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4. How will 476897 in the decimal scale be expressed in the duodecimal scale ?
Ans. ltee95. 593.° To change any number into the decimal scale, when expressed in any other scale of notation.
Ex. 1. Change 377t0 from the duodecimal to the decimal scale.
377 + 0 12
We multiply the left-hand figure by the ra
dix, and add to the product the next figure; 43
then we multiply this sum by the radix, and 12
add to the product the next figure, and so 5 2 3
proceed till all the figures have been em12
ployed; and we thus have, as the values of the
several figures collected into one sum, 75432, 6 2 8 6
obtained in a manner similar to the reduction 1 2
of compound numbers. 7 5 4 3 2
RULE. – Multiply the left-hand figure of the given number by the given radix, and to the product add the next figure ; then multiply this sum by the radix, and add to this product the next figure; and so proceed till all the figures of the given number have been added. The result will be the given number in the decimal scale.
NOTE. — When it is required to change a number from a scale other than decimal to another scale also other than decimal, first change the number as given into the decimal scale, and then the result into the required scale. 2. Reduce 234 from the quinary to the decimal scale.
Ans. 69. 3. Change 21122 from the ternary to the decimal scale.
Ans. 206. 4. Change 100101 in the binary scale to a number in the decimal scale.
Ans. 37. 5. Reduce 13579 in the duodecimal scale to the undecimal scale.
Ans. 190t3. 6. How will 123454321 in the senary scale be expressed in the duodenary scale ?
Ans. 9873t1. 594. To perforni addition, subtraction, multiplication, division, &c. in a scale of notation whose radix is other than 10, we may
Proceed as in the common scale of notation, except that the radix of he given scale must be used in the cases wherein the number 10 would be applied in the decimal system.
Ex. 1. Required the sum and difference of 45324502 and 25405534 in the senary scale, or scale whose radix is 6.
Ans. Sum, 115134440 ; difference, 15514524. 2. Multiply 2483 by 589 in the undenary scale, or scale whose radix is 11.
Ans. 13122t5. 3. Divide 1184323 by 589 in the duodenary scale, whose radix is 12.
Ans. 2483. 4. Extract the square root of 11122441 in the senary scale.
595. DUODECIMALS are numbers expressed in a scale whose radix is 12, so that 12 units of each lower order make a unit of the next higher.
596. In finding the contents of surfaces and solids, however, it is customary to apply the term duodecimal to a mixture of the decimal and duodecimal scales. Thus, in admeasurements in which the foot is the leading unit, though the different orders of units are expressed according to the duodecimal scale, the number of units in each order is usually expressed according to the decimal scale.
597. According to this mixed scale, the foot is divided into 12 equal parts, and each of these parts into 12 other equal parts, and so on indefinitely, giving li, ita, &c. In writing these fractions without their denominations, to distinguish their orders, or denominations, accents, called indices, are written on the right of the numerators. Thus, inches are called primes, and are marked '; the next subdivision is called seconds, marked "; the next is thirds, marked "'; and so on.
Note. Numbers expressed by the mixed scale of feet, primes, seconds, &c. may be changed to the pure duodecimal scale, and the operations of addition, subtraction, multiplication, division, and so on, then be performed with them, as in Art. 594, observing to place a point between the unit and its lower duodecimal orders, and in the result changing the figures on the left of the point into the decimal scale, and marking those on the right as primes, seconds, &c., according to their places from the order of units.
But the operations of adding, subtracting, &c. are usually performed by other methods, such as are given in the articles that follow.
ADDITION AND SUBTRACTION OF DUODECIMALS.
598. Duodecimals may be added and subtracted in the same manner as compound numbers.
Ex. 1. Add together 121ft. 3' 9", 105ft. 11' 8", 80ft. 0'6", and 15ft. 10' 0" 4".
Ans. 323ft. 1' 11" 4". 2. From 462ft. 4' 9" take 307ft. 9' 1".
3. What is the value of 92ft. 0' 6" 21ft. 9' 10" +19ft. 10' 3" 6"?
Ans. 90ft. 0' 11" 6".
MULTIPLICATION OF DUODECIMALS.
The index of the unit of a product of any two duodecimal orders is equal to the sum of the indices of those factors. That is, feet multiplied by a number denoting feet produces feet ; feet by a number denoting primes produces primes ; primes by a number denoting primes produces seconds, &c.
NOTE. — In multiplication of duodecimals, or in other multiplication, the multiplier is always regarded as an abstract number, though the notation of feet, primes, &c. is usually retained, in order the better to note the different orders of units. For the same reason, in division of duodecimals, the divisor usually retains the notation of feet, primes, &c.
600. To multiply one duodecimal by another.
Ex. 1. Required the number of square feet in a platform 6 feet 8 inches long, and 4 feet 5 inches wide.
Ans. 29 sq. ft. 5' 4". We first multiply each of the terms in the multi6ft. 8' plicand by the 5' in the multiplier; thus, 8'X 5'=40" 4ft. 5'
3 and 4". Writing the 4" under the multiplier,
we reserve the 3' to add to the next product. 2 9' 4" Then 6ft. X 5' = 30'; and 30' + 3' - 33' = 2ft. 26 8' and 9', which we write in their order beneath the
multiplier. We next multiply by the 4ft., thus : 29sq.ft. 5' 4" 8' x '4 feet 32' : 2ft. and 8'. We write the 8'
under the primes in the other partial product, and reserve the 2ft. to add to the next product; and 6ft. X 4ft. 24ft.; 24ft. + 2ft. 26ft., which we write under the feet in the other partial product. The two being added together, we have 29 sq. ft. 5'4"; or (the primes and seconds being changed to fraction of a foot), 29.800 sq. ft.
In the second operation the work 6.8
is performed as in pure duodecimals
(Art. 594). The point ) separates 4.5
lower duodecimal orders from those be2 94
ginning with feet. As to the number 2 2 8
of feet in the multiplicand and multi
plier, no change is required in that 2 5.5 4 29sq. ft. 5' 4".
part of either of the given numbers in
expressing them according to the duodecimal scale. In performing the multiplication, we make the several reductions required according to the radix 12; and have, after pointing off, 25.54 square feet, expressed according to the duodecimal scale. The units, or feet, at the left of the point, are readily changed to the decimal scale by multiplying the left-hand figure, 2, by 12, the number of units in the radix, and adding the right-hand figure, 5, and giving the figures to the right of the point their proper notation, we have then 29sq. ft. 5' 4" for the answer, as before.
RULE. Write the multiplier under the multiplicand, so that units of the same orders shall stand in the same column.
Beginning at the right, multiply each term in the multiplicand by each term of the multiplier, and write the first term of each partial product under its multiplier, observing to carry a unit for every twelve from each lower order to the next higher.
The sum of the partial products will be the product required.
2. How many square feet in a floor 48 feet 6 inches long, and 24 feet 3 inches broad?
Ans. 1176 sq. ft. 1' 6". 3. The length of a room being 20 feet, its breadth 14 feet 6 inches, and height 10 feet 4 inches, how many square yards of painting are in it, deducting a fireplace of 4 feet by 4 feet 4 inches, and 2 windows, each 6 feet by 3 feet 2 inches?
Ans. 734 square yards. 4. Required the solid contents of a wall 53 feet 6 inches long, 10 feet 3 inches high, and 2 feet thick.
5. There is a house with four tiers of windows, and four windows in a tier; the height of the first is 6 feet 8 inches; the second, 5 feet 9 inches; the third, 4 feet 6 inches; the fourth, 3 feet 10 inches ; and the breadth is 3 feet 5 inches; how many square feet do they contain in the whole ?
Ans. 283sq. ft. 7. 6. How many cords in a pile of wood 97 feet 9 inches long, 4 feet wide, and 3 feet 6 inches high? Ans. 10117 cords.
7. Required the number of cords of wood in a pile 100 feet long, 4 feet wide, and 6 feet 11 inches high.
DIVISION OF DUODECIMALS.
601. To divide one duodecimal by another.
A board in the form of a rectangle, whose area is 27 sq. ft. 8' 6", is 1ft. 7in. wide; what is its length ?
Ans. 17ft. 6in.
We find how many times 27 1ft. 7') 27sq. ft. 8' 6" (17ft. 6' square feet contains the divisor, 26 11
and obtain 17 feet for the quo
tient, we multiply the entire di9 6
visor by the 17ft., and subtract 9 6
the product, 26ft. 11', from the
corresponding portion of the dividend, and obtain 9, to which remainder we bring dcwn the 6", and, dividing, obtain 6' for the quotient. Multiplying the entire divisor by 6', we obtain Oft. 9'6", which, subtracted as before, leaves no remainder. Therefore 17 feet 6 inches is the length required.
In the second operation we re1.7) 23.86 ( 15.6
17ft. 6 duce the feet of the given multi17
plicand and multiplier to the duo
decimal scale, and thus obtain 88
23.86 and 1.7. We then conduct
the division with reference to the 96
radix 12, as is ordinarily done with
respect to 10 (Art. 594). The re96
sult obtained is 15.6 in the duo
decimal, which, on changing the figures to the left of the point to the decimal scale, and giving the proper notation to the figure on the right of the point, becomes transformed to 17ft. 6' 17ft. 6in., the answer, as before.
RULE. — Find how many times the highest term of the dividend will contain the divisor. By this quotient multiply the entire divisor, and subtract the product from the corresponding terms of the dividend. To the remainder annex the next denomination of the dividend, and divide as before, and so continue till the division is complete.
2. It required 834 sq. ft. 3' of board to cover the side of a certain building. The height was 17ft. 9in.; what was the length of the side ?
Ans. 47 feet. 3. How many feet wide is a plank of uniform width, whose length is 18ft. Sin., thickness 3 inches, and solid contents 84ft. 4' 6:1 ?
4. An alley has an area of 792ft. 6' 9" 21". Its width is 12ft. 7' 8". Required its length.
Ans. 62ft. 8' 6".