41. Making the general expressions a, b, the factors of a product, that product is indicated in any of the three forms, axb, ab, ȧb. The product of two numerical factors can be indicated in the first form only, as 7 × 8. If one factor is a particular, and the other a general expression, the product is written in any of the forms, 3 × a, 3'a, 3 a. In like manner, the product of three factors, a, b, c, may be indicated in any of the forms, a× b×c, a·b'c, abc. Also by Art. 35. a., abc-acb-bac-bca=cab=cba. The product of any factors, a, b, c, d, . . is termed a mulfiple of each factor, or of any combination of the factors, not comprehending the whole; conversely the factors are termed submultiples, or parts of the whole product. When numeral and literal factors are combined in the same expression, the former may be multiplied together (Art. 35. a.), and their product written before the expression of the product of the literal factors. Thus 3a x 56 x 8xc=3x5 x 8 x abc=120abc. 42. When a general expression is repeatedly added to itself, the number of repetitions is marked by a coefficient (Art. 18. a.), as 3a, instead of a +a+a. By an analogous convention, when a general expression is multiplied by itself, the number of repetitions of the factor is expressed by a figure or letter written to the right of that factor, and above the horizontal line on which it is placed. Thus : The figure or letter which expresses the number of repetitions of the factor, is called an index or exponent. Whence a coefficient expresses the repeated additions of the number to which it is prefixed; and an exponent, the repeated multiplications of the number to which it is annexed. If the number multiplied by itself is made 2, 3, n times a factor, the product is named the 2nd, 3rd . . power of the number; and the number the 2nd, 3rd, nth root of the corresponding power or product. The coefficient and exponent, 1, are not expressed. Thus a signifies l'a1. The product of two in the usual manner. powers of the same root may be indicated Thus the product of a2 by a3 is a2 a3. This may, however, be reduced to a more simple form. But the product of m+ n factors each equal to a is amta. Therefore aman = am+n. Whence the product of two powers of the same root is equal to that root raised to a power indicated by the sum of the exponents of the factors. 43. Let a and b be two general expressions of number, then a+b is the sum of these numbers. Also (a+b)+(a + b) = 2 (a + b) =2a+2b. (Art. 18. a) And generally, (a + b) + (a + b) + na+nb. ..... (to n repetitions)=n (a+b)= Let c, d, be other two general expressions, and let it be required to multiply (a+b) by (c+d). Making (c+d)=n; (a + b) (c + d)=(a+b) n=an+bn. But n=c+d, therefore an=a (c+d)=ac+ad; bn=b (c+d)=bc+bd. * .. (a+b) (c + d)=ac+ad+bc+bd. Whence the product of two binomial expressions is equal to the sum of the partial products of each term of the multiplicand by each term of the multiplier. The product of a+b by c+d may be also obtained as follows: - a+b c+d ac+bc. ad+bd =product of (a + b) by c ; ac+ad+be+bd: = sum of these partial products. .. is put for the word "therefore," and . for "since" or "because." And the product of a+b+c by d+e+f, as follows: a+b+c ad+bd+cd. ae+be+ce af +bf+cf =product of (a+b+c) by d; =product of (a+b+c) by e; =product of (a+b+c) by f; ad+ae+af+ d+be+bf+cd+ce+cf=sum of partial products. Whence, also, the product of two trinomial expressions is equal to the sum of the partial products of each term of the multiplicand by each term of the multiplier. The number of terms of either or of both factors may evidently be increased indefinitely, and a corresponding result obtained. a. If a+b is multiplied by a + b, the product (which is also called the square of a+b) is a2+2ab+b2. And if a2+2ab+b2 is again multiplied by a + b, the product (which is called the cube of a + b) is a3+3a2b+3ab2+b3. b. In (a+b)2=a2+2ab+b2, let b=1. Then (a + 1)2= a2 + 2 a + 1. Also in (a+b)3 = a3 +3a2b+3ab2 + b3, let b = 1. Then (a+1)3a3 +3a2 + 3a + 1. 45. In Division two numbers, the dividend and the divisor, are given, and it is required to find into how many parts, each equal to the divisor, the dividend can be decomposed. The method of solution, by continual subtraction of the divisor from the dividend and successive remainders, is noticed in Art. 33. As an illustration, let it be required to divide 35 by 7. The dividend being exhausted by 5 subtractions of the divisor, it follows that the quotient is 5. It also follows from this method that the quotient contains. unity as often as the dividend contains the divisor. Now whether the divisor be subtracted from the dividend and successive remainders, or the sum of as many repetitions of the divisor as there are ones in the quotient be at once subtracted from the dividend, the result is the same (Art. 27.). But the sum of these repetitions is equal to the product of the divisor and quotient. Consequently, when the last subtraction leaves no remainder, the dividend is the product of two factors, the divisor and the quotient. Whence, in Division, there are given the product of two factors and one of these factors to find the other factor. In Multiplication two factors are given to find their product. Multiplication and Division are, consequently, reverse operations. 46. From the preceding considerations it is evident that the division of a number not exceeding 9 x9 by a number not exceeding 9, may be made with the help of the Multiplication Table. For example, let it be required to divide 35 by 7. The divisor, 7, is found in the left vertical column. Tracing horizontal column 7 to the right, the dividend, 35, is found, over which, in the uppermost horizontal line, is placed the other factor, 5, which is the required quotient. In like manner it is found that 48 6 8 and 63+9=7. Let it be required to divide 32 by 5. By inspection of the table, it is found that 32 falls between 5×6=30, and 5 × 7=35. Whence 32 contains 5 more than 6, and less than 7 times; and since 32-30=2, it is to be concluded that the quotient is 6, and the remainder 2. Writing the remainder and divisor thus, (Art. 33.), the result of the division of 32 by 5 may be expressed as follows: 32+5=63. When the division is exact (that is, when the last remainder =0), the dividend is equal to the product of the divisor and quotient. If the division of the remainder (when there is one) by the divisor is indicated, and the result annexed to the other part of the quotient, as in the instance 32÷5=6%, then also the product of the divisor and quotient is equal to the dividend. For 63 x 56 x 5+ × 5. Now 6 x 5=30. And signifies 2÷5..3 × 5=2÷5 and multiplied by 5. d But if any number be both divided and multiplied by the same number, it is not changed by the process, which (since =1, and generally 2=1) amounts to the multiplication of the number by unity. This reasoning being general, it follows that, in any case, the product of the divisor and quotient is equal to the dividend. The expression of an unexecuted division, such as }, may be read, two divided by five, two over five, or two-fifths. Numbers thus written are called fractions. Between any two consecutive multiples of a divisor, there are as many numbers, less one, as the divisor contains ones. But no number between two consecutive multiples of a divisor can be exactly divided by that divisor. Whence, most commonly, a remainder greater than O, and less than the divisor, must terminate the operation of division. 47. If the divisor is less than 10, and the dividend less than 9 × 10, the quotient is found by inspection of the Multiplication Table (supposing this to contain only the products from 1x1 to 9 x9). Let the divisor be less than 10, and the dividend any number whatever; also let the divisor, dividend, and quotient be denoted by d, D, q, respectively. Then since dxq=D, and d contains one figure, q must contain as many figures as D, or as many less one (Art. 38.); and D is composed of the partial products of d by these quotient |