PERIODICAL DECIMALS.

The reduction of vulgar fractions to decimals presents two cases, which it may be proper to examine. The first case is where the operation is terminated. For example, if it be required to reduce to decimals; the operation is as follows.

3 8

30 0,375

60

40

Last remainder 0

Our object here being to obtain a quotient expressed in tenths, hundredths, thousandths, &c., the partial dividends must be converted into the same denominations. As the divisor 8 is not contained in 3, the dividend, we reduce this last to tenths, and have 30 tenths to be divided by 8, which gives 3 tenths for quotient, and 6 tenths, or 60 hundredths, for remainder; dividing 60 hundredths by 8, we have 7 hundredths for quotient, and 4 hundredths, or 40 thousandths, for remainder; then dividing 40 thousandths by 8, we have 5 thousandths for quotient, and cipher for the last remainder. The value of in decimals, then, is 0,375 or 375 thousandths. This result may be verified by reducing the fraction 375 to its most simple expression, for will then be reproduced.

1000

The second case is where the operation is not terminated. For example, if the fraction be reduced to decimals, the operation and result are as follows.

As in the preceding case, the alternate remainders 3 and 8, express, successively, tenths, hundredths, thousandths, ten thousandths, hundred thousandths, &c.These remainders diminish in value very rapidly; hence, by adding a sufficient number of terms to the quotient 0,272727, &c., we can obtain the value of the fraction approximated to any required degree of exactness. This is a general remark. For by precisely the same operation we find, that the value of, expressed in decimals, is 0,142857 142857, &c., and that of 14, 0,524 324, &c.

Decimal fractions of this kind, in which a certain number of figures are repeated periodically in the same order, without terminating, are called, from this property, periodical fractions.

As a property of this kind of numbers, it may be observed, that the number of different remainders, obtained by division, can never exceed the number of units, contained in the divisor. For, by the nature of the operation, the remainders, being necessarily less than the divisor, will be found in the series of natural numbers between cipher and that which precedes the divisor. Thus, for example, the divisor being 7, we can only obtain these 7 different remainders, 0, 1, 2, 3, 4, 5, 6. Hence, after a number of divisions equal to, or less than, the number of units in the divisor, there will result a remainder, which is cipher, or a remainder already obtained. In the former case, the fraction will be terminated; but in the latter, the figures of the quotient returning in the same order, without terminating, the fraction will be periodi

cal.

To know whether a vulgar fraction can be expressed exactly in decimals, or not,-it may be remarked, that every fraction, the denominator of which is not divisible either by 2 or by 5, cannot be expressed exactly in decimals, and, consequently, gives rise to a periodical fraction. As this is an important property, we shall give two general demonstrations of it.

1st. In performing the division of the numerator by the denominator, in order that the operation may terminate, one of the remainders, (none of which are divisible by the divisor,) must be made so; it is necessary, then, that the multiplication of these remainders by 10,

should introduce some factor common with those in the divisor; which can take place only when the divisor contains one of the factors 2 and 5, that compose the number 10.

2d. In performing the division of the numerator by the denominator, that the operation may terminate, some one of the partial dividends must be divisible by the divisor, and, consequently, must contain all its factors in the same number. But, the successive remainders being smaller than the divisor, can never contain the same factors in the same number; it is necessary, then, that the multiplication of these successive remainders should introduce the factors, that are wanting; that is, the factors of the dividend, which are not found in the remainder. But, 10 being divisible only by 2 and 5, cannot introduce these factors. Then, if no one of them is contained in the divisor, on multiplying the remainders by 10 to form the partial dividends, we do not introduce any factors, which are common with those in the divisor; consequently the division will not be exact, and the quotient will be periodical.

When a periodical fraction is given, if we neglect the whole series after the first period, we shall have a second decimal fraction composed of the first period, and, consequently, equal to a common fraction having this period, considered as a whole number, for its numerator, and for a denominator, unity, followed by as many ciphers as there are figures in the period. Hence, the numerator of this last fraction remaining always equal to the period, the denominator is diminished by unity. And as this denominator is equal to unity followed by as many ciphers as there are figures in the period, it will then become a number composed of as many nines as there are figures in the period. The new fraction which results will then have the period for numerator, and for denominator, as many nines as there are figures in the period. This last fraction, having the same numerator as the preceding, and a denominator less by unity, on performing division, will give a quotient, that is the same as the preceding, that is, the first period; and the remainder, instead of being a cipher, will be equal to the quotient, or that same period. The remainder being the

same as the first dividend, the operation will be recommenced in the same manner, since it will be reduced to a continual multiplication by 10. The figures already obtained in the quotient (the same that compose the period) will then be continually reproduced in the same order, and, consequently will form a periodical fraction equal to the given one, for it will be composed of the same period. But this periodical fraction is derived from a common fraction, having the period for numerator, and for denominator, a number composed of as many nines as there are figures in the period. Hence, we deduce this general rule. Every periodical fraction is equal to a common fraction, having the period for numerator, and for denominator, a number composed of as many nines as there are figures in the period. If we apply this rule to the following periodical fractions,

0,142857 142857, &c. and 0,324 324, &c.

for their respective values, we shall have the fractions 142857 and 324

9999999

999*

These fractions, reduced to their most simple expressions, are and 2.

In the preceding operations, we have supposed the period to commence with the first figure; when it does, not commence with the first figure, the following method is to be followed. Let 0,48 324 324 be a periodical fraction, of which the period 324 does not commence until after the second decimal. We multiply this fraction by 100, by removing the separatrix two places towards the right, which gives 48,324 324; or 48 plus 0,324 324, &c.; but, the periodical fraction 0,324 324, &c., of which the period commences with the first figure, has for its value the fraction 334. Then the periodical fraction 48,324 324 has for its value 48 whole numbers and 324 or 48276, and as this last fraction is 100 times great

999

9999

999

er than the given one, the given fraction will be equal to

48276

99900

If we generalize the reasoning, that has conducted to this result, the following rule may be deduced.

When the period does not commence until after several decimals, the separatrix must be removed towards the right, unto the first figure of the period, which is the same as to multiply the given periodical fraction by unity, followed by as many ciphers as there are figures before the period. We then have a new decimal fraction, composed of a whole number, plus a periodical fraction, whose period commences with the first figure. This last fraction, joined to the whole number, gives a fractional expression, which is equal to the given periodical fraction, multiplied by unity, followed by as many ciphers as there are figures before the period. Then dividing this expression by unity, followed by as many ciphers as there are figures before the period, the result will be the value of the given periodical fraction.

LOGARITHMS.

216. Logarithms are a series of numbers in arithmetical progression, corresponding, term for term, to a similar series in geometrical progression. For example, in the two following series:

.

2:48: 16: 32: 64: 128: 256: &c.

Each term in the lower, or arithmetical series, is called the logarithm of the term corresponding to it, in the upper, or geometrical series.

217. Any number may have a great variety of different logarithms; because, to any geometrical progression, there may be formed a great number of corresponding, different, arithmetical progressions. But as our design is to consider logarithms only in relation to their use in numerical calculations, we shall not consider the different arithmetical and geometrical progressions, that might be compared together; but pass to the consideration of such as are used in the formation of Tables of Logarithms.

218. For the geometrical progression, the decuple or tenfold progression is chosen; and for the arithmetical