Convert these ratios so as to have a common consequent T (V. 17, 114), and let them become R: T and S: T, then hence R: TS: T, [IV. 29 Divide T successively into 10, 100, 1000, ... parts, until a part (say the one thousandth) is found which is less than the difference between R and S. Take a sufficient number (say m) of these parts, so that m of the parts shall be less than R, and not less than s. Then R contains more than m thousandths of T, while S contains not more than m thousandths of T. Hence Hence the thousandth multiples of the antecedents occupy different positions in the two scales. Therefore the abbreviated scales are unlike, which is contrary to the hypothesis. Hence the supposition made is false; that is to say, the complete scales are everywhere alike, and A: BP: Q. then some decimal multiple of A occupies a more advanced position among the multiples of B than the like decimal multiple of P occupies among the multiples of Q. (This is proved in the course of the proof of theorem 1.) 5. NOTE. It follows from 3 and 4 that the abbreviated scale will serve the same purpose as the complete scale, and is sufficient to characterize the corresponding ratio. MCM. ELEM. GEOM.-22 ASSOCIATED NUMERICAL RATIOS 6. The abbreviated scale may be used to write down two sets of numerical ratios, such that the ratios of one set are each less than the given ratio, and those of the other set each greater than the given ratio. E.g., from the abbreviated scale in Art. 2 Thus the ratio A: B is greater than each of the numerical ratios 2:1, 21:10, 215: 100, 2159: 1000, ..., and less than each of the ratios 3:1, 22:10, 216: 100, 2160: 1000, .... Each ratio of the first set is called an inferior decimal proximate of the given ratio, and each ratio of the second set a superior decimal proximate. The successive proximates are said to be of the first order, the second order, and so on. E.g., the ratio 216: 100 is the third superior proximate of the ratio : B above. A general definition will now be given. 7. Definition. If a certain ratio lies between two numerical ratios whose consequents are each equal to the nth power of 10, and whose antecedents differ by unity, then the less of the two ratios is called the inferior (and the greater the superior) decimal proximate, of the (n + 1)st order, to the ratio that lies between them. From this definition and Arts. 3, 4 the following corollaries are immediate inferences. 8. Cor. 1. If two ratios are equal, then their corresponding decimal proximates are equal. 9. Cor. 2. If one ratio is greater than another, then some inferior decimal proximate of the first is greater than any inferior decimal proximate of the second. 10. While the use of decimal proximates is especially applicable to irrational ratios, it is to be observed that rational ratios also have their inferior and superior decimal proximates. E.g., the ratio 1:3 has the inferior proximates 3:10, 33: 100, 333: 1000, ....., and the superior proximates 4:10, 34: 100, 334: 1000, .... The series of proximates to a certain ratio A: B will terminate if it happens that some decimal multiple of 4 is exactly equivalent to some multiple of B. E.g., if the magnitudes A and B mentioned above are such which is both a rational ratio and a decimal ratio. This ratio would be the last of the series of decimal proximates to the ratio A: B. 11. Definition. The ratio of any two magnitudes of the same kind is called a decimal ratio if it can be exactly expressed as a numerical ratio whose consequent is a power of 10. If it cannot be so expressed it is called a non-decimal ratio. A non-decimal ratio may be either rational or irrational. 12. Ex. 1. If a given non-decimal ratio is greater than any other given ratio, then some inferior decimal proximate of the first ratio is greater than the second ratio. (The line of proof is as in Arts. 2, 3.) 13. Ex. 2. If a given non-decimal ratio is less than any other given ratio, then some superior decimal proximate of the first ratio is greater than the second. 14. Definition. NUMBER-CORRESPONDENT If the antecedent of a numerical ratio is divided by its consequent, the quotient is called the number-correspondent of the given ratio, or of any ratio equal to it. E.g., the ratio 10: 5 has the number-correspondent 2; the ratio 5:10 has the number-correspondent or 1; the ratio 9:5 has the number-correspondent . If two commensurable magnitudes A and B have the common measure P, and if P is contained m times in 4, and n times in B, then A: BmPnP = m : n. [IV. 38 Hence the number-correspondent of the ratio : B is Any number that can be expressed as the quotient of two whole numbers is called a rational number. Hence the number-correspondent of the ratio of any two commensurable magnitudes is a rational number. For this reason such a ratio is called a rational ratio (2). Comparison of two ratios. 15. THEOREM 2. According as one rational ratio is greater than, equal to, or less than another rational ratio, so is the number-correspondent of the first greater than, equal to, or less than the numbercorrespondent of the second. Let the two ratios be respectively equal to the numerical m:n and p: q. ratios therefore, by reducing the fractions to lowest terms, If the sign > is replaced by either < or =, the proof is similar. Addition of ratios. 16. THEOREM 3. The number-correspondent of the sum of two rational ratios is equal to the sum of their number-correspondents. For the numerical ratios m:n and p: q are respectively equal to the ratios mq : nq, np: nq; hence their sum is equal to the ratio mq + np: nq, [V. 115 whose number-correspondent is mq+np, which equals the sum of the numbers and P m ng 17. Cor. The number-correspondent of the difference of two ratios equals the difference of their number-correspondents. Compounding ratios. 18. THEOREM 4. The ratio compounded of two rational ratios has a number-correspondent equal to the product of their number-correspondents. For the ratio compounded of the numerical ratios m: n and pq equals mp: nq (V. 26); and the number-correspondent of this ratio is mp, which equals the product of the nq' m numbers and n тр 19. It follows from 15, 16, 18 that any two rational ratios can be compared, added, compounded, etc., by means of their number-correspondents. Hence the number-correspondent of a rational ratio is sufficient to characterize it. |