any number proposed; that is to say, that to every digit is given its local as well as its original or natural value: thus, in the number 76034, the second digit from the right is 3, but we consider it as representing 30, on account of its local situation, being the second figure from the right, and so on of the rest: so that the value of each digit is estimated according to its local situation, and its original value, the former indicating a certain power of 10, and the latter representing the number of those powers that are intended to be expressed. 462. It is evident, therefore, that any number may in the same manner be represented by another value of the radix (r), and hence arise the different scales of notation, which receive their particular denominations, as binary, ternary, &c., according to the value of the radix r. And since it has been already shown that the coefficients a, b, c, &c. are always less than the radix of any system into which they enter, it follows that, for every scale, we must have as many characters, including the cypher, as are equal to the number which expresses the radix of the system. In the duodenary system, the first nine numbers may be expressed as in the common system; but, as the period ought to terminate only at twelve, it will be necessary to represent ten and eleven by some new characters, as ʊ and ¤, and thus the scale will be, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, π, . 463. Now, in order to transform any number from the denary to another scale of notation, let us resume our equation, Nar br-1+ cp 2+, &c..... pr2 + gr+w, and we shall readily perceive that in this proposition the numbers N and r are given to find the coefficients a, b, c, &c. and the exponent 2. Now, if we divide the equation by r, we shall have, This last quotient being again divided by r, gives, ar +brn−3+cpnt+ +p, with remainder q, and so on. Here it is evident, then, that the successive remainders will be the coefficients w, p, q, &c.; or the digits, which express any number in the scale of which r is the radix. If it were proposed, for example, to convert the number 17486 into a corresponding number in the senary scale, we should then have, 17486a.6"+b 6n−1+c.6n−2+, &c. .. 212542 is the number required. EXAMPLE 2. To convert 215855 into a number in the duodenary scale. 464. To transform a number from any other scale of notation to the denary, or common scale, is evidently the converse of the preceding proposition, and is readily effected by a reverse operation. Let in Let arn+br.n−1+, &c. +pr2+qr+w, represent a number any known scale of notation, whose radix is r. Then, since a, b, c, &c. are known, we have only to collect the successive values of the different terms, and their sum will be the number transformed. EXAMPLE 1. To transform 1054 from the senary to the denary scale. Hence r=6, .. 1054=1.63 + 0. 62 + 5. 6+4 =216+0+30+4 =250. EXAMPLE 2. Express 2379 by a number whose local value is 2. |