By increasing the factor introduced in both terms, we may make the resulting root true to any degree of SQUARE ROOT OF DECIMALS. 112. The denominator of a decimal fraction will be a perfect square when the number of decimal places is even, and the number of decimal places in the root of the decimal will be half the number of decimal places in the given decimal. Hence, from the preceding principle, we have the following rule for finding the square root of any decimal fraction, to any degree of exactness. RULE. Annex O's to the decimal till the number of decimal places is twice the number of decimal places · required in the root; extract the square root of the result as though it were a whole number, and point off the required number of decimal places in the root. If we have a vulgar fraction, it may be converted into an equivalent decimal fraction, and then the above rule is applicable. The rule is also applicable to whole numbers. EXAMPLES. Find the square roots of the following decimals, approx imately : 113. It is to be observed, that if a whole number is not a perfect square its exact square root cannot be found even decimally. That is, the square root of a whole number can never be a fraction. To prove this principle, let n represent any whole number, and a suppose that its square root is an irreducible fraction, h that is, Squaring both members (Axiom 5), a Since is irreducible, there are no common factors in b α a and b; and because the square of is formed by b taking each factor of a twice for a new numerator, and each factor of b twice for a denominator, it follows that a2 and 72 have no common factors; or, in other a2 words, is an irreducible fraction. We have, therefore, b2 a2 b29 a whole number n, equal to an irreducible fraction, which is impossible; therefore, the supposition that the square root of a whole number can be an irreducible fraction, is absurd. This kind of demonstration is called "Reductio ad absurdum." CUBE ROOT OF NUMBERS. 114. The cube root of a number is one of its three equal factors. Thus, 27 3 × 3 × 3; hence, 3 is the cube root of 27; also, 4 is the cube root of 64, be cause 4 x 4 × 4 = 64. The following table, verified by actual multiplication is employed in finding the cube root of any numbe less than 1000: TABLE. 1 1 8 27 64 125 216 343 512 729 1000 To employ the table in finding the cube root of a number less than 1000. Look for the number in the first line, if found there, its cube root is immediately, below it; if the number is not in the first line, it wil be between two numbers in that line, and its root will fall between the corresponding numbers in the second line: the lesser of the two is the entire part of the cube root required, that is, it is the cube root to within less than 1. If the number is greater than 1000, its cube root will be greater than 10, that is, it will be made up of tens and units. Denoting the number by N, the tens in its cube root by a, and the units by b, we shall have, (a + b)3 = a3 + 3a2b + зal2 + b3. N = That is, a number is equal to cube of the tens of its cube root, plus three times the product of the square of the tens by the units, plus three times the product of the tens by the square of the units, plus the cube of the units. We first find the tens of the root. Since the cube of the tens can contain no significant figure of a less denomination than thousands, we point off three figures from the right hand, and find the cube root of the remaining number to within less than 1: this will be the tens required. We next subtract the cube of the tens from the period on the left, and bring down the period on the right for a remainder, This remainder is made up of 3 times the square of the tens by the units, plus other parts. If, therefore, we divide it by 3 times the square of the tens, the quotient will be the units or some greater number. Since 3 times the square of the tens by the units can give no significant figure less than hundreds, in making the division, the last two figures in the remainder may be pointed off and not used. To test the accuracy of the units found, annex it to the tens and form the cube of the result; if this cube is equal to, or less than the given number, the root found is correct; if it is greater, the last figure must be diminished until the cube of the root found is equal to or less than the given bumber. If the number to the left of the first period of three figures on the right contains more than three figures, the tens of the root will be made up of tens of tens, or hundreds, and units of tens, or tens; and for the same reason as before, a second period of three figures must be pointed off, and so on until the first period on the left contains three figures or less; it may contain but 2 figures, or it may contain but 1. The root is then found by a repetition of the process above given. Hence, the RULE. I. Separate the number into periods of three figures each, beginning at the right hand; the left hand period may contain less than three figures. II. Find the greatest perfect cube in the first period on the left, and write its cube root on the right, after the manner of a quotient in division. Subtract the cube of the root found from the first period on the left, and bring down the next period for a remainder. III. Divide the remainder, exclusive of the two right hand figures, by three times the square of the root found, and annex the quotient to the root. Cube this number (diminishing it by 1, 2, &c., if necessary), and subtract the result from the first two periods, and bring down the next period for a second remainder. IV. Divide this remainder by three times the square of the root already found, proceeding as before, and con |