CHAP. VII. OF THE MULTIPLICATION OF COMPOUND QUANTITIES. 64. WHEN we wish merely to represent the multiplication of two or more quantities, we have only to place between parentheses each of the quantities intended to be multiplied together, sometimes without any sign, but generally with the sign × between them. For instance, in order to represent the multiplication of the formulas ab + c and d e +f, we may write (ab+c). (de +f) or This method of expressing products is much used, because it shows immediately of what factors those products are composed. - 65. In order however to show how multiplication is actually performed, we must first observe that to multiply, for example, the formula a b+c by 2, we must multiply each term separately by that number, when we shall obtain the product, 2 a 26+2 c. 66. The same method will of course apply to all other numbers, and generally therefore the same formula multiplied by d, would give ad bd + cd. 67. We have hitherto supposed d to be a positive number; but if it were required to perform the multiplication by a negative number e, we must recur to the Rule that we have given in Art. 30, viz., that unlike signs multiplied together give and like signs +; we should therefore have 68. In order to show how any quantity A, either simple or compound, is to be multiplied by a compound quantity, as de, we will first take an example in numbers, and suppose that A is to be multiplied by 73; now it is evident that the product will be four times A. For we must first take A 7 times, and then subtract 3 times A from the product. 69. In general then, if it be required to multiply A by d e, we must multiply first by d and next by e, and then subtract the last product from the first; the result of which will be, 71. Since therefore we know precisely the product of (a-b) x (d-e) we shall now exhibit the same example of multiplication in the following form, a-b d-e ad-bd-ae+be. 72. From this it will be seen, that we must multiply each term of the upper formula by each term of the lower, and that with regard to the signs we must observe strictly the rule above given. 73. According to this rule, it will be easy, for example, to calculate the following example, viz. a+b× a−b. 74. We know that for a and b may be substituted any known numbers whatever, so that the example which we have given establishes the following Theorem, viz., that The sum of any two numbers multiplied by their difference gives a product equal to the difference of their squares, which may be expressed also in the following form, (a + b) × (a−b) = a2 —b2 75. From hence also we arrive at another Theorem, viz., that The difference of the squares of any two numbers is always a product, divisible either by the sum or by the difference of their square roots; and consequently that The difference between two squares can never be a prime number. (c) 76. Now if it were required to multiply the quantity a+b into itself, the operation would be as follows, a + b a2+ab +ab+b2 Prod. a2+2ab+b2 Since therefore a+b may be supposed to represent any two parts into which a number, as x, may be divided, we obtain the following Theorem, presenting an algebraical proof of Euclid's Prop. IV. Book II., viz., that If a number be divided into any two parts, the square of the whole number is equal to the squares of the two parts, together with twice the product (or rectangle) of the two parts. For (c) Prime numbers are such as cannot be represented by factors, which are therefore not divisible by any other numbers without remainder. This theorem however is not universally true, although it has but one single exception, viz., where the difference between two numbers is only 1, and their sum is a prime number: for example 7+6=13, and 72—62= 13; 9+8=17, and 92-82=17, &c. if x be divided into two parts, as a and b, xa+b, therefore x2=a2+2ab+b2. 77. If we have more than two compound quantities to multiply together, we shall easily understand that after having multiplied two of them together, we must then multiply that product by the other, and so on of the rest, and that it is wholly indifferent in what order the different operations succeed each other. If it were required, for example, to find the value or product of the four following factors, (a+b) • (a2+ab+b2). (a - b) · (a2 — a b+b2) we we should first multiply together the two first factors, viz., Prod. II. a3-2 a2b+2 a b2-b3 It then only remains to multiply the product I. by II., viz., Change the order of this multiplication in any manner, and the product will be the same. 78. The truth of the example may be further illustrated by a numerical application. Let us assume a3 and b±2. Then a+b=5 and a-b1. Moreover a2=9, ab=6, b2=4. Therefore a2+ab+b2=19, and a2-ab+b2=7. the product required is 5 × 19 × 1 × 7, which is 665. Now a6729, and b664. Therefore ao-b6665, as we have before seen. Ex. 8. Thus Ex. 7. Multiply 5a+2ab+c2 +d x + y + x2-y2 x−y+x2+ y2 Prod. |