DUODECIMALS. ART. 164. A Duodecimal (Latin duodecim, twelve) is a number whose scale is 12; hence, 12 units of any order make one unit of the next higher order. This system of numbers is used by artificers in finding the contents of surfaces and solids. For this purpose the foot is divided into 12 equal parts called inches or primes, marked '; the inch or prime is divided into 12 equal parts called seconds, marked", &c. The accents used to mark the different orders are called indices. Note.-Duodecimals may be added and subtracted like Denominate Numbers. MULTIPLICATION OF DUODECIMALS. ART. 165. To multiply one duodecimal by another. Ex. 1. How many square feet in a board 9 ft. 5 in. long, and 2 ft. 8 in. wide? 29 Explanation. Since 8' of a foot, and 5' 8' x 5' ===40"= 3' 4". Write 4" in seconds order. Again, since 8', 9 ft. x 8' 9 ft. x = }}=72′ and 72'3' (above) = 75'6 sq. ft. 3'. 6 sq. ft. 3' 4". Again 5' or ×2 ft.=;;; 2 ft.=18 sq. ft. Hence 9 ft. 5′ ×2 ft.=18 = sq. ft. 10'. Adding these two products, the total product is 25 sq. ft. 1' 4". It will be observed that the denomination of the product of any two denominations is denoted by the sum of their indices; thus 5' x 8"-40"", 6" x 4"-24""", &c. " In the above process, the notations of feet, primes, &c., are used for convenience. The multiplier is, however, really an abstract number. RULE. Write the Multiplicand under the Multiplier, placing ft. under ft., primes under primes, &c. Beginning at the lowest order, multiply each order of the multiplicand by each order of the multiplier, adding their indices to ascertain the denomination of the product, and carrying one for every twelve from a lower order to the next higher. Add the several partial products for the product required. Examples. Ans. 61 sq. ft. 2′ 8′′. Ans. 39 sq. ft. 2' 10" 8'". 2. Multiply 12 ft. 8' by 4 ft. 10'. 3. Multiply 4 ft. 6' 4" by 8 ft. 8'. 4. Multiply 10 ft. 6' 6" by 4' 8". 5. How many square feet in a board 12 ft. 9 in. long and 11' 4" wide ? Ans. 12 sq. ft. 2′ 6′′. 6. How many cubic inches in a block 2 ft. 9' long, 1 ft. 8' wide, and 2 ft. 4' high? 7. Required the solid contents of a block 4 ft. 4' long, 2 ft. 3' wide, and 10' high. 8. How many square feet in 60 boards, each board being 15 ft. 4' long, and 1 ft. 2' wide ? 9. Divide 10 sq. ft. 2' 10" by 5 ft. 7'. Ans. 1 ft. 10'. Remark.-By observing that division is the reverse of multiplication, the following process will be readily understood. The divisor is placed at the right of the dividend for convenience. 10. Divide 62 sq. ft. 11" 3" by 8 ft. 6′ 9′′. Ans. 7 ft. 3'. INVOLUTION. ART. 166. Involution is the method of finding the powers of numbers or quantities. The power of a number (except the first) is the product obtained by multiplying the number by itself one or more times. The first power of a number is the number itself. It is also called the root. The second power, or square, is the product of the number multiplied by itself once. The third power, or cube, is the product of the number multiplied by itself twice. The different powers derive their names from the number of times the number is taken as a factor. Thus, the first power contains the number as a factor once; the second power, twice; the third power, three times, &c. The index or exponent of a power is a small figure placed at the right and a little above the number, to show the degree of the power, or how many times the number is taken as a factor. The O power of any number or quantity results from dividing the number by itself and is equal to unity or 1. Thus, 6°=1,.25° 1, 50°=1, &c. The following table will illustrate the above definitions and remarks. 5°(5÷5)=1, the 0 power of 5. 5' 5, the first power or root of 5. 5=5×5-25, the second power or square of 5. 5' 5x5x5=125, the third power or cube of 5. Remark. The second power of a number is called its square, because the area of a geometrical square is obtained. by multiplying the number of linear units in one of its sides. by itself once. The third power is called the cube, because the solid contents of a geometrical cube is obtained by multiplying the number of linear units in one of its sides by itself twice. Ex. 1. What is the cube or third power of 24 ? 96 48 It is evident from the definition that the cube of a number is obtained by mul576, 2d power. tiplying the number by itself twice, or by taking it three times as a factor. 24 2304 1152 13824, 3d power. The RULE.-Multiply the number by itself as many times as there are units in the exponent of the power, LESS ONE. last product will be the required power. NOTE. The power of a fraction, either common or decimal, SUGGESTION.-Since the product of two or more powers of a given number is the power denoted by the sum of their ex ponents, the 9th power of 12 may be found by multiplying the the 3d power by itself twice; thus, 12' x 12' x 12'=12o. 9. What is the 4th power of 21 ? 10. What is the 3d power of 2.04 ? 11. What is the value of 15a ? 12. What is the value of (3)*? 13. What is the value of 201' ? 14. What is the value of .001' ? 15. What is the square of 9? Ans. 39. Ans. 3. 33 Ans. 85. EVOLUTION. ART. 167. Evolution is the method of finding the roots of numbers or quantities. Evolution is the reverse of involution. In the latter, the root is given to find the power. In the former, the power is given to find the root. The root of a number is such a number as multiplied by itself a certain number of times, will produce the given number. The first root of a number is the number itself. It is also called the first power. The second, or square root of a number is that number which, multiplied by itself once, will produce the given number. The third, or cube root must be multiplied by itself twice to produce the given number. The different roots take their names from the number of times they are taken as factors to produce the given number. The first root is taken once as a factor; the second or square root, twice; the third or cube root, three times, &c. A root of a number may be defined to be a factor which taken a certain number of times, will produce the given number. The root of a number is usually indicated by the radical sign placed before it, with the index of the root written above it. |