on the first two or three figures in the left of the corresponding divisor.

Instead of annexing 00 to the remainder, we reject two figures in the right of the divisor, namely, the right hand one of the preceding divisor, and the one obtained from dividing. This does not affect the value of the resulting root or quotient, since the divisor and dividend thus taken are each reduced to one hundredth of the value that would accrue to them from the additional figures on the right of each, (§ 57.)

The same method of contraction is applicable to the cube root, and for similar reasons.

Note. The number of figures that may be found in the root after the contraction commences, will evidently be one less than the number in the last complete divisor that was used; but the last figure thus found may be incorrect. In the preceding example, the root carried to the same extent by the Common Rule, is

1.414213'.

This uncertainty in regard to the last figure, is of little importance when the root contains several decimal figures.

The preceding method of contraction is to be applied to the following

EXERCISES.

1. Extract the square root of 3 to five decimal figures.

Ans. 1.73205'. .

2. Extract the square root of 5.13 to five decimal figures.

Ans. 2.26495'.

3. Extract the square root of 7.35 to six decimal figures.

Ans. 2.711088'.

4. Extract the square root of 21.345 to six decimal figures.

Ans. 4.620065'.

5. Extract the square root of 7342.5 to six decimal figures.

Ans. 85.688393.

6. Extract the cube root of 4 to five decimal figures.

Ans. 1.58740'.

7. Extract the cube root of 5.18 to five decimal figures.

Ans. 1.73025'.

8. Extract the cube root of 285.75 to five decimal figures.

Ans. 6.58669'.

THE HIGHER ROOTS.

$326. The fourth root of a number is that number whose fourth power is equal to the given number; the fifth root of a number is that number whose fifth power is equal to the given number; and so on.

Thus 3 is the 4th root of 81, since the fourth power of 3 is 81; and 2 is the 5th root of 32, since the fifth power of 2 is 32.

§ 327. The fourth root may be obtained, most readily, by extracting the square root of the square root of the given number.

Thus the square root of 625 is 25, and the square root of 25 is 5; then 5 is the 4th root of 625, since, from the manner in which the 5 has been obtained, its 4th power must be equal

to 625.

The square root of 6561 is 81, and the square root of 81 is 9; then 9 is the fourth root of 6561.

§ 328. Any root whose fractional exponent is resolvable into two factors, may be found by extracting such root of the given number as is denoted by one of those factors, and then such root of that root as is denoted by the other factor.

The 6th root is equal to the cube root of the square root, or the square root of the cube root,-the fraction, which is the exponent of the 6th root, being equal to .

The 5th root cannot be extracted in this way, since the exponent cannot be resolved into two factors. A similar remark is applicable to the 7th root, &c.

Note. General Rules might be given for extracting the higher roots of numbers. These rules, however, are very tedious in their application; and these higher roots are not required in any calculations which properly come within the sphere of Arithmetic. At a more advanced stage in his mathematical studies, the pupil will meet with the proper applications of these roots, and the best methods of extracting them.

REMARK.

The preceding, it is believed, comprise most, if not all, of the metheds of abbreviating the operations under the General Rules of Arithmetic, that can be considered as practically useful.

By pursuing these methods in connection, as has been done in the present work, they may more conveniently be made a subject of distinct study by the pupil; while, by the same arrangement, the most, important directions relating to them, have been given in brief, but comprehensive terms.

REPEATING DECIMALS.

§329. A repeating decimal, also called a Repetend, is one in which the same figure or figures recùr in immediate and continual succession.

In reducing to a decimal, we obtain the repetend .333 and so on, which is denoted by .3, a point over the repeating figure.

In reducing to a decimal, we obtain the repetend .181818 &c., which is denoted by .18, a point over the first and the last repeating figure.

§ 330. A mixed repetend is a decimal in which other figures precede a repetend or repeating decimal. These precedent figures are called finite figures,—the number of figures in the repetend itself being infinite.

Thus if we reduce 15 to a decimal, we shall obtain the mixed repetend .416, in which .41 are the finite figures.

In what cases Repetends occur.

(331. Decimal division will always produce a repetend when the divisor and dividend are prime to each other, and the divisor contains any other prime factors than 2 or 5.

Suppose that 13 is to be divided, decimally, by 15; these numbers having no common measure greater than a unit.

We annex decimal Os to 13, and divide the result as an integer. The Os annexed, multiply the 13 successively by 10, or 2×5; and this introduces the factors 2 and 5 into the dividend.

But the factors of 15 are 3 and 5, the first of which is not a factor of the dividend. Hence in dividing 13 with Os annexed by 15 there will always be a remainder; that is, the division will never terminate.

Every case of interminable division will result in a repeating decimal. For as the remainders, to which Os are annexed, are always less than the divisor, some one remainder must occur a second time before the number of divisions is equal to the divisor; and then the same figures will recur in the quotient that succeeded the first occurrence of that remainder; that is, the quotient will become a repetend.

Repeating Decimals Reduced to Equivalent Vulgar Fractions.

332. A Repetend is always equal to a vulgar fraction whose numerator is the repeating figure or figures, and denominator as many 9s as there are repeating figures.

For, by reducing to a decimal, we obtain the repetend .i; then .i=}; and consequently .2=3; .3=3, &c.

By reducing to a decimal, we obtain the repetend .0i; then .01; .02=3; .03=3; .04—,,, &c.

By reducing to a decimal, we obtain the repetend .001; then .001; .002; .003, and so on.

The same method of illustration may be applied to a repetend consisting of four, five, or more figures.

Thus we should find .0001; .0002-, &c.

§ 333. The value of a mixed repetend may be expressed by a complex or mixed decimal, (§ 132); and this decimal may then be reduced to a vulgar fraction.

Thus in .416 the repeating figure 6 annexed to .41, is equivaJent to or annexed. (§ 332.)

412

Then .416 .413=

100

135-100-125-5.

These principles are to be applied in the following

EXERCISES.

1. Reduce .5 to an equivalent vulgar fraction. 2. Reduce .15 to an equivalent vulgar fraction. 3. Reduce .24 to an equivalent vulgar fraction. 4. Reduce .513 to an equivalent vulgar fraction. 5. Reduce .412 to an equivalent vulgar fraction. 6. Reduce .503 to an equivalent vulgar fraction. 7. Reduce .264 to an equivalent vulgar fraction. 8. Reduce .3036 to an equivalent vulgar fraction. 9. Reduce .2412 to an equivalent vulgar fraction. 10. Reduce .3330 to an equivalent vulgar fraction.