of inquiry may consult Legendre "Sur la Theorie des Nombres," the "Disquisitiones Arithmetica" of Gauss, or Barlow's "Elementary Investigation of the Theory of Numbers." Also, for the highly interesting properties of Circulating Decimals, and their connexion with prime numbers, consult the curious works of the late Mr. H. Goodwyn, entitled "A First Centenary," and "A Table of the Circles arising from the Division of a Unit by all the Integers from 1 to 1024." A useful Numerical Problem, to reduce a given fraction or a given ratio, to the least terms; and as near as may be of the same value. RULE 1.-Let A, B, be the two numbers. Divide the latter B by the former A, and you will have 1 for A; and some number and a fraction annexed, for B, call this C. Place these in the first step. Then subtract the fractional parts from the denominator, and what remains put after C+ 1, with a negative sign. Then throw away the denominator, and place 1 and that last number in the second step. This is the foundation of all the rest. If the fractional parts in both be nearly equal, add these two steps together; if not, multiply the less by such a number as will make the fractional parts, in both, nearly equal, and then add. And this multiplier is found by dividing the greater fraction by the less, so far as to get an integer quotient. When you add the steps together, you must subtract the fractional parts from one another, because they have contrary signs. The process is to be continued on, the same way, adding the last step, or its multiple, to a foregoing step, viz. to that which has the least fraction. Note. The ratios thus found will be alternately greater and less than the true one, but continually approaching nearer and nearer. And that is the nearest in small numbers, which precedes far larger numbers: and the excess or defect of any one is visible in the operation. 9 29. If a cube number be divisible by 7, it is also divisible by the cube of 7. 30. The difference between any integral cube and its root is always divisible by 6. 31. Neither the sum nor the difference of two cubes can be a cube. 32. A cube number may end with any of the natural numbers 1, 2, 3, 4, 5, 6, 7, 8, 9, or 0. 33. If any series of numbers, beginning from 1, be in continued geometrical proportion, the 3d, 5th, 7th, &c. will be. squares; the 4th, 7th, 10th, &c. cubes; and the 7th, of course, both a square and a cube. 34. All the powers of any number that end with either 5 or 6, will end with 5 or 6, respectively. 35. Any power, n, of the natural numbers, 1, 2, 3, 4, 5, 6, &c. has as many orders of differences as there are units in the common exponent of all the numbers; and the last of those differences is a constant quantity, and equal to the continual product 1 × 2 × 3 × 4 × x n, continued till the last factor, or the number of factors be n, the exponent of the powers. Thus, ...... The 1st powers 1, 2, 3, 4, 5, &c. have but one order of differences 1 1 1 1 &c. and that difference is 1. The 2d powers 1, 4, 9, 16, 25, &c. have two orders of differences 3 5 7 9 222 of which the last is constantly 2 = 1 x 2. The 3d powers 1, 8, 27, 64, 125, &c. have three orders of differences 7 19 37 61 12 18 24 6 6 of which the last is 6 = 1 X 2 X 3. In like manner, the 4th, or last, differences of the 4th powers, are each = 24 = 1 x 2 x 3 x 4; and the 5th, or last differences of the 5th powers, are each 125 = 1 x 2 x 3 x 4 x 5. 36. If unity be divided into any two unequal parts, the sum of one of those parts added to the square of the other, is equal to the sum of the other part added to the square of that. Thus, of the two parts and,+()2 = 3 + (})2 = 2; so, again, of the parts and 3, + (3)2= } + (3)2 = 1 3. 2 5 19 For the demonstrations of these and a variety of other properties of numbers, those who wish to pursue this curious line of inquiry may consult Legendre "Sur la Theorie des Nombres," the "Disquisitiones Arithmetica" of Gauss, or Barlow's "Elementary Investigation of the Theory of Numbers." Also, for the highly interesting properties of Circulating Decimals, and their connexion with prime numbers, consult the curious works of the late Mr. H. Goodwyn, entitled "A First Centenary," and "A Table of the Circles arising from the Division of a Unit by all the Integers from 1 to 1024.” A useful Numerical Problem, to reduce a given fraction or a given ratio, to the least terms; and as near as may be of the same value. RULE 1.-Let A, B, be the two numbers. Divide the latter B by the former A, and you will have 1 for A; and some number and a fraction annexed, for B, call this C. Place these in the first step. Then subtract the fractional parts from the denominator, and what remains put after C+ 1, with a negative sign. Then throw away the denominator, and place 1 and that last number in the second step. This is the foundation of all the rest. If the fractional parts in both be nearly equal, add these two steps together; if not, multiply the less by such a number as will make the fractional parts, in both, nearly equal, and then add. And this multiplier is found by dividing the greater fraction by the less, so far as to get an integer quotient. When you add the steps together, you must subtract the fractional parts from one another, because they have contrary signs. The process is to be continued on, the same way, adding the last step, or its multiple, to a foregoing step, viz. to that which has the least fraction. Note. The ratios thus found will be alternately greater and less than the true one, but continually approaching nearer and nearer. And that is the nearest in small numbers, which precedes far larger numbers: and the excess or defect of any one is visible in the operation. 9 Example 1. To find the ratio of 10000 to 7854, in small numbers. The ratio of 10000 to 7854 is the same as 1 to 0 + 7854 or 1 to 12146; here 1 and 1 is the first ratio. But 2146 being less than 7854, divide the latter by the former, and you get 3 in the quotient, then multiply 1 and 12146 by 3, produces 3 and 36438 as in the 3d step. This third step added to the first step produces 4 and 3 for the integers, and subtracting the fractional parts, leaves 1416. So the 4th step is 4 and 31416; and the integers 4 and 3 is the 2d ratio. In this manner it is continued to the end; and the several ratios approximating nearer and nearer are 1, 3, 4, 7, 11, 172, 133, 353, 1137 and 5009 ៖៖៖. Here is the nearest in small numbers, the defect being only roo 893 44 9 14 219 233 452 Example 2. To find the ratio of 268.8 to 282 in the least numbers. 20 41 61 224 41 So the several ratios are 1, 33, 84, 233, And the defect or excess is plain by inspection, e. g. 4 differs from the truth only 3 parts; and 29, but 48 such parts. 36 RULE 2. Divide the greater number by the less, and the divisor by the remainder, and the last divisor by the last remainder, and so on until 0 remains. Then 1 divided by the first quotient, gives the first ratio. And the terms of the first ratio multiplied by the second quotient, and 1 added to the denominator, give the second ratio. And in general the terms of any ratio, multiplied by the next quotient, and the terms of the foregoing ratio added, give the next succeeding ratio. |