proposed, either reduce it to an improper fraction, or reduce the vulgar fraction to a decimal, and proceed by the rule. THE ROOT of any given number, or power, is such a number as, being multiplied by itself a certain number of times, will produce the power; and it is denominated the first, second, third, fourth, &c. root, respectively, as the number of multiplications made of it to produce the given power is o, 1, 2, 3, &c. that is, the name of the root is taken from the number, which exceeds the multiplications by 1, like the name of the power in involution, NOTE I. The index of the root, like that of the power in involution, is I more than the number of multiplications, necessary to produce the power or given number. NOTE 2. Roots are sometimes denoted by writing before the power, with the index of the root against it : 3 so the third root of 50 is 50, and the second root of it is 50, the index 2 being omitted, which index is always understood, when a root is named or written without one, But if the power be expressed by several numbers with the signor, &c. between them, then a line is drawn from from the top of the sign of the root, or radical sign, over all the parts of it so the third root of 47-15 is 3 √47-15. : And sometimes roots are designed like powers, with the reciprocal of the index of the root above I the given number. So the 2d root of is 3; the 2d root of 50 is 50; and the third root of it is 503; also the third root of I 47—15 is 47—153. And this method of notation has justly prevailed in the modern algebra; because such roots, being considered as fractional powers, need no other di'rections for any operations to be made with them, than those for integral powers. NOTE 3. A number is called a complete power of any kind, when its root of the same kind can be accurately extracted; but if not, the number is called an imperfect power, and its root a surd or irrational number: so 4 is a complete power of the second kind, its root being 2; but an imperfect power of the third kind, its root being a surd number. Evolution is the finding of the roots of numbers either accurately, or in decimals, to any proposed extent. The power is first to be prepared for extraction, or evolution, by dividing it from the place of units, to the left in integers, and to the right in decimal fractions, into periods containing each as many places of figures, as are denominated by the index of the root, if the power contain a complete number of such periods: if it do not, the defect will be either on the right, or left, or both; if the defect be on the right, it may be supplied by annexing cyphers, and after this, whole periods of cyphers may be annexed to continue the extraction, if necessa ry; but if there be a defect on the left, such defective period must remain unaltered, and is accounted the first period of the given number, just the same as if it were complete. Nove Now this division may be conveniently made by writing a point over the place of units, and also over the last figure of every period on both sides of it; that is, over every second figure, if it be the second root; over every third, if it be the third root, &c. Thus, to point this number 21035896′12735; for the second root, it will be 21035896 1273501 but for the third root and for the fourth 21035896*127350; 21035896*12735000. NOTE. The root will contain just as many places of figures, as there are periods or points in the given power; and they will be integers, or decimals respectively, as,the periods are so, from which they are found, or to which they correspond; that is, there will be as many integral or decimal figures in the root, as there are periods of integers or decimals in the given number. TO EXTRACT THE SQUARE Root. RULE.* 1. Having distinguished the given number into periods, find a square number by the table or trial, either equal to, or * In order to shew the reason of the rule, it will be proper to premise the following LEMMA. The product of any two numbers can have at most but as many places of figures, as are in both the factors, and at least but one less. DEMONSTRATION. Take two numbers consisting of any number of places, but let them be the least possible of those places, viz. unity with cyphers, as 1000 and 100; then their product will be with as many cyphers annexed as are in both the numbers, or the next less than, the first period, and put the root of it to the right hand of the given number, after the manner of a quotient figure in division, and it will be the first figure of the root required. 2. Subtract the assumed square from the first period, and to the remainder bring down the next period for a dividend. 3. Place. viz. 100000; but 100000 has one place less than 1000 and 100 together have and since 1000 and 100 were taken the least possible, the product of any other two numbers, of the same number of places, will be greater than 100000; consequently the product of any two numbers can have, at least, but one place less than both the factors. Again, take two numbers of any number of places, that shall Now be the greatest of those places possible, as 999 and 99. 999 × 99 is less than 999 × 100; but 999 × 100 (=99900) contains only as many places of figures, as are in 999 and 99; therefore 999 X 99 or the product of any other two numbers, consisting of the same number of places, cannot have more places of figures than are in both its factors. COROLLARY I. A square number cannot have more places of figures than double the places of the root, and, at least, but one less. COR. *2. A cube number cannot have more places of figures than triple the places of the root, and, at least, but two less. The truth of the rule may be shewn algebraically thus: Now, it appears from the lemma, that there will be always as many places of figures in the root, as there are points or periods in the given number, and therefore the figures of those places may be represented by letters. Suppose 3. Place the double of the root, already found, on the left hand of the dividend for a divisor. 4. Consider what figure must be annexed to the divisor, so that if the result be multiplied by it, the product may be equal to, or the next less than, the dividend, and it will be the second figure of the root. 5. Subtract the said product from the dividend, and to the remainder bring down the next period for a new dividend. 6. Find Suppose N to consist of two periods, and let the figures in the root be represented by a and b. 2 Then a+ba2+2ab+b2=N≈ given number; and to find the root of N is the same as finding the root of a2+2ab+b2, the method of doing which is as follows: 1st divisor a)a2+2ab+b2 (a+b= root. a2 2d divisor za+b)2ab+b2 2ab+b2 Again, suppose N to consist of 3 periods, and let the figures of the root be represented by a, b and c. Then 2 a+b+c=a2+2ab+b2+2ac+2bc+c2, and the man ner of finding a, b and c, will be as before: thus, 1st divisor a)a2+2ab+b2+2ac+2bc+c2 (a+b+c= root. a2 2d divisor 2a+b)2ab+b2 2ab+b2. 3d divisor 2a+2b+c) 2ac+2be+c2 2ac+2bc+c2 Now, the operation, in each of these cases, exactly agrees with the rule, and the same will be found to be true when N consists of any number of periods whatever. |