Obs. In the ordinary system of Notation, the value of a figure increases ten-fold with every increase of its distance from the units' place. This is the convention in common use. But there is no reason why this should necessarily be the convention; nor why it should not be agreed that the value of a figure shall increase five-fold, six-fold, or a hundred-fold with every increase of its distance from the units' place. In these cases respectively 10 would stand for the numbers five, six, or one hundred. Def. 1. The number which 10 stands for is called the radix of the scale of notation. Def. 2. Scales are termed the binary, ternary, quaternary, quinary, senary, septenary, octenary, nonary, denary, undenary, duodenary, according as 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 is the radix of the scale. Obs. As many different figures are required in any scale, as are equal to the radix of the scale. Therefore in the undenary and duodenary scales, let t stand for ten, e for eleven. I. TO CHANGE AN INTEGER FROM ONE SCALE TO ANOTHER. Rule. Divide the given number by the radix of the new scale; divide again the quotient, and repeat the division as often as possible. The successive remainders are the successive digits, the first being the unit's digit. EXAMPLES. Change 814 from the denary to the senary scale; and the result to the 11. TO CHANGE A FRACTION FROM ONE SCALE TO ANOTHER. Rule. Multiply by the radix of the scale, and reduce the result to a mixed number: multiply the fractional part again by the radix, and reduce the result in like manner proceed as long as any fractional part remains. The several integers are the digits of the number in the new scale, the first being the digit immediately following the separating point. EXAMPLES. 1. Change from the denary to the ternary scale. 2. Change from the quinary to the ternary scale. III. TO ADD, SUBTRACT, MULTIPLY, OR DIVIDE IN ANY SCALE. Rule. Perform the operations precisely as in the denary scale, carrying always the radix of the scale or its multiples, instead of 10 and its multiples. EXAMPLES. 1. Add together in the nonary scale 4567, 803, 111: and subtract in the senary scale 35432 from 43215. 2. Multiply 476te8 by 56 in the duodenary; and divide the product by 56. PART II. COMMERCIAL ARITHMETIC. I. REDUCTION OF INTEGERS. Def. 1. Concrete quantities are measured, or estimated, by considering how often they contain certain fixed quantities of the same kind, which are called units of magnitude. Def. 2. The unit being represented numerically by 1, other quantities of the same kind are represented by the number, which shews how often they contain the unit, which must be specified in every case. Def. 3. Since all quantities do not contain the unit an exact number of times, and are not therefore expressible by whole numbers, several units of the same kind are used. The process of converting the numbers, expressing a given quantity in terms of one or more units, into others expressing it in other units, is called Reduction. Def. 4. When a quantity is made up of several others, or expressed by two or more numbers with different units, it is called a compound quantity. I. TO CHANGE NUMBERS FROM A HIGHER DENOMINATION TO A LOWER. Rule. Multiply the number of the highest denomination by the number of the next lower denomination, which make up one of the higher; and if the quantity be compound, add to the product the number of the next lower denomination. In the same manner change this number to the next lower denomination, and so on, till the denomination required is arrived at. EXAMPLES. 1. Reduce £87: 16: 10 to farthings. £ s. d. 87:16: 102 20 1756 s. 12 21082 d. 4 Ans. 84331 farthings. 2. Reduce 411 guis. 18s. 2d. to two-pences and farthings. guis. s. d. 411: 18:2 11. TO CHANGE NUMBERS FROM A LOWER DENOMINATION TO A HIGHER. Rule. Divide by the number of the lower denomination which make up one of the next higher; the quotient is the number of the higher, and the remainder, if any, is of the lower denomination. In the same way change to the next higher, and so on, till the highest required is arrived at. |
