It frequently happens that a quantity consists of several quantities of the same kind, as of two or more distances to make one distance; these must, therefore, be joined by addition, or by addition and subtraction. In order to indicate this junction, two distinct signs will be necessary. The sign+ (plus) implies that the quantity which follows it is to be added to that which goes before, and that all the quantities, when more than one, are to be added together into one sum. Thus, a+b shows that b is to be added to a, or that a and b are to be added together. Again, a+b+c+d implies that b is to be added to a; c, to the sum of a and b; d, to the sum of a, b, c. The sign (minus) placed between two quantities, denotes that the quantity which follows it is to be subtracted from that which precedes it. Thus, m-n denotes that the quantity represented by n is to be subtracted from that represented by m. Suppose, for instance, that m is 7, and n 3; then 7-3 will be 4. Therefore, m-n denotes the remainder arising by subtracting n from m. Hence SUBTRACTION is an opposite operation to ADDITION; and, therefore, if any quantity be both added to and subtracted from the same quantity, the quantity thus added and subtracted may be taken away entirely, by which the expression will be in its most simple form: thus, m+a-a is equivalent to m. When two quantities are equal to each other, this equality is implied by the interposition of the double bar = between each of the quantities. Thus, m+a-a=m; as, also, 4+3+6-2=11. Equal quantities, thus connected, are called Equations. TERMS are all those parts of an expression that are separated by the signs of Addition and Subtraction. Thus, a+b+c-d, is a quantity consisting of four terms. Quantities which contain two terms are called binomials: thus, a+b, or, a-b, are binomials. The MULTIPLICATION of two or more factors is indicated by connecting the letters representing the factors; as, ab denotes a product of two factors; abx a product of the three factors, a, b, and x. Thus, let a=2, b=3, and x=5; then abx=30: Again, mmmm signify a product of four factors, of which all are equal suppose m=2, then mmmm=16. When any number of factors are equal to each other, instead of repeating them to that number in the representation of the product, the product will be indicated with less trouble by writing only one of the equal factors and a digit; the latter containing as many units as the factors are in number, over the right-hand side of the factor so written. Thus, instead of aa, bbb, xx, xxx, the same idea will be more conveniently expressed thus, a2, b3, x2, x3. The continued product of equal quantities is called a Power; the quantity itself is called the Base of that power; and the digit, which indicates the number of factors, is called the Index or Exponent of that power. When factors of a product consist of compound terms, each compound factor is enclosed within brackets, or parentheses, and the factors thus included are joined to each other by bringing the bracket on the left hand of the one nearly close to that on the right-hand of the other. Thus, the product of a+b+c, x+a+c+d, and a+x, is represented by (a+b+c)(x+a+c+d) (a+x). Let a=1, b=2, c=3, d=4, and x=5, then will (a+b+c) (x+a+c+d) (a+x) = (1+2+3) (5+1+3+4) (1+5)=468. Any Power of a compound quantity is represented in a similar manner to that of representing a simple quantity, by inclosing the compound within brackets, and writing the number which indicates the power over the righthand bracket, and on the right-hand side of that bracket: thus, (a+b)3 denotes the cube of a + b, and (x+y+x)* denotes the fourth power of x+y+x. DIVISION is represented by placing the dividend above the divisor, with a short line between them, as; which expression shows how often the quantity a contains the quantity b; or how often the dividend contains the divisor. Let a be 12, and b be 3, then will be 4. m n FRACTIONS are represented in the same manner as Division, by placing the numerator above and the denominator below a short line. Thus, indicates a fraction, whose numerator is m, and denominator n. Let m be equal to 2, and n equal to 3; then is equivalent to or two-thirds: or, if we suppose m n m to be equal to 17, and n equal to 5, then would be 17-fifths of unity, or by dividing the numerator m, which is equivalent to 17, by the denominator n, which is equivalent to 5: the quotient will be 3 and 3. All expressions of quantity are said to be Simple when the operations are indicated by one or more letters, either in Multiplication or Division, without the intervention of the signs + or -, as in the following: a, ab,,; are all Simple expressions. ab ; which Known quantities are generally represented by the initial letters, a, b, c, &c. of the alphabet, or by numbers; and the unknown quantities by the final letters, v, W, X, Y, Z. A Co-efficient is the number prefixed to any quantity. Thus, in the expression 5x, the number 5 is the co-efficient of x; or, if x represent a quantity to be discovered by an operation, and a a quantity already known, then, in the expression ax, the quantity a is called the co-efficient of x. Having explained the forms which indicate the operations of Simple Quantities, we shall now explain the rules for those performed upon Compounds. ADDITION OF ALGEBRA. 97. To add any number of simple affirmative quantities, which are of the same kind, together, or any number of quantities that have a common factor: Prefix the sum of the co-efficients to the quantity, and the product will represent the sum; observing that, when no co-efficient is written, the coefficient is understood to be unity: and, when the co-efficients are expressed by letters, these letters are to be joined with the sign + within brackets, and the common quantity adjoined or subjoined. In the following examples let the sum be put equal to S. Example 2.-Add ax, 2 ax, 3 ax, together; then 6ax = S. Ex. 3.—Add ax, bx, cx, dx, ex, fx, together; then (a+b+c+d+e+f)x=S. 98. To add any number of simple affirmative quantities of different kinds together: Connect the whole to be added by the sign + ; and, if two or more of these quantities are to be found, of the same kind, they must be united into one simple quantity, or term, as above. Example 1.—Add a, b, c, d, together; then a+b+c+d=S. Example 2.- Add a, b, b, c, b, dx, ey, together; a+3b+c+dx+ey = S. 99. To add quantities together which have different signs: Join all the quantities into one expression for the sum; observing to prefix the same sign to each quantity that it had before the whole were united. Example.—Add a,- bx, cd,− 2 bx, together; then, a + cd − 3 bx = S. 100. To add Compound Quantities together: Connect all the quantities, in the several parts, to be added into one expression, giving each quantity the same sign that it had before, in each separate part, and observing to unite such terms as may be found of the same kind. Example.-Add 5 bx +4, 5 ab-, 2 bx - 3 ab+3ex, together. The answer will be 7bx+2ab++3ex = S. 4 c c SUBTRACTION OF ALGEBRA. 101. To subtract one simple quantity from another: Join the quantity to be subtracted to that from which the subtraction is to be made, with a different sign to the original one. Let the difference be put equal to D. Example 1.-Subtract n from m; then m-n=D. Example 2.-Subtract 3 ab from 7ab; then 7ab-3ab4ab= D. 102. To subtract a compound quantity either from a simple or compound quantity. Subjoin the terms of the quantity to be taken away, with their signs changed, to that from which they are to be taken; observing that, when two terms are of the same kind, they must be united. Ex. From xy+4b-3c, subtract bx-5b+4c; then xy+9b-7c – bx = D. It is evident that changing the signs of the terms of the quantity to be taken away cannot affect its aggregate or value, considered independently of its signs. For, if they are of different kinds, the difference must be the same after the change as before it took place. Thus, let 5-2+7-3, be a quantity to be subtracted. Hence we see the reason for changing the signs of the quantity to be subtracted. MULTIPLICATION OF ALGEBRA. 103. To find the algebraic product of two compound factors, or of one simple and the other compound. If one of the factors be a simple quantity, let that factor be made the multiplier; then join the multiplier to every term of the multiplicand, and prefix the sign + to each product when its factors have like signs; but prefix the sign — when its factors have unlike signs; then the sum of all the products is the total product. If the multiplier consist of more than one term, proceed with every term of the multiplier in the same manner as if it had but one term; then the sum of all the simple products is the whole product; observing that, all such simple products, as have a common factor, may be united together. Example 1.-Multiply a + b by a +b. Operation... a + b a + b a2 + ab + ab + b2 a2 +2ab+b2 = (a + b)2 104. So that the square of any binomial consists of the square of each part and twice their product. Hence, if we see such a quantity as a2+2ax + a2, we shall immediately know that it is the square of the binomial x + a. |