NOTE.-If, in any case, the dividend will not contain the divisor, the right hand figure of the former being omitted, place a zero in the root, and also at the right of the divisor, and bring down the next period. ART. 190.—In Division, when the remainder is greater than the divisor, the last quotient figure may be increased by at least 1; but in extracting the square root, the remainder may sometimes be greater than the last divisor, while the last figure of the root can not be increased. To know when any figure may be increased, the pupil must be acquainted with the relation that exists between the squares of two consecutive numbers. Let a and a+1 be two consecutive numbers. Then (a+1)2=a2+2a+1, is the square of the greater. is the square of the less. (a)2=a2 Their difference is 2a+1. IIence, the difference of the squares of two consecutive numbers, is equal to twice the less number, increased by unity. Consequently, when the remainder is less than twice the part of the root already found, plus unity, the last figure can not be increased. Extract the square root of the following numbers. EXTRACTION OF THE SQUARE ROOT OF FRACTIONS. ART. 191.-Since, therefore, the square root of is, perfect squares, its square root will be found, by extracting the square root of both terms. Before extracting the square root of a fraction, it should be reduced to its lowest terms, unless both numerator and denominator are perfect squares. The reason for this, will be seen by the following example. Find the square root of 1. Now, neither 12 nor 27 are perfect squares; REVIEW.--189. What is the rule for extracting the square root of numbers? 190. What is the difference between the squares of two consecutive numbers? When may any figure of the quotient be increased? but, by canceling the common factor 3, the fraction becomes, of which the square root is. When both terms are perfect squares, and contain a common factor, the reduction may be made either before, or after the square root is extracted. Thus, √!==; or, §§=}, and √}=}. Find the square root of each of the following fractions. 36 ART. 192.-A number whose square root can be exactly ascertained, is termed a perfect square. Thus, 4, 9, 16, &c., are perfect squares. Comparatively, these numbers are few. A number whose square root can not be exactly ascertained, is termed an imperfect square. Thus, 2, 3, 5, 6, &c., are imperfect squares. Since the difference of two consecutive square numbers, a2 and a2+2a+1, is 2a+1; therefore, there are always 2a imperfect squares between them. Thus, between the square of 4(16), and the square of 5(25), there are 8(2a=2×4) imperfect squares. A root which can not be exactly expressed, is called a surd, or irrational root. Thus 2 is an irrational root; it is 1.414+. The sign +, is sometimes placed after an approximate root, to denote that it is less, and the sign that it is greater than the true root. It might be supposed, that when the square root of a whole number can not be expressed by a whole number, that it might be found exactly equal to some fraction. We will, therefore, show, that the square root of an imperfect square, can not be a fraction. Let c be an imperfect square, such as 2, and if possible, let its square root be equal to a fraction which is supposed to be in its lowest terms. a a2 a Then c= ; and c=- by squaring both sides. Now, by supposition, a and b have no common factor, therefore, their squares, a2 and b2, can have no common factor, since to square a2 a number, we merely repeat its factors. Consequently, must b2 be in its lowest terms, and can not be equal to a whole number. Therefore, the equation C= is not true; and hence, the suppo α sition is false upon which it is founded; that is, that c; there fore, the square root of an imperfect square can not be a fraction. APPROXIMATE SQUARE ROOTS. ART. 193.-To illustrate the method of finding the approximate square root of an imperfect square, let it be required to find the square root of 2 to within 3. Reducing 2 to a fraction whose denominator is 9 (the square of 3, the denominator of the fraction ), we have 2=18. 4 Now, the square root of 18 is greater than 4, and less than 5 therefore, the square root of 18 is greater than 3, and less than §; therefore, is the square root of 2 to within less than }. Hence the RULE, FOR EXTRACTING THE SQUARE ROOT OF A WHOLE NUMBER TO WITHIN A GIVEN FRACTION. Multiply the given number by the square of the denominator of the fraction which determines the degree of approximation; extract the square root of this product to the nearest unit, and divide the result by the denominator of the fraction. EXAMPLES. 1 1 23. 1 30. Ans. 21. Ans. 213. 8 Ans. 338. Ans. 5. Ans. 3.7. Ans. 3.87. 1. Find the square root of 5 to within . 2. Find the square root of 7 to within 73. 3. Find the square root of 15 to within 4. Find the square root of 27 to within 5. Find the square root of 14 to within 70. 6. Find the square root of 15 to within 700. Since the square of 10 is 100, the square of 100, 10000, and so on, the number of ciphers in the square of the denominator of a dec imal fraction is equal to twice the number in the denominator itself. Therefore, when the fraction which determines the degree of approximation is a decimal, it is merely necessary to add two ciphers for each decimal place required; and, after extracting the root, to point off from the right, one place of decimals for each two ciphers added. 7. Find the square root of 2 to six places of decimals. Ans. 1.414213. 8. Find the square root of 5 to five places of decimals. Ans. 2.23606. REVIEW.-191. How is the square root of a fraction found, when both terms are perfect squares? 192. When is a number a perfect square? Give examples. When is a number an imperfect square? How can you determine the number of imperfect squares between any two consecutive perfect squares? What is a root called, which can not be exactly expressed? Prove that the square root of an imperfect square can not be a fraction. 193. How do you find the approximate square root of an imperfect square to within any given fraction? What is the rule, when the fraction which determines the degree of approximation, is a decimal? ART. 194.—To find the approximate square root of a fraction. 1. Let it be required to find the square root of to within 4. 21 Now, since the square root of 21 is greater than 4, and less than 5, therefore, the square root of is greater than 4, and less than; hence is the square root of to within less than 4. Hence, if we multiply the numerator of a fraction by its denomi nator, then extract the square root of the product to the nearest unit, and divide the result by the denominator, the quotient will be the square root of the fraction to within one of its equal parts. to within 7 to within. to within 13. Ans.. Ans.. Ans. 2. Find the square root of 3. Find the square root of 4. Find the square root of 1 Since any decimal may be written in the form of a fraction having a denominator a perfect square, by adding ciphers to both terms (thus, .4100=10000, &c.), therefore, the square root may be found, as in the method of approximating to the square root of a whole number, by annexing ciphers to the given decimal, until the number of decimal places shall be equal to double the number required in the root. Then, after extracting the root, pointing off from the right, the required number of decimal places. Ans. .538516. 6. Of .29 to six places of decimals. The square root of a whole number and a decimal, may be found in the same manner. Thus, the square root of 2.5 is the same as the square root of 258, which, carried out to 6 places of decimals, is 1.581138+. 100, 7. Find the square root of 10.76 to six places of decimals. Ans. 3.280243. 8. Find the square root of 1.1025.. When the denominator of a fraction is a perfect square, its square root may be found by extracting the square root of the numerator to as many places of decimals as are required, and dividing the result by the square root of the denominator. Or, by reducing the fraction to a decimal, and then extracting its square REVIEW. 194. How do you find the approximate square root of a fraction to within one of the equal parts of the denominator? How do you extract the square root of a decimal? How do you extract the square root of a fraction, when both terms are not perfect squares? root. When the denominator of the fraction is not a perfect square, the latter method should be used. 9. Find the square root of to five places of decimals. √3=1.73205+, √4=2, √3=1.73205+=.86602+. 7 Or, 3.75, and 1.75=.86602+. Ans. 1.795054+. Ans. .661437+. Ans. 1.802775+. Ans. 2.426703+. Ans. .377964+. Ans. .935414+. Ans. 1.527525+. EXTRACTION OF THE SQUARE ROOT OF MONOMIALS. ART 195. From the principles in Art. 179, it is evident, that in order to square a monomial, we must square its coëfficient, and multiply the exponent of each letter by 2. Thus, (3ab2)2=3ab23ab2=9a2b1. Therefore, 9a2b-3ab2. Hence, the RULE, FOR EXTRACTING THE SQUARE ROOT OF A MONOMIAL. Extract the square root of the coëfficient, and divide the exponent of each letter by 2. Since +a+a+a2, and —aX—a=+a2, Therefore va2=+a, or —a. Hence, the square root of any positive quantity is either plus, or minus. This is generally expressed, by writing the double sign before the square root. Thus, √4a2=±2a, which is read, plus or minus 2a. If a monomial is negative, the extraction of the square root is impossible, since the square of any quantity, either positive or negative, is necessarily positive. Thus, √−9, √—4a2, √ —b, are algebraic symbols, which indicate impossible operations. Such expressions are termed imaginary quantities. They occur, in attempting to find the value of the unknown quantity in an equation of the second degree, where some absurdity or impossibility exists in the equation, or in the problem from which it was derived. See Art. 218. REVIEW.-195. How do we find the square of a monomial? How, then, do we find the square root of a monomial? What is the sign of the square root of any positive quantity? Why is the extraction of the square root of a negative monomial impossible? Give examples of algebraic symbols that indicate impossible operations. What are they termed? Under what circumstances do they occur? |