DUODECIMALS. DUODECIMALS or CROSS MULTIPLICATION, is a rule used by workmen and artificers, in computing the contents of their works. Dimensions are usually taken in feet, inches, and quarters; any parts smaller than these being neglected as of no consequence. And the same in multiplying them together, or casting up the contents. The method is as follows. SET down the two dimensions to be multiplied together, one under the other, so that feet may stand under feet, inches under inches, &c. Multiply each term in the multiplicand, beginning at the lowest, by the feet in the multiplier, and set the result of each straight under its corresponding term, observing to carry 1 for every 12, from the inches to the feet. In like manner, multiply all the multiplicand by the inches and parts of the multiplier, and set the result of each term one place removed to the right-hand of those in the multiplicand; omitting, however, what is below parts of inches, only carrying to these the proper number of units from the lowest denomination. Or, instead of multiplying by the inches, take such parts of the multiplicand as there are of a foot. Then add the two lines together after the manner of Compound Addition, carrying 1 to the feet for 12 inches, when these come to so many. INVOLUTION. INVOLUTION is the raising of Powers from any given number, as a root. A Power is a quantity produced by multiplying any given number, called the Root, a certain number of times continually by itself. Thus, 2= 2X2= 2 X 2 X 2 = 2 × 2 × 2 × 2 = 2 is the root, or 1st power of 2. And in this manner may be calculated the following Table of the first nine powers of the first 9 numbers. 7 49 343 2401 16807 117649 823543 5764801 40353607 8 64 512 4096 32768 262144 2097 15216777216 134217728| 9/81 729/6561 59049 531441 4782969 43046721 387420489 The ! 3. Reduce, and to a common denominator. Ans. 40 36 45 60 60 60° 4. Reduce, 23, and 4 to a common denominator. 302 78 120 Ans.. Note I. When the denominators of two given fractions have a common measure, let them be divided by it; then multiply the terms of each given fraction by the quotient arising from the other's denominator. 5 and = 1 and 2 by multiplying the former by 7, and the latter by 5. 2. When the less denominator of two fractions exactly divides the greater, multiply the terms of that which has the less denominator by the quotient. = Ex. and and, by mult. the former by 2. 2 3. When more than two fractions are proposed, it is sometimes convenient, first to reduce two of them to a common denominator; then these and a third; and so on till they be ali reduced to their least common denominator. Ex. and and = } and & and CASE VII. To find the value of a Fraction in Parts of the Integer. MULTIPLY the integer by the numerator, and divide the product by the denominator, by Compound Multiplication and Division, if the integer be a compound quantity. Or, if it be a single integer, multiply the numerator by the parts in the next inferior denomination, and divide the product by the denominator. Then, if any thing remains, multiply it by the parts in the next inferior denomination, and divide by the denominator as before; and so on as far as necessary; so shall the quotients, placed in order, be the value of the fraction required*. The numerator of a fraction being considered as a remainder, in Division, and the denominator as the divisor, this rule is of the same nature as Compound Division, or the valuation of remainders in the Rule of Thre before explained. EXAMPLES. EVOLUTION. EVOLUTION, or the reverse of Involution, is the extracting or finding the roots of any given powers. The root of any number, or power, is such a number, as being multiplied into itself a certain number of times, will produce that power. Thus, 2 is the square root or 2d root of 4, because 22 2 X 2 = 4; and 3 is the cube root or 3d root of 27, because 33 = 3X 3X 3 = 27. Any power of a given number or root may be found exactly, namely, by multiplying the number continually into itself. But there are many numbers of which a proposed root can never be exactly found. Yet, by means of decimals, we may approximate or approach towards the root, to any degree of exactness. Those roots which only approximate, are called Surd roots; but those which can be found quite exact, are called Rational Roots. Thus, the square root of 3 is a surd root; but the square root of 4 is a rational root, being equal to 2: also the cube root of 8 is rational, being equal to 2; but the cube root of 9 is surd or irrational. Roots are sometimes denoted by writing the character ✔ before the power, with the index of the root against it. Thus, the 3d root of 20 is expressed by 3/20; and the square root or 2d root of it is 20, the index 2 being always omitted, when only the square root is designed. When the power is expressed by several numbers, with the sign+ or between them, a line is drawn from the top of the sign over all the parts of it: thus the third root of 45 12 is 45 — 12, or thus (45-12), inclosing the numbers in parentheses. 3 But all roots are now often designed like powers, with fractional indices: thus, the square root of 8 is 82, the cube root of 25 is 25, and the 4th root of 45 18 is 45 18 or (45 18)*. ΤΟ 2. Reduce & of a penny to the fraction of a pound. 4 × 11 × 10 = the Answer. 6. Reduce 3 dwt to the fraction of a lb troy. 7. Reduce crown to the fraction of a guinea. Ans. 32d. Ans. Ans. 32. Ans. Ans. 5 Ans. 25. Ans. ADDITION OF VULGAR FRACTIONS. If the fractions have a common denominator; add all the numerators together, then place the sum over the common denominator, and that will be the sum of the fractions required. If the proposed fractions have not a common denominator, they must be reduced to one. Also compound fractions must be reduced to simple ones, and fractions of different denominations to those of the same denomination. Then add the numerators as before. As to mixed numbers, they may either be reduced to improper fractions, and so added with the others; or else the fractional parts only added, and the integers united afterwards. * Before fractions are reduced to a common denominator, they are quite dissimilar, as much as shillings and pence are, and therefore cannot be incorporated with one another, any more than these can. But when they are reduced to a common denominator, and made parts of the same thing, their sum, or difference, may then be as properly expressed by the sum or difference of the numerators, as the sum or difference of any two quantities whatever, by the sum or difference of their individuals. Whence the reason of the Rule is manifest, both for Addition and Subtraction. When several fractions are to be collected, it is commonly best first to add two of them together that most easily reduce to a common denominator; then add their sum and a third, and so on. EXAMPLES. |