Ex. 10. The 11th term of the series for (a3 — x 3)3 is = 115. The binomial theorem is also employed to determine approximate values of the roots of numbers. If we wished to form a new term, it would manifestly be obtained by multi the rest, the terms after the first being alternately positive and negative. This being premised, let it be required to extract the cube root of 31. The greatest cube contained in 31 is 27; in the above formula let us make r = 3, z=27, a = 4, and we shall then have 2560 In like manner, we shall find the next term by multiplying by 43046721 Let us however confine our attention to the first five terms of the series, and reduce them to decimals, we shall have, for the sum of the additive terms, a result which we shall proceed to show is within 0.00001 of the truth. 116. When the expression for a number is expanded in a series of terms, the Bumerical value of which go on decreasing continually, we easily perceive that the greater the number of terms which we take, the more nearly shall we approach to the real value of the proposed expression. But if, in addition to this, we suppose that the terms of the series are alternately positive and negative, we Can, upon stopping at any particular term, determine precisely the degree of approximation at which we have arrived. Let there be a series ab+c―d+e-f+g—h+k—l+m— - - - composed of an indefinite number of terms, in which we suppose that the quantities a, b, c, d, go on diminishing in succession, and let us designate by N the number repro sented by this series, we shall prove that, the numerical value of N lies between any two consecutive sums of any number of the terms of the above series. For let us take any two consecutive sums, a b + c d + e -f, and, a-b+c―d+e-f+9 Upon considering the first of these, we perceive that the terms which follow —fare, + (g — h) + (k − 1) + -----; but since the series is a decreasing one, the positive differences g · h, k — l, &c. are all positive numbers, hence it follows, that, in order to obtain the complete value of N, we must add to the sum a − b + c —-de-f some positive number. Hence, a−b+c―d+e-ƒ< N With regard to the second sum, the terms which follow + g are, — (h — k), -; but the partial differences, h — k, l — m, &c. are positive, · (l — m), · hence, · (h — k), — (l — m), --- are all negative, and therefore, in order to obtain the complete value of N, we must subtract some positive number from - b + c―d+e-f+g. Hence, the sum a a-b+c-d+e—f+g>N and it has been shown that ab+c-d+e-f therefore N lies between these two sums. <N From this it follows, that, since g is the numerical value of the difference of these two sums, the error committed when we assume a certain number of terms a b + c d + e f for the value of N, is numerically less than the term which immediately follows that at which we stopped. - In the preceding example, all the terms after the first being alternately positive and negative, we may conclude that the numerical value of the first five 117. From what has been said above it will be seen, that, in order to obtain an Approximate value of the n' root of any number N by the method of series, we may make use of the following RULE. Resolve the given number N into two parts of the form p" + q, where pTM is the highest n' power contained in N, and in the developement of (x +a)= make s=p", a = = 9. The number of terms to be taken in the resulting series will depend upon the degree of accuracy required, and can be determined by the prin ciple just explained. Convert all the terms of which account is taken into decimals, and then effect the reduction between the additive and subtractive terms. This method cannot be employed with advantage except when is a small n P fraction; for unless this be the case the terms of the series will not diminish with sufficient rapidity, and it will be necessary to take account of a great number of terms in order to arrive at a near approximation. It may happen that p" isq, we must then modify the above process, for then P a or - is greater than unity, and therefore all the powers of crease in numerical value as the degree of the power increases. a x will in Suppose, for example, that the cube root of 56 is sought, 27 being the greatest cube contained in 56, we shall have and the terms of the series will go on increasing instead of diminishing, (we do not speak of the coefficients, which are fractions differing but little from unity). 8 64, or, g But we may resolve 56 into 64-8, or, 43 — 8; but is a small fraction. On the other hand, if we substitute —a for a in the expression for If we put x64, a 8, we shall obtain a series of terms which will decrease with great rapidity. Here all the terms, with the exception of the first, are negative, and we cannot apply to this series the criterion established in Art. (116.) for fixing the degree of approximation. But we shall approach very nearly to the required degree of approximation if we take into account such a number of terms that the first which we neglect shall be less, by one tenth, for example, than the decimal place to which we wish to limit the approximation. The student may take the following examples as exercises : Ex. 1. /39 = √32+7 = 2.0807.... true to 0.0001. √64+1 = 4.02073... 0.00001. 0.00001. 3. √260 = 4. √108 = √256+4 = 4.01553... RATIOS AND PROPORTION, 118. Numbers may be compared in two ways. When it is required to determine by how much one number is greater or less than another, the answer to this question consists in stating the difference between these two numbers. This difference is called the Arithmetical Ratio of the two numbers. Thus, the arithmetical ratio of 9 to 7 is 9—7 or 2, and if a, b designate two numbers, their arithmetical ratio is represented by a - b. When it is required to determine how many times one number contains, or, is contained in, another, the answer to this question consists in stating the quotient which arises from dividing one of these numbers by the other. This quotient is called the Geometrical Ratio of the two numbers. The term Ratio, when used without any qualification, is always understood to signify a geometrical ratio, and we shall, at present, confine our attention to ratios of this descrip tion. 119. By the ratio of two numbers, then, we mean the quotient which arises from dividing one of these numbers by the other, Thus the ratio of 12 to 4 is repre 12 5 2 sented by or 3, the ratio of 5 to 2 is or 2.5, the ratio of 1 to 3 is or 3 .333... We here perceive that the value of a ratio cannot always be expre sed exactly, but that, by taking a sufficient number of terms of the decimal, we can approach as nearly as we please to the true value. It may happen that one or both terms of the ratio can only be expressed in decimal fractions which do not terminate; thus, in the ratio of 1 to 2, and in the ratio of 3 to 7, the quantities √2, √3, ₺7 can only be expressed in decimals which do not ter minate, and therefore the values of the above ratios cannot be exactly expressed, although we can approach to them as nearly as we please. 120. If a, b, designate two numbers, the ratio of a to b is the quotient arising from dividing a by b, and will be represented by writing them a: 6, or, a 121. A ratio being thus expressed, the first term, or a, is called the antecedent of the ratio, the last term, or b, is called the consequent of the ratio. 122. It appears, therefore, that, in arithmetic and algebra, the theory of ratios becomes identified with the theory of fractions, and a ratio may be defined as a fraction whose numerator is the antecedent, and whose denominator is the consequent of the ratio. 123. When the antecedent of a ratio is greater than the consequent, the ratio is called a ratio of greater inequality; when the antecedent is less than the consequent it is called a ratio of less inequality; and when the antecedent and 12 consequent are equal it is called a ratio of equality. Thus, is a ratio of 4 12 3 greater inequality, 144 is a ratio of less inequality, or 1 is a ratio of equality. It is manifest that a ratio of equality may always be represented by unity. 124. When the antecedents of two or more ratios are multiplied together to form a new antecedent, and their consequents multiplied together to form a new consequent, the several ratios are said to be compounded, and the resulting ratio is called the sum of the compounding ratios. Thus, the ratio is compounded с a with the ratio by multiplying the antecedents a, c, for a new antecedent, d and the consequents b, d, for a new consequent, and the resulting ratio is called the sum of the ratios a c bd In like manner, the ratios of all the antecedents together for a new antecedent, and all the consequents for |