9. Given quantities have their values all known, and are generally expressed by the early letters of the alphabet; as, a, b, c, d, &c. 10. Unknown quantities are those whose values are unknown, and are generally expressed by the final letters of the alphabet; as, x, y, or z. 11. The sign + is read plus, or more. 12. The sign is read minus, or less. 13. Simple quantities are composed of one term only; as, a, b, 6ab, 7ax2, &c. 14. Compound quantities consist of several terms, connected by the sign plus or minus; as, a+b, 7ax—3b, 3ab +b-c, &c. 15. Positive or affirmative quantities are such as have the sign+before them; as, a, + 6xy. 16. Negative quantities are those which have the sign before them; as, a, 6xy. 17. Like signs are all affirmative (+) or all negative (-). 18. Unlike signs are composed of affirmative (+) and negative (-) signs. 19. A binomial quantity consists of two terms, as, a+b; a trinomial of three terms, as, a + b − c; and a quadrinomial of four, as, a + b − c + d, &c. 20. A residual quantity is a binomial, in which one of the terms is negative; as, a— b. 21. A surd or irrational quantity has no exact root, as √ɑ. or Va2, or abt. 22. A rational quantity has no radical sign (✔) or index annexed to it; as, a or ab. : 23. The reciprocal of any quantity is that quantity inverted, or unity divided by it; as, f is, and of is Further Explanation of Algebraic Characters. The signis employed to connect, or add, one quantity with another it is, therefore, the symbol, or sign, of addition. The sign indicates that the number before which it is placed, is to be taken from the number which precedes the symbol. It is, therefore, the sign of subtraction. x or a dot (.) is the sign of multiplication. of division. of proportion. of the square root.* of the cube root. of equality, and read equal to. Thus, a + b shews that the number represented by b is to be added to that represented by a. a- shews that the number represented by b is to be subtracted from that represented by a. Roots are also expressed by fractional indices, as explained in Def. 6. ab represents the difference of a and b when it is not known which is the greater. a xb, or ab, or ab, denotes the product of the numbers represented by a and b, a a+b, or indicates that the number represented by a is to be b divided by that denoted by b. abcd expresses, that a is in the same ratio, or proportion to b, that c is to d. I = a b+c exhibits an equation, shewing that x is equal to the difference of a and b, added to c. (a+b)c, or a+bx c, is the product of the compound quantity a+b multiplied by the simple quantity c; a+b+a-b, or (a+b) + a+b (ab), or — ab is the quotient of a+b divided by a-b; and the bar thus placed over two or more quantities, to connect them, is called a vinculum. a+b- or (a+b-c)' is the cube, or third power, of the quantity att-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, 7x2 + 9xa— 16x2. #2 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. 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 7040. 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 (2 a); 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 proposition, 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 — 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. 2 This rule may be thus illustrated: If it were required to subtract 5 (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 bc were subtracted from a, the remainder would be greater by e than if b only were taken away. Now if+bbe 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 b a positive one, as in the expression just given, we change the signs of both, thus: -b+c, when we would take them from a. |