15 is third root of it is 50; also the third root of 47 47—15.. And this method of notation has justly prevailed in the modern algebra; because such roots, being considered as fractional powers, need no other directions lor 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 ac curately, 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, each containing 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 necessary; 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. 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. 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 next less than the first period, and put the root of it on the right of the given number, in the manner of a quotient figure in division, and it will be the first figure of the root required. * In order to show the reason of the rule, it will be proper t» 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, namely, unity with cyphers, as 1000 and 100; then their product will be 1 with as many cyphers annexed, as are in both the numbers, namely, 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 be the greatest of these places possible, as 999 and 99. Now 999x99 is less than 999x100; but 999 × 100 (99900) contains 0nly as many places of figures, as are in 999 and 99; therefore 999x99, or the product of any other two numbers, consisting of 2. Subtract the assumed square from the first period, and to the remainder bring down the next period for a dividend. 3. Place the double of the root, already found, on the left of the dividend for a divisor. the same number of places, cannot have more places of figures than are in both its factors. Corollary 1. 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 hay« more places of figures than triple the places of the root, and at Least but two less. The truth of the rule may be shown algebraically thus: Let N= the number, whose square root is to be found. 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 to consist of two periods, and let the figures in the root be represented by a and b. Then a+b a2+2ab+b2=N= given number; and to find the root of Nis the same, as finding the root of o2+2a4+4s, the method of doing which is as follows; 1st divisor a)2+2ab+b2(a+b= root. a2 2d divisor 2a+b)2ab+b* 2a4+d* Again suppose to consist of 3 periods, and let the figures of the root be represented by a, 6, and t. Then 2 a+b+c=as+2ab+b2+2ac+24c+c2, and the manner «f finding a, b, and c will be, as before thus, 1st divisor a)a2+2ab+b2+2ac+24c+c2 (a+4+c= root. a2 2d divisor 2a+b)2ab+b2 2ab+b2 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 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 a divisor as before, by doubling the figures already in the root; and from these find the next figure of the root, as in the last article; and so on through all the periods to the last. Note 1. When the root is to be extracted to a great number of places, the work may be much abbreviated thus: having proceeded in the extraction by the common method till you have found one more than half the required number of figures in the root, the rest may be found by dividing the last remainder by its corresponding divisor, annexing a cypher to every dividual, as in division of decimals; or rather, without annexing cyphers, by omitting continually the first figure of the divisor on the right, after the manner of contraction in division of decimals. Note 2. By means of the square root we readily find the fourth root, or the eighth root, or the sixteenth root, &c. that is, the root of any power, whose index is some power of the number 2; namely by extracting so often the square root, as is denoted by that power of 2; that is, twice for the fourth root, thrice for the eighth root, and so on. 3d divisor 2a+2b+c)2ac+2bc+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 No consists of any number of periods whatever. TO EXTRACT THE SQUARE ROOT OF A RULE. First prepare all vulgar fractions by reducing them to their least terms, both for this and all other roots. Then 1. Take the root of the numerator and that of the denominator for the respective terms of the root required. And this is the best way, if the denominator be a complete power. But if not, then 2. Multiply the numerator and denominator together; take the root of the product: this root, being made the numerator to the denominator of the given fraction, or the denominator to the numerator of it, will form the fractional root repuired. Vab And this rule will serve, whether the root be finite or infi nite. Or 3. Reduce the vulgar fraction to a decimal, and extract its root. EXAMPLES. 1. Required the square root of 5499025. 5499025(2345 the root. 4 43 149 464/2090 4685/23425 |