IV. The logarithm of the root of a number is equal to the logarithm of the number divided by the index of the root. 319. An exponential equation, that is, an equation in which the exponent is the unknown quantity, is easily 320. Logarithms of numbers to any base a may be converted into logarithms to any other base b by dividing the computed logarithms by the logarithm of b to the base a. CHAPTER XX. RATIO, PROPORTION, AND VARIATION. 321. The relative magnitude of two numbers is called their ratio, and is expressed by the fraction which the first is of the second. Thus, the ratio of 6 to 3 is indicated by the fraction §, which is sometimes written 6:3. 322. The first term of a ratio is called the antecedent, and the second term the consequent. When the antecedent is equal to the consequent, the ratio is called a ratio of equality; when the antecedent is greater than the consequent, the ratio is called a ratio of greater inequality; when less, a ratio of less inequality. 323. When the antecedent and consequent are interchanged, the resulting ratio is called the inverse of the given ratio. Thus, the ratio 3:6 is the inverse of the ratio 6:3. 324. The ratio of two quantities that can be expressed in integers in terms of a common unit is equal to the ratio of the two numbers by which they are expressed. Thus, the ratio of $9 to $11 is equal to the ratio of 9:11; and the ratio of a line 2 inches long to a line 3 inches long, when both are expressed in terms of a unit of an inch long, is equal to the ratio of 32 to 45. 325. Two quantities different in kind can have no ratio, for then one cannot be a fraction of the other. 326. Two quantities that can be expressed in integers in terms of a common unit are said to be commensurable. The common unit is called a common measure, and each quantity is called a multiple of this common measure. Thus, a common measure of 2 feet and 3 feet is of a foot, which is contained 15 times in 2 feet, and 22 times in 3 feet. Hence, 2 feet and 3 feet are multiples of of a foot, 2 feet being obtained by taking of a foot 15 times, and 3 by taking of a foot 22 times. 327. When two quantities are incommensurable, that is, have no common unit in terms of which both quantities can be expressed in integers, it is impossible to find a fraction. that will indicate the exact value of the ratio of the given quantities. It is possible, however, by taking the unit sufficiently small, to find a fraction that shall differ from the true value of the ratio by as little as we please. Thus, if a and b denote the diagonal and side of a square, 1.41421356....., a value greater than 1.414213, but less If, then, a millionth part of b be taken as the unit, the value of the ratio lies between 1414313 and 1414214, and therefore differs from either of these fractions by less than 1000000 100 By carrying the decimal farther, a fraction may be found that will differ from the true value of the ratio by less than a billionth, trillionth, or any other assigned value whatever. 328. Expressed generally, when a and b are incommensurable, and b is divided into any integral number (n) of equal parts, if one of these parts be contained in a more than m times, but less than m+1 times, then a The error, therefore, in taking either of these values for 1 is <1. But by increasing n indefinitely, can be made n n to decrease indefinitely, and to become less than any assigned value, however small, though it cannot be made absolutely equal to zero. 329. The ratio between two incommensurable quantities is called an incommensurable ratio. 330. As the treatment of Proportion in Algebra depends upon the assumption that it is possible to find fractions which will represent the ratios, and as it appears that no fraction can be found to represent the exact value of an incommensurable ratio, it is necessary to show that two incommensurable ratios are equal if their true values always lie between the same limits, however little these limits differ from each other. m n Let ab and c: d be two incommensurable ratios.. : Suppose the true values of the ratios a b and c d lie between Then the difference between the true values of these I and m+1 n ratios is less than however small the value of 1 may be. n n 1 But since n 328. can be made to approach zero at pleasure, can n be made less than any assumed difference between the ratios. Therefore, to assume any difference between the ratios is to assume 1 it possible to find a quantity that for the same value of shall be both greater and less than ; which is a manifest absurdity. Hence, a:bc: d. n 331. It will be well to notice that the word limit means a fixed value from which another and variable value may be made to differ by as little as we please; it being impossible, however, for the difference between the variable value and the limit to become absolutely zero. 332. A ratio will not be altered if both its terms be multiplied by the same number. α For the ratio a: b is represented by the ratio ma mb is repre b' = a: b. : 333. A ratio will be altered if different multipliers of its terms be taken; and will be increased or diminished according as the multiplier of the antecedent is greater or less than that of the consequent. 334. A ratio of greater inequality will be diminished, and a ratio of less inequality increased by adding the same num 335. A ratio of greater inequality will be increased, and a ratio of less inequality diminished, by subtracting the same number from both its terms. |