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7. A gentleman hired a laborer for 90 days on these conditions : that for every day he wrought he should receive 60 cents, and for every day he was absent he should forfeit 80 cents. At the expiration of the term he received $ 33. How many days did he work, and how many days was he idle ?

Ans. He labored 75 days, and was idle 15 days. 8. There is a fish whose head weighs 15 pounds, his tail weighs as much as his head and of his body, and his body weighs as much as his head and tail. What is the weight of the fish ?

Ans. 721b. 9. If 12 oxen eat 34 acres of grass in 4 weeks, and 21 oxen eat 10 acres in 9 weeks, how many.oxen would it require to eat 24 acres in 18 weeks, the grass growing uniformly ?

Ans. 36 oxen. 10. What number exceeds three times its square root by 11 ? (Note 2.)

Ans. 26.4201648.

SCALES OF NOTATION.

589. The scale of any system of notation is the law of relation existing between its units of different orders.

590.° The radix of any scale is the number of units it takes of one order to make a unit of the next higher. Thus, 10 is the radix of the decimal or denary system, 2 of the binary, 3 of the ternary, 4 of the quaternary, 5 of the quinary, 6 of the senary, 7 of the septenary, 8 of the octary, 9 of the nonary, 11 of the undenary or undecimal, 12 of the duodenary or duodecimal, 20 of the vigesimal, 30 of the trigesimal, 60 of the sexagesimal, and 100 of the centesimal.

591.° In writing any number in a uniform scale, as many distinct characters or symbols are required as there are units in the radix of the given system. Thus, in the decimal or denary scale 10 characters are required, in the binary scale 2 characters, in the duodenary or duodecimal 12 characters, and so on. In the binary scale use is made of the characters 1 and 0, in

OPERATION.

6)349 1 6)58 1

the ternary, 1, 2, and 0, &c., the cipher being one of the characters in each scale. In the duodenary scale, eleven characters being required beside the cipher, the first nine may be supplied by the nine digits, the tenth by t, the eleventh by e, and the twelfth by 0.

592.° To change any number expressed in the decimal scale to any other required scale of notation.

Ex. 1. Express the common number 75432, in the senary and duodenary scales.

Ans. 1341120 and 377e0.

By dividing the given 6)75432 12)75432

number by 6, it is distribut

ed into 12572 classes, each 6)125720 12) 6286 0

containing 6, with O remain62095 2 12)523 10, or t der. By the second division 12)437

by 6, these classes are dis

tributed into 2095 classes, 3 7

each containing 6 times 6, 6)9 4.

or the second power of 6,

with a remainder of 2 of the 1 3

former class, each containing

6. By the third division the classes last found are distributed into 349 classes, each containing 6 of the latter, which were each the second power of 6, and therefore these are the third power of 6, with a remainder 1 time the second power of 6 In like manner, the next quotient expresses 58 times the fourth power of 6, with a remainder 1 time the third power of 6; the next quotient expresses 9 times the fifth power of 6, with a remainder 4 times the fourth power of 6; and the last quotient expresses 1 time the sixth power of 6, with a remainder 3 times the fifth power of 6. Hence, the given number is found to be equal to 1 x 66 + 3 X 61 + 4 X 64 + 1 X 63+1 X 6+ 2 x 6 +0, or according to the senary system of notation 1341120.

By proceeding in like manner, we find the given number to be equal to 3 X 124 + 7 x 123 + 7 x 122 + 10 X 12 + 0, or, according to the duodenary scale, 377e0.

RULE. - Divide the given number by the radix of the required scale repeatedly, till the quotient is less than the radix; then the last quotient, with the several remainders in the retrograde order annexed, placing ciphers where there is no remainder, will be the the given number expressed in the required scale. 2. Change 37 from the decimal to the binary scale.

Ans. 100101. 3. Reduce 1000000 in the decimal scale to the ternary and also to the nonary.

Ans. 1212210202001, and 1783661.

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.

Ans. 75432.

OPERATION.

37 7 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 em1 2

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 i 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 perform 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 the 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.

Ans. 2405.

DUODECIMALS.

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 th, T., &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. O' 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.

599 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.

PIRST OPERATION.

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'X5'=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' 21t. 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 Gft. 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. 5o4"; or (the primes and seconds being changed to a fraction of a foot), 294 sq. ft.

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