as b represents the difference of a and b when it is not known which is the greater. a x b, or a.b, or ab, denotes the product of the numbers represented by a and b, a + b, or indicates that the number represented by a is to be divided by that denoted by b. a:b:: c:dexpresses, that a is in the same ratio, or proportion to b, that c is to d. x = a - b + c exhibits an equation, shewing that a is equal to the difference of a and b, added to c. (a+b)c, or a+bxc, is the product of the compound quantity a+b multiplied by the simple quantity c; a+b+a-b, or (a+b) + a+b (a-b), or is the quotient of a + b divided by a-b; and the bar a-b thus placed over two or more quantities, to connect them, is called a vinculum. a+b-cor (a+b-c) is the cube, or third power, of the quantity a+b-c. 5a indicates that the quantity a is to be taken 5 times; likewise, 7 (b+c) is 7 times (b+c). Note. The axioms of Geometry apply also to several branches of Algebra, and should be studied in connection with these definitions. ADDITION. (24.) From the twofold division of algebraic quantities into positive and negative, like and unlike, there arise three cases of Addition, which must be separately considered. CASE I. To add like quantities with like signs. Rule. Add all the co-efficients, annex the common letter, or letters, and prefix the common sign. Note. When a leading quantity has no sign prefixed to it, the sign + plus is always understood; and a quantity without any co-efficient is supposed to have 1, or unity before it. Thus a = once a (1a). • Obs. Quantities with any kinds of exponents are, in all respects, to be considered as if they were represented by a single letter. Thus, + 9x2 - 16x2. 4 CASE II. To add like quantities with unlike signs. Rule 1. Collect the positive co-efficients into one sum, and the negative ones into another. 2. From the greater of these sums subtract the less, and to the remainder prefix the sign of the greater, and annex the common letters. Note. If the aggregate of the positive terms be equal to that of the negative ones, their sums will be = 0. Examples. -7 ab+3 bc- ху ab+2 bc +4 xy 3 -2 y+2 ax bc+2xy -7 у-3 ах -2ab+4bc-3 ху 4 -5 a3+13x -2-47 7x+x 9-142 3 ab 5 ab-8 bc + xy +5y+3 ax -9 y- ax -13x-2.r +2x-4 -4x+13 -9x+9 Illus. 1.-The manner in which we generally calculate a person's property, is an apt illustration of the foregoing Note, and consequently of the rule. We denote what a man really possesses by positive numbers, using the sign +; whereas his debts are represented by negative numbers, or by understanding the sign, as affecting those numbers. 2. Thus, when it is said of any one that he has 100 crowns, but owes 50, this means that his real possession amounts to 100-50, or which is the same thing + 100-50, that is to say, 50. And if he has in possession 20 crowns, but owes 20, his real possession amounts to 20-20, or + 20-20-0. In fine, he has nothing; but then he owes nothing. But, on the other hand, if he owes 70 crowns, and has in possession only 40, his real possession would be expressed thus - 70 + 40. Here his debt is fairly represented by the negative number - 70, while his real possession is represented by the positive number + 40. It is certain, therefore, that he has 30 crowns less than nothing; and we might, consequently, express the state of his finances - 30; for if any one were to make him a present of 30 crowns to pay this debt, or-30, he would only be at the point nothing (0), though really richer than when - 30 stood against his future prospects and exertions. 3. Debts, or sums of money owing, are therefore as much real sums, or quantities, of money, or real numbers, as credits are; and the sign +, or -, governs the quantity or number that follows it. * Scholium. In the language of Algebra, a and b may stand for any two numbers whatever (Notes 1 and 2, p. 1); and, therefore, a + b stands for u made more by b. Again, a - b stands for a made less by b; that is, for the difference of a and b (where b is supposed less than a). Now, by the rule of Case II. a + b and a - b, added together, make twice a (2a); therefore we derive this (25.) † THEOREM. If the sum and difference of any two numbers be added together, the whole will be twice the greater number; for if a + b be added to a-b, the sum is 2a. CASE III. To add unlike quantities. Rule. Collect all the like quantities, by the last rules, and set down those which are unlike, one after another, with their proper signs. • A Scholium is a remark or observation made on some foregoing propοsition, or other premises. + A Theorem is a demonstrative proposition, in which some property is asserted and the truth of it required to be proved. SUBTRACTION. (26.) Rule 1. Write, in one line, those quantities from which the subtraction is to be made, and which we call the minuend; then anderneath write all the quantities to be subtracted, which we call the subtrahend, ranging under each other the quantities of the same denomination. 2. Change the signs of the quantities to be subtracted, or conceive them changed; then collect the different terms, and place them as directed by the rules of Addition.* Scholium. In the scholium to Case II. of Addition, we shewed that a + b may represent the sum, and a - b the difference of any two numbers, of which a is the greater and b the less. Now here it appears (in Example 1. of Subtraction), that if a - b be taken from a + b, the remainder will be twice b (2b); whence we derive this (27.) Theorem. If the difference of any two numbers be subtracted from their sum, the remainder will be twice the less number. * This rule may be thus illustrated: If it were required to subtract 5 - 2 (i. e. 3) from 9, it is evident that the remainder would be greater by 2, than if 5 only were subtracted. For the same reason, if b - c were subtracted from a, the remainder would be greater by e than if b only were taken away. Now if + b be subtracted from + a, the remainder will be a - b; and consequently, if b-c be subtracted from a, the remainder will be a - b + c. If b were a negative quantity (-b) to be taken from + a (or a), we should obtain a + b. For the same reason when c is a negative quantity (c), and ba positive one, as in the expression just given, we change the signs of both, thus: -b+c, when we would take them from a. MULTIPLICATION. (28.) In the multiplication of algebraic quantities, four circumstances are to be considered. 1. The signs of the quantities : 2. Their co-efficients: 3. The letters of which they are composed; and 4. The indices or exponents of those letters. (29.) In performing any operation in multiplication, we must, therefore, observe the four following rules. 1. When quantities having like signs are multiplied together, the product will be +. On the contrary, if their signs are unlike, the sign of the product will be -.* 2. That the co-efficients of the factors must be multiplied together, to form the co-efficient of the product. 3. That the letters of which the factors are composed must be set down, one after another, according to their order in the alphabet. 4. That if the same letter be found in both factors, the indices of this letter must be added, to form its index in the product. • That like signs make +, and unlike signs in the product, may be illustrated thus: First. When + a is to be multiplied by + 6, this denotes that + a is to be taken as many times as there are units in b; and because the sum of any number of affirmative terms is affirmative, it is obvious that + ax + b = + ab. Secondly. If two quantities are to be multiplied together, the result will be actually the same, in whatever order they are placed for a times b is the same as b times a; and, therefore, when-a is to be multiplied by + b, or +bby-a, it is the same thing as taking a as many times as there are units in + b; and as the sum of any number of negative terms is negative, it is plain that-ax + b, or + bx-a ab. Lastly. When - a is to be multiplied by -b, we have ab for the product at first sight, but still we must determine whether the sign + or - is to be placed before the product. Now it cannot be the sign -, for + x -, or which is the same thing, ax - b gives - ab, and -aby-t cannot produce the same result as - ax + b; but must produce a contrary result, to wit, ab; consequently we have the following rule: - multiplied by - produces +, in the same manner as + X + give +. But this illustration may be demonstrated thus: When the compound quantity + a - b is to be multiplied by + c, we repeat or add + a - b to itself as often as there are units in c; hence, since the sum of any number of affirmative terms is affirmative, and the sum of any number of negative terms is negative, it is obvious, that + a - b moltiplied by + c produces + ac-be; for the same reason, + a - b multiplied by + d produces + ad - bt. Whence, if from a times (a - b) = + ac - And x-d 5-bx + c = be bd bc 01 + ac C { al + ad be unlike sigus produce minus. |