CHAPTER VII. SCALES OF NOTATION. 76. The ordinary numbers with which we are acquainted in Arithmetic are expressed by means of multiples of powers of 10; for instance This method of representing numbers is called the common or denary scale of notation, and ten is said to be the radix of the scale. The symbols employed in this system of notation are the nine digits and zero. In like manner any number other than ten may be taken as the radix of a scale of notation; thus if 7 is the radix, a number expressed by 2453 represents 2 × 73 + 4 × 7o + 5 × 7+3; and in this scale no digit higher than 6 can occur. Again in a scale whose radix is denoted by r the above number 2453 stands for 2r3 + 4r2 + 5r + 3. More generally, if in the scale whose radix is r we denote the digits, beginning with that in the units' place, by a, a,, α,,...a; then the number so formed will be represented by where the coefficients a, a,-1,...a, are integers, all less than "', of which any one or more after the first may be zero. Hence in this scale the digits are in number, their values ranging from 0 to r- 1. 77. The names Binary, Ternary, Quaternary, Quinary, Senary, Septenary, Octenary, Nonary, Denary, Undenary, and Duodenary are used to denote the scales corresponding to the values two, three,...twelve of the radix. In the undenary, duodenary,... scales we shall require symbols to represent the digits which are greater than nine. It is unusual to consider any scale higher than that with radix twelve; when necessary we shall employ the symbols t, e, T as digits to denote 'ten', 'eleven' and 'twelve'. It is especially worthy of notice that in every scale 10 is the symbol not for 'ten ', but for the radix itself. 78. The ordinary operations of Arithmetic may be performed in any scale; but, bearing in mind that the successive powers of the radix are no longer powers of ten, in determining the carrying figures we must not divide by ten, but by the radix of the scale in question. Example 1. In the scale of eight subtract 371532 from 530225, and multiply the difference by 27. Explanation. After the first figure of the subtraction, since we cannot take 3 from 2 we add 8; thus we have to take 3 from ten, which leaves 7; then 6 from ten, which leaves 4; then 2 from eight which leaves 6; and so on. Again, in multiplying by 7, we have 3 x 7 twenty one=2x8+5; we therefore put down 5 and carry 2. Next 7x7+2=fifty one=6x8+3; put down 3 and carry 6; and so on, until the multiplication is completed. Example 2. Divide 15et20 by 9 in the scale of twelve. 9)15et20 Explanation. Since 15=1×T+5= seventeen=1×9+8, we put down 1 and carry 8. Also 8 x T +e=one hundred and seven-ex9+8; we therefore put down e and carry 8; and so on. Example 3. Find the square root of 442641 in the scale of seven. 442641(546 34 134 1026 1416 12441 EXAMPLES. VII. a. 1. Add together 23241, 4032, 300421 in the scale of five. 4. From 3te756 take 2e46t2 in the duodenary scale. 5. Divide the difference between 1131315 and 235143 by 4 in the scale of six. 6. Multiply 6431 by 35 in the scale of seven. 7. Find the product of the nonary numbers 4685, 3483. 8. Divide 102432 by 36 in the scale of seven. 9. In the ternary scale subtract 121012 from 11022201, and divide the result by 1201. 10. Find the square root of 300114 in the quinary scale. 11. Find the square of tttt in the scale of eleven. 12. Find the G. C. M. of 2541 and 3102 in the scale of seven. 13. Divide 14332216 by 6541 in the septenary scale. 14. Subtract 20404020 from 103050301 and find the square root of the result in the octenary scale. 15. Find the square root of eet001 in the scale of twelve. 16. The following numbers are in the scale of six, find by the ordinary rules, without transforming to the denary scale: (1) the G. C. M. of 31141 and 3102; (2) the L. C. M. of 23, 24, 30, 32, 40, 41, 43, 50. 79. To express a given integral number in any proposed scale. Let N be the given number, and r the radix of the proposed scale. Let a, a, a,,...a be the required digits by which N is to be expressed, beginning with that in the units' place; then N=a_r" +α-"~1+ ... + α ̧?12 + α ̧r +α ̧· We have now to find the values of a, a, a,,... An Divide N by r, then the remainder is and the quotient is If this quotient is divided by r, the remainder is a,; if the next quotient and so on, until there is no further quotient. Thus all the required digits a, a,, a,...a, are determined by successive divisions by the radix of the proposed scale. Example 1. Express the denary number 5213 in the scale of seven. and the number required is 21125. Example 2. Transform 21125 from scale seven to scale eleven. .. the required number is 3t0t. e)21125 Explanation. In the first line of work 21=2x7+1 = fifteen = 1xe+4; therefore on dividing by e we put down 1 and carry 4. Next 4 x7+1=twenty nine=2xe+7; therefore we put down 2 and carry 7; and so on. Example 3. Reduce 7215 from scale twelve to scale ten by working in scale ten, and verify the result by working in the scale twelve. 7215 12 Thus the result is 12401 in each case. Explanation. 7215 in scale twelve means 7 × 123+2 × 122+1 × 12+5 in scale ten. The calculation is most readily effected by writing this expression in the form [{(7 × 12+2) } × 12+1] × 12+5; thus we multiply 7 by 12, and add 2 to the product; then we multiply 86 by 12 and add 1 to the product; then 1033 by 12 and add 5 to the product, 80. Hitherto we have only discussed whole numbers; but fractions may also be expressed in any scale of notation; thus Fractions thus expressed in a form analogous to that of ordinary decimal fractions are called radix-fractions, and the point is called the radix-point. The general type of such fractions in scale r is where b1, b2, b or more may ; are integers, all less than 7, of which any one be zero. 81. To express a given radix fraction in any proposed scale. Let F be the given fraction, and r the radix of the proposed scale. Let b1, b, b,... be the required digits beginning from the left; then Multiply both sides of the equation by r; then Hence b, is equal to the integral part of rF; and, if we denote the fractional part by F1, we have Multiply again by r; then, as before, b, is the integral part of r; and similarly by successive multiplications by r, each of the digits may be found, and the fraction expressed in the proposed scale. |