AN INTRODUCTION TO ALGEBRA. CHAPTER I. DEFINITIONS AND PRELIMINARY RULES. 1. ALGEBRA is a science in which the letters of the alphabet, as well as the figures of arithmetic, are employed as the symbols or representatives of quantity, and in which certain signs or marks are used to indicate numerical operations. In common arithmetic, one, two, three, &c. are represented by the marks or symbols 1, 2, 3, &c. and by no others; so that each of these marks or symbols conveys the same idea to every body's mind: the symbol 5 always stands for five, 7 for seven, 8 for eight, and so on. The notation of arithmetic, that is, the system of characters used in it, is thus a notation of fixed and unvarying meaning: all have agreed that the symbol, or mark, or character 9, shall represent nine, and nothing else; that 4 shall stand for four, and for nothing else, and so on. Now the demands of mathematical investigation require that, in addition to a system of symbols thus fixed in meaning, we should also have another system entirely free from all such restriction; so that we may be at liberty at any time to take any symbol of the latter system, and make use of it as the representative of whatever we please. In other words, the demands of mathematical science require that we should have a system of symbols of fixed interpretation-the symbols of arithmetic and a system of symbols of arbitrary interpretation-the symbols of algebra, which, as just noticed, are no other than the letters of the alphabet. The important advantages of this latter system cannot as yet be fully explained to the learner. One of these advantages, B however, he may even now form some conception of; it is this, namely, that by aid of the symbols of algebra, we may represent to the eye quantities of which the actual values are really unknown to us; a thing that the characters of arithmetic could never enable us to do. The answer to an arithmetical question for instance, could not be represented to the eye, by aid of the notation of that science, till the question was worked out and the answer actually discovered; but by the notation of algebra we might represent the unknown answer at once, as we might take, from that notation an x, or a y, or a z, or any other letter, to stand for it: a privilege of very great consequence in the solution of questions. Moreover, as algebra teaches us how to perform operations with these letters, just as arithmetic teaches us how to perform operations with common figures, we have the means of arriving at important conclusions respecting quantities of whose individual or particular values we may be ignorant, or rather at conclusions which are true universally, or for all values of the symbols concerned: another thing beyond the powers of arithmetic, because the symbols of this science are all of particular interpretation. The first thing that a learner has to do, in commencing the study of algebra, is to acquire the ability of performing those operations with letters, which, in arithmetic, he has been accustomed to perform with figures. In arithmetic, he always knows both the operations and the things operated upon; in algebra, with the things operated upon he has, at first, no concern, the operations alone are all that he has to attend to. In order to carry on these, it is not necessary that he should know what the symbols he is dealing with represent; yet beginners often perplex themselves on this head, require satisfaction as to the nature or value of the quantities with which they have to deal, and imagine a difficulty in having to add together a set of things to which they can attach no meaning. But a practical algebra, analagous to this, is in every-day operation in the commercial affairs of life; and no schoolboy, if placed in a merchant's warehouse, would think it a hard task if required to enumerate or collect into one sum all the bales or packages stamped with a particular mark or symbol, an x for instance, and to collect into another sum all marked with a different symbol, as the symbol y, and so on; he would surely not require that the bales should be opened and the contents shown to him before he could perform the operation of adding those bearing distinct marks into so many distinct sums, he would at once see that the interpretation of the marks or hieroglyphics on the bales could have nothing at all to do with the operation in which he was engaged. It is just so with the operations of algebra: these may all be learnt, and expertly performed, in reference to things the nature of which we know nothing. Learners are also apt to fancy that processes performed with letters are more difficult than the corresponding processes with figures: the contrary is however the case; and in going through this little work, it will be discovered that the difficulty or labour of any algebraic operation will almost always be proportional to the merely arithmetical work involved in it; for as already observed, algebra combines in its processes both figures and letters: the fewer the figures the easier, in general, will the operations be. As before remarked, there are certain signs for these operations: so that the notation of algebra consists of signs of operation and signs of quantity: the latter, as already stated, are arbitrary, the former are fixed and settled by common consent: the principal of them are those which follow. DEFINITIONS. 2. The sign+, called plus, indicates the operation of addition, implying that the quantity before which it is written is to be added. Thus 2+4, which is read two plus four, means that 4 is to be added to 2; and, in like manner, a+b, which is read a plus b, means that b is to be added to a; or, to be more explicit, it means that the quantity or figure, represented by b, is to be added to that represented by a. 3. The sign called minus, indicates subtraction; and implies that the quantity before which it stands is to be subtracted. Thus 6-2, which is read six minus two, means that 2 is to be subtracted from 6; and in like manner, a-b, which is read a minus b, means that b is to be subtracted from a; observing that when we thus speak of the addition or subtraction of letters, we in reality refer to whatever those letters may represent. So long as the things represented by the letters a and b are unknown to us we cannot perform the operations implied in the expressions a+b, and a-b: we can do no more than thus indicate them. If a were known to represent 6 and b, 4, then the operations indicated could be actually executed: the result of the operation a+b would be 10, and the result of the operation a-b would be 2. The result of the operation implied in 5a+ 3a, which is read 5 times a plus 3 times a, could however be written down, without the value or interpretation of a being known: it would be 8a, because 3 things, of whatever kind, added to 5 things of the same kind, make 8 of those things; and in like manner, 5a-3a would be 2a. It is scarcely necessary to inform the learner, that in every problem or operation of algebra the same letter always represents the same thing throughout the process in hand. 4. The sign- placed between two quantities denotes the difference of those quantities, or the remainder which arises from subtracting the less from the greater: thus, 4-2 or 2-4 equally mean the difference between 4 and 2, that is 2; and a b, or ba, means in like manner the difference between a and b. NOTE. To avoid an unnecessary expenditure of words it is usual to call the letters employed in algebra quantities, although in reality they are only the representatives of quantities; and algebraists always speak of these symbols as if they were really the things for which they stand. 5. The sign placed between two quantities implies that those quantities are to be multiplied together: it is called the sign of multiplication: thus 4x3 or 3x4 means that 3 and 4 are to be multiplied together, and a×b or b× a means that a and b are to be multiplied together. We can actually perform the operation thus indicated only when the numerical value of the quantity taken for the multiplier is known. Instead of the sign x between the quantities to be multiplied together, a dot merely is often used for the same thing: thus a.b equally means the product of a and b, and still more frequently the letters or factors are written close together without any interposed sign at all, so that a× b, a.b and ab all mean the same thing: the product of a and b. In reading either of these forms we may say either a multiplied by b, or a into b, or the product of a and b. The term factors, which we have just employed is the name given, both in arithmetic and algebra, to all quantities which are multiplied together, thus if 2, 3, and 5, are multiplied together, the product will be 30; and 2, 3 and 5 are said to be factors of 30: so also are 10 and 3; 6 and 5; and 15 and 2. 6. The sign placed before a quantity supplies the place of the words divided by; so that ab means a divided by b: but the operation of division is also indicated, and indeed more frequently, by placing the divisor under the dividend with a α horizontal line between them: thus: is the same as ab. 7. The sign means equal to: it is called the sign of equality: thus 3+5=8, means that 3 plus 5 are equal to 8: and 4b+6b =106, means that 4 times b plus 6 times b are equal to 10 times b. 8. The figure which is thus prefixed to a letter, or to any algebraic expression, as a multiplier, is called the coefficient of that letter or expression: thus 4 is the coefficient of b in the quantity 4b; and 10 is the coefficient of b in 10b. In like manner 7 is the coefficient of xyz in the quantity 7xyz. 9. A quantity which, like each of the quantities just considered, is not separated into parts by the intervention of a plus or minus sign, is called a simple quantity; thus each of the following is a simple quantity: it is said to consist of but one term; the component parts into which algebraic expressions are separated by plus and minus signs being called terms. 10. A quantity or expression consisting of two or more terms, or simple quantities, is called a compound quantity: the following are compound quantities: a+b, 3z-2y, 5x+3y-2 ab, 4 abx-3c+2d, &c. The learner should early acquire the power of reading such compound expressions as these. Agreeably to the foregoing directions the expressions above would be read as follows:a plus b; 3 times z minus twice y; 5 times x plus 3 times y, minus twice a into b; 4 times a into b into x, minus 3 times c, plus twice d, &c. But this mode is unnecessarily formal, and an algebraist would read the expressions thus: - a plus b, 3 z minus 2 y, the words "times" and "twice" being omitted: 5x plus 3y, minus 2 a,b; 4 a, b, x, minus 3c, plus 2d; &c. It would be well, too, if the beginner were to give numerical interpretations to the letters in algebraic expressions like the above, that is, if he were to assume figures, any he pleases, |