233. Theorem. Any positive integer can be expressed in any scale of notation, and this can be done in only one way. For divide N by r, and let Q, be the quotient and d the remainder. Then N = d2+r× Q1 Now divide Q, by r, and let Q, be the quotient and d the remainder. Then Q,d,+rQ; therefore N = d2+rd1 + r2Q By proceeding in this way we must sooner or later come to a quotient, Q, d, which is less than r, when the process is completed, and we have n N = d2+rd2+ r2d2 + r3d ̧ +................ r"d„, so that the number would in the scale of r be written d...... d.d.d.d. Each of the digits do, d1, d,...... is less than r, and any one or more of them, except the last, d,, may be zero. Since at every stage of the above process there is only one quotient and one remainder the transformation is unique. The given number N may itself be expressed either in the common or in any other scale of notation. Ex. 1. Express 2157 in the scale of 6. The quotients and remainders of the successive divisions by 6 are as under: Thus 2157 when expressed in the scale of 6 is 13553. Ex. 2. Change 13553 from the scale of 6 to the scale of 8. We have the following successive divisions by 8, remembering that since 13553 is in the scale of 6 each figure is six times what it would be if it were moved one place to the left, so that to begin with we have to divide 1 × 6+3, and so on. Ex. 3. Change 4155 from the scale of 8 to the scale of 10. Proceeding as before, we have Since 4155=4×83+1x82+5×8+5={(4×8+1)8+5}8+5, the required result may be obtained as follows: : Multiply 4 by 8 and add 1; multiply this result by 8 and add 5; then multiply again by 8 and add 5. Ex. 4. Express 3166 in the scale of 12. eleven by e.] 234. [Represent ten by t, and Ans. 19et. Radix Fractions. Radix fractions in any scale correspond to decimal fractions in the ordinary scale, so that α b с abc... stands for + + +. To shew that any given fraction may be expressed by a series of radia fractions in any proposed scale. Let F be the given fraction; and suppose that, when expressed by radix fractions in the scale of r, we have where each of a, b, c...... is a positive integer (including zero) less than r. b Hence a must be equal to the integral part, and ++...... must be equal to the fractional part of Fr. (If Fr be less than 1, a is zero.) Let F, be the fractional part of Fr; then Hence b must be equal to the integral part of F ̧r. Thus a, b, c,...... can be found in succession. Ex. 3. Change 324-26 from the scale 8 to the scale 6. The integral and fractional parts must be considered separately. Thus the required result is 552.20213. Ex. 4. In the scale of 8 express 16315 as a vulgar fraction. Ex. 5. In the scale of 7 express 231 as a vulgar fraction. Ex. 6. Change 314.23 from the scale of 5 to the scale of 7. Ans. 150-3564. 235. Theorem. The difference between any number and the sum of its digits is divisible by r — 1, where r is the radix of the scale in which the number is expressed. Let N be the number, S the sum of the digits, and let do, d, d...... be the digits. and Then N = d + rd1 + r2d2 +.......... +r”dn' 0 .. N − S = (r − 1) d ̧ + (r2 − 1) d ̧ +...... + (?” — 1) d„. 1 Now each of the terms on the right is divisible by r-1 [Art. 86]. Hence NS is divisible by r 1. Since NS is divisible by r —– 1, N and S must leave the same remainder when divided by r− 1. Ex. 1. The difference of any two numbers expressed by the same digits is divisible by r 1. For the sum of the digits is the same for both; and since N1-S and N-S are both divisible by r-1, it follows that N1-Ñ2 is divisible by r− 1. Ex. 2. 2 Shew that in the ordinary scale a number is divisible by 9 if the sum of its digits be divisible by 9, and by 3 if the sum of its digits is divisible by 3. N-S is a multiple of 9; hence, if S be a multiple of 9, so also is N; and, if S be a multiple of 3, so also is N. Ex. 3. Shew that any number is divisible by r+1 if the difference between the sum of the odd and the sum of the even digits is divisible by r+1. and Let Then N-D=d1 (r+1) + d2 (r2 − 1) + dg (r3 + 1) + . Each of the terms on the right is divisible by r+1; .. N-D is divisible by r+1. Hence if D is divisible by r+1 so also is N. 2 1 Ex. 4. If N1 and N, be any two whole numbers, and if the remainders left after dividing the sum of the digits in N1, N2 and in N1 × N2 by 9 be n1, n, and p respectively; then will n1n be equal to p, or differ from p by a multiple of 9. For N1n+a multiple of 9, and N=n,+ a multiple of 9; therefore N1x N2 = n1 × n2+ a multiple of 9. Hence nn2+ a multiple of 9 is equal to p+ a multiple of 9. If the above is applied in any case of multiplication, and it is found that n ̧n does not equal p, or differ from it by a multiple of 9, there must be some error in the process of multiplication. This gives a method of testing the accuracy of multiplication; the test is not however a complete one, for although it is certain that there must be an error if n×n, does not equal p, or differ from it by a multiple of 9, there may be errors when the condition is satisfied, provided that the errors neutralize one another so far as the sum of the digits in the product is concerned. This is called the "Rule for casting out the nines." Ex. 5. A number of three digits in the scale of 7 has the same digits in reversed order when it is expressed in the scale of 9: find the number. Let a, b, c be the digits; then we have 49a+7b+c=81c+9b+a, where a, b, c are positive integers less than 7. |