original decimal, that will remain unaltered, and the required answer will be a mixed number, which may be reduced to an improper fraction, if necessary.

EXAMPLE.-3.1415.

Taking the decimal part separately, 1415 = 1415-14

1401

=

9900'

Hence 31415 = = 300 = 31101 expressed as an

9900

improper fraction.

9900

Or it may be expressed as an improper fraction, as follows, at once :—

The truth of this latter method may be established exactly in the same way as the two cases we have already explained.

29. The learner is recommended at first, in reducing circulating decimals to vulgar fractions, to perform the operation in the way we have indicated in the examples we have given-i.e., by multiplying by the requisite powers of ten, subtracting, &c. He will thus better appreciate the truth of the rule, which he will afterwards employ. The equivalent fractions found by the rule will evidently often not be in their lowest terms.

EXERCISE XXIII.

REDUCTION OF CIRCULATING DECIMALS TO VULGAR FRACTIONS. Reduce to their equivalent Vulgar Fractions

30. Approximation. Decimals correct to a given number of places, &c.

We have already remarked that if we take only a limited number of the figures of a decimal, we approach nearer and nearer to the true result as we continue to take in more figures.

We give an example, taken from De Morgan's "Arithmetic," which shows this clearly :

= 142857 a circulating decimal.

Now taking successively one, two, three, &c., figures of the decimal, we have

We thus see that the difference between the decimal and the true value of the fraction continually diminishes. In the case of a terminating decimal this difference becomes zero when we have taken all the figures in. In the case of a circulating decimal, it never actually becomes zero, but we can make it as small as we please by taking a sufficient number of decimal places.

31. When a result is required correct only to a certain number of decimal places, it is better, as we have already explained (Art. 14), to find one figure more of the result than is actually required, so as to ascertain whether this figure is greater or less than 5. If it is greater, we increase the figure in the last place which is required in the result by 1.

The following is an example of a decimal continually

approximated to in this way, by taking successive figures, and increasing, where necessary, the last figure. by unity :

Let 489169 be the decimal. The successive approximations would be

5, 49, 489, 4892, 48917, 489169.

Here 5 is nearer to the true value than 4 would be

32. Operations in which Circulating Decimals occur are better conducted by reducing the circulating decimals to their equivalent vulgar fractions, if absolute accuracy is required. If an approximate result is desired true to a certain number of decimal places, then, in additions and subtractions, it will be sufficient to take in two or three figures of the period beyond the number of places required, and then add or subtract. For instance, in adding 4567 to 3124689 correctly to 9 decimal places, we should write the decimals as follows:

*45675675675
*31246894689

*769225703

In all cases, however, where circulating decimals are involved as multipliers or divisors, it will be best to reduce them to their equivalent vulgar fractions, before performing the multiplications or divisions, and then afterwards to reduce the resulting fractions to decimals.

EXERCISE XXIV.

(1.) Write down the decimals containing respectively one, two, three, four, five, and six decimal places which are the nearest approximation to the decimals 67819473, 203781947.

Find the value correctly to 7 decimal places

(2.) Of 2·0127 + 89.3897 + .003704.

(3.) Of 15 379 +213459+18+ 70 2178 +5.34567.

1. WE have already stated that when any number is multiplied by itself any number of times, the products are called the second, third, fourth powers, &c., of the number respectively.

The second and third powers of any number are generally called the square and cube of that number (compare Chap. X., Arts. 7, 10). Thus, 81 is the square of 9, 27 is the cube of 3.

Any power of a number is expressed by writing the number of the power in small figures above the number, a little to the right.

Thus, the square of 9 would be written 92; the cube of 3, 33; the 5th power of 7, 75; and so on.

Conversely, the number which, taken twice as a factor, will produce a given number, is called the square root of that number; that which, taken three times as a factor, will produce a given number, is called the

cube root of it; that which, taken four times as a factor, will produce a given number, is called the fourth root of it; and so on.

Any root of a number is represented by writing the sign over the number, and placing the number corresponding to the number of the root on the left of the symbol, thus: /g indicates the cube root of 8, 81 the fourth root of 81.

The square root of a number is generally expressed by merely writing the symbol over the number, without the figure 2.

Thus,

means the square root of 3; √84 the square root of 84.

2. Every number has manifestly a 2nd, 3rd, 4th, &c., power. But every number has not conversely an exact square, cube, third root, &c. For example, there is no whole number which, when multiplied into itself, will produce 7, and since any fraction in its lowest terms multiplied into itself must produce a fraction, 7 cannot have a fraction for its square root. Hence 7 has no exact square root. But although we cannot find a whole number or fraction which, when multiplied into itself, will produce 7 exactly, we can always, as will be shown hereafter, find a decimal which will be a very near approximation to a square root of 7, and we can carry the approximation as nearly to√ as we please. And the same will be true of every number which has no exact square root, third root, &c.

It is desirable that the student should know by heart the squares and cubes of the successive numbers from I up to 12.

He is recommended to draw up a table of them for himself.

In finding the square of any number which is not very large-under a hundred, say- the following method will be found useful.

3. Short Method for finding the Square of a number.