IRRATIONAL NUMBERS 20. If A and B are incommensurable, the ratio : B has no rational number-correspondent. Such a ratio has been shown, however, to have two series of proximate numerical ratios, and each proximate has its own number-correspondent. These number-correspondents collectively characterize the ratio. By general agreement it is usual to say that the irrational ratio AB has then an irrational number-correspondent, characterized or defined by the two categories of rational numbers, the decimal proximates, just as the ratio itself is characterized or defined by the order of certain multiples. These categories of rational numbers are called the two decimal categories belonging to the irrational number. The number-correspondent of any ratio A: B is denoted by the symbol A B It is now necessary to give definitions of the words equal, greater, less, sum, product, etc., when applied to the irrational numbers just defined. The definitions, and certain inferences from them, are given in the following articles. 21. Definition. An irrational number is said to be greater than, equal to, or less than another number (whether rational or irrational) according as the ratio to which the first number corresponds is greater than, equal to, or less than the ratio to which the second number corresponds. 22. Theorem 2 may now be restated without restriction: According as one ratio is greater than, equal to, or less than another ratio, so is the number-correspondent of the first greater than, equal to, or less than the number-correspondent of the second. I.e., according as 4: B >= < C: D, so is B >= 012 23. Cor. I. When two irrational numbers are equal, their decimal categories are identical, respectively. For the abbreviated scales of their corresponding ratios are similar (def. and 3), hence the decimal proximates are alike (6). 24. Cor. 2. If two irrational numbers have identical decimal categories, then the irrational numbers are equal. For then the decimal proximates are alike, hence the abbreviated scales are alike (6), and hence the corresponding ratios are equal. 25. Cor. 3. If one irrational number is greater than another, then some inferior decimal proximate of the first is greater than any inferior decimal proximate of the second (9). Ex. 1. If an irrational (or a non-decimal number) is greater than any other given number, then some inferior decimal proximate of the first number is greater than the second (12). Ex. 2. If an irrational number (or a non-decimal number) is less than any other given number, then some superior decimal proximate of the first number is less than the second (13). 26. Definition. The sum of two irrational numbers is defined as the number-correspondent of the ratio which is the sum of the two ratios corresponding to the two irrational numbers. A similar definition applies to the difference of two irrational numbers, and also to the sum (or difference) of a rational and an irrational number. 27. Theorem 3 may now be restated without restriction: The number-correspondent of the sum of any two ratios is equal to the sum of their number-correspondents. 28. Cor. I. The sum of two irrational numbers is greater than the sum of any two numbers that are inferior proximates to them, respectively, and less than the sum of any two superior proximates. (Use 22, 27; and V. 120.) Addition commutative. 29. Cor. 2. The addition of numbers is a commutative operation, i.e. the sum of any two or more numbers is the same, in whatever order they may be taken (V. 118). 30. Definition. The product of two irrational numbers (or of a rational number and an irrational number) is defined as the number-correspondent of that ratio which is compounded of the ratios corresponding to the given numbers. 31. Theorem 4 may now be stated without restriction: The ratio compounded of any two or more ratios has a number-correspondent equal to the product of their number-correspondents. The process of finding the product of two or more numbers is called multiplication. Multiplication commutative. 32. Cor. I. Multiplication is a commutative operation, i.e. the product of any two or more numbers is the same, in whatever order they may be taken. (Use definition, and V. 27.) Multiplication distributive. 33. Cor. 2. Multiplication is distributive as to addition, i.e. the product of any number by the sum of any other numbers is equal to the sum of the products of the first number by the other numbers separately (V. 121). MEASURE-NUMBER 34. Definitions. The ratio which any magnitude bears to a standard magnitude of the same kind is called the measure-ratio of the first magnitude. The number-correspondent of the measure-ratio of any magnitude is called the measure-number of that magnitude. The measure-number of a magnitude is rational or irrational according as the magnitude is or is not commensurable with the standard magnitude (14). If M is any magnitude, and s the standard magnitude of the same kind, then the measure-ratio of M, and the measurenumber of M, are respectively The measure-number of a straight line is called its length with reference to the standard line. The measure-number of a polygon is called its area with reference to the standard polygon. The square described on the standard line is usually taken as the standard polygon. The universal standard of line-magnitude adopted by scientific men is the meter. It is the largest dimension of a certain standard bar of platinum when taken at the temperature of melting ice. This standard bar is carefully preserved in the Paris observatory. The first three decimal multiples of the meter are denoted by prefixes formed from the Greek words for 10, 100, 1000. The first three decimal submultiples of the meter are denoted by prefixes formed from the Latin words for 10, 100, 1000. A straightedge on which are marked divisions equal to a meter and to its decimal submultiples is called a measuring-line. With such a measuring-line the successive decimal proximates to the measurenumber of any other accessible line can be found as follows: Apply the meter in succession as often as it will go until the remainder is less than a meter. Suppose the meter goes 3 times. Then 3 is the first inferior proximate, and 4 the first superior proximate, to the measure-number of the given line. Next, to the remainder apply the decimeter as often as it will go until there is a remainder less than a decimeter. Suppose the decimeter goes 5 times. Then the proximates of the second order 5 3+1, 3+10. are Again, to the last remainder apply the centimeter until the remainder is less than a centimeter. Suppose it goes 8 times. Then the proximates of the third order are 5 3+10+180, 3 + 10 + 180· Next, to the last remainder apply the millimeter, and suppose it goes twice with a remainder less than a millimeter. Then the proximates of the fourth order are 5 FRAGMENT OF METER RULE (Graduated to millimeters, numbered in centimeters; 10cm. 1dm.) 3+10+180 +1000, 3 + 10 + 180 + 1000• 10 6 8 3 4 6 7 2 1 Again, to the last remainder apply the tenth of the millimeter; suppose it goes four times with a remainder less |