by 9, setting down the remainder, which we mark as a repeater: then, carrying the quotient, we proceed as usual. For example, to multiply

30

9

we first say, 6 times 5 is 30. Now this is and as the division gives the same figure for quotient and remainder, we write 3 and carry 3, after which we proceed as with ordinary decimals.

We may prove the work thus:

6,35×10=635472: 472 106,72

572 6 572

ДФ 1 15

=38,13 as before.

Again, if we have 5,758 to multiply by 7:

5,758

7

40,312

we say 7 times 8 is 56; then to find the remainder, we cast out the nines, saying 5 and 6 is 11; 1 and 1 is 2; which we write and mark. Then adding the 9 cast out as a unit to the