8

NOTE 2.-A vulgar fraction which-like, or or 15cannot be represented by a decimal fraction of EXACTLY the same value, can, nevertheless, be represented by a decimal fraction of as NEARLY equal value as may be required for any practical purpose: all that is necessary being, to multiply the terms of the vulgar fraction by a sufficiently high power of 10, divide the terms of the resulting fraction by the original denominator, and reject the fractional part of the numerator so obtained. Multiplying the terms of the fraction, for example, by 10, 100, 1,000, 10,000, 100,000, and 1,000,000, successively; dividing the terms of the resulting fraction, in each case, by the original denominator, 7; and rejecting, in each case, the fractional part of the new numerator, we obtain for approximations 10, 100, 1000, 10000, 100000, and 1000, respectively:

714 7142 71428

5_5×10_50_50÷7_7

7 7 × 10 70 70÷7 IO

5_5×100_500_500÷7_ 71 X

7 7 × 100 700 700÷7 100

714285 1000000

1000

&c.

It will be seen that each of these approximations is closer than the preceding one: the first,, being less than the given fraction () by the tenth part of; the second,, by the hundredth part of; the third, 7, by the thousandth part of ; the fourth, 76, by the ten-thousandth part of; the fifth, 71428, by the hundredth-thousandth part of; and the sixth, 714285 by only the millionth part of 5. There obviously is no limit to the number of such approximations; and the paradox is thus presented of our being able to go on continually

100000

10000001

144 TERMINATE AND INTERMINATE DECIMALS. approaching, without ever reaching, such a fraction as, which for a reason already explained--is not EXACTLY equal to any decimal fraction.

145. Terminate and Interminate Decimals.— To convert a VULGAR fraction into a simple number of EXACTLY or NEARLY (as the case may be) the same value, we divide the numerator by the denominator -reducing units to tenths, tenths to hundredths, hundredths to thousandths, &c. [See NOTE, p. 57.] The decimals thus obtained divide themselves into two classes: (1) those which terminate, and (2) those which do not.

The division of the numerator by the denominator converts ,, and, for example, into the "terminate" decimals '4, 75, and 875, respectively :

5)2'0

'4

4)3'00
*75

8)7.000
.875

If, however, we take , or, or 15, and divide the numerator by the denominator, we shall for a reason to be explained presently-obtain an "interminate" decimal; so that, in writing such a fraction as under the form of a simple number, we are obliged to be satisfied with a sufficiently close approximation.

146. When the denominator of a fraction-that is, a fraction in its simplest form-contains any prime factor which is neither 2 nor 5, the decimal resulting from the division of the numerator by the denominator will be INTERMINATE; when no such factor occurs, the decimal will be TERMINATE.

This follows from § 144, as we shall find on reflecting that (§ 141) whatever can be converted into a terminate decimal is convertible into a decimal fraction, and that (§ 143) whatever is convertible into a decimal fraction can be converted into a terminate decimal. The fraction, or, or 15, if convertible into a terminate decimal, could be converted into a decimal fraction; but we have seen that such a fraction as §, or, or 15 is not convertible into a decimal fraction. On the other hand, , or, or, being convertible into a decimal fraction, can be converted into a terminate decimal.

147. Circulating Decimals.-If carried sufficiently far, every interminate decimal obtained as a quotient would "circulate:" that is, some particular figure, or combination of figures, would be continually repeated.

:

In order to understand this, let us take the fraction, and observe what occurs when the numerator is divided by the denominator. We first reduce the numerator to (5 × 10=) 50 tenths, the division of which by 7 gives 7 tenths for quotient, and I tenth for remainder: we then reduce the remainder to (IX 10=) 10 hundredths, the division of which by 7 gives I hundredth for quotient, and 3 hundredths for remainder we next reduce this second remainder to (3 × 10=) 30 thousandths, the division of which by 7 gives 4 thousandths for quotient, and 2 thousandths for remainder and so on--the first remainder (1), when a cipher is annexed to it, becoming the second partial dividend (10); the second remainder (3), when a cipher is annexed to it, becoming the third partial dividend (30); the third remainder (2), when a cipher is annexed to it, becoming the fourth partial dividend (20); &c. Now, as the remainder is always less than the divisor, we can never, when dividing by 7, have a larger remainder than 6; and the only other remainders which can possibly arise are 5, 4, 3, 2, and I. [The remainder can never become o, because (§ 146) the fraction is not convertible into a terminate decimal.] Annexing a cipher, therefore, to each of these numbers, we find that at every stage of the work—even the very first stage, being a 66 proper fraction-the partial dividend must be 60, 50, 40, 30, 20, or 10. So that after setting down the first six figures of the decimalthe list of partial dividends being then exhausted—we are able, without making any calculation, to say with certainty that the seventh partial dividend will be some one of the six already employed, and that, as a necessary result, there will be a repetition of one or more figures in the quotient. The seventh partial dividend being found to be the same as the first, it is evident that however far we continue the work-the first six partial dividends will be repeated again and again, and in the same order; and that, consequently, the decimal will be a continual repetition of the combination 714285:

L

Let us next take the fraction, and divide the numerator by the denominator. Before proceeding with the division, we are able-after what has already been explained-to say that

Be

not more than (11-1=)10 quotient
figures will have been set down when
the decimal begins to circulate.
cause, as the remainder will always be
10, 9, 8, 7, 6, 5, 4, 3, 2, or 1, the par-
tial dividend will always be one of these
numbers with a cipher annexed: 100,
90, 80, 70, 60, 50, 40, 30, 20, or IO; so
that the eleventh partial dividend must
be one of the ten previously employed.
In the case under consideration, how-
ever, we do not exhaust our list of
partial dividends; the third being the
same as the first, and the first two
figures of the quotient, consequently,
being continually repeated.

)80(72 72 72 &c.

77

30

22

80

77

30

22

80

77

30

22

&c.

If, as a third illustration, we take the fraction, and divide the numerator by the denominator, we shall

have only one figure (6) continually re- 3)2.00000 &c.

peated: =66666 &c.

•66666 &c.

In each of the preceding examples, the decimal circulates from the beginning-the first partial dividend being, in all three cases, the first to repeat itself. Sometimes, however, the decimal circulates (not from the beginning, but) from the second, or third, or fourth, &c. figure--the corresponding partial dividend being the first to repeat itself. The fractions 1, 2, and 104. 43 afford illustrations of this. Converting these fractions into decimals, we find that the circulation begins--in the first case, from the second figure (8); in the second case, from the third figure (3); and in the third case, from the fourth figure (4) :-

1)150(6 81 81 81 &c.

132
180

176

15

12)70('58333 &c.

60

100

96