FIRST PRINCIPLES.] ALGEBRA something more than mere numbers), is found to be inade quate, taken by itself, to the more difficult cases of mathe matical investigation; and it is therefore necessary, in many inquiries concerning the relations of magnitude, to have recourse to that more general mode of notation, and more extensive system of operations, which constitute the science of algebra. In algebra quantities of every kind may be denoted by any characters whatever, but those commonly used are the letters of the alphabet; and as in the simplest mathe matical problems there are certain magnitudes given, in order to determine other magnitudes which are unknown, the first letters of the alphabet, a, b, c, &c., are used to denote known quantities, while those to be found are represented by v, x, y, &c., the last letters of the alphabet. efinitions 2. The sign + (plus) denotes, in arithmetic, that the [ signs. quantity before which it is placed is to be added to some other quantity. Thus, a + b denotes the sum of a and b ; 3 + 5 denotes the sum of 3 and 5, or 8. The sign - (minus) signifies that the quantity before which it is placed is to be subtracted. Thus, a b de notes the excess of a above b; 6 2 is the excess of 6 above 2, or 4. Quantities which have the sign + prefixed to them are called positive, and such as have the sign - are called negative. When no sign is prefixed to a quantity, + is always understood, or the quantity is to be considered as posi tive. Quantities which have the same sign, either + or , are said to have like signs. Thus, + a and + b have like signs, but + a and - c have unlike signs. 3. A quantity which consists of one term is said to be simple; but if it consist of several terms, connected by the signs + or , it is then said to be compound. Thus, + a and -c are simple quantities; and b + c, and a + b-d, are compound quantities. 4. To denote the product arising from the multiplica tion of quantities, they are either joined together, as if intended to form a word, or else they are connected to gether, with the sign x or . interposed between every two of them. Thus, ab, or a x b, or a . b, denotes the pro duct of a and jb; also abc, or a x b x c, or a . b . c, denotes the product of a, b, and c. If some of the quantities to be multiplied be compound, each of these has a line drawn over it called a vinculum, and the sign x is interposed, as before. Thus, a x c + d x e f denotes that a is to be con sidered as one quantity, the sum of c and d as a second, and the difference between e and / as a third ; and that these three quantities are to be multiplied into one another. Instead of placing a line over such compound quantities as enter a product, we may enclose each of them between two parentheses, so that the last product may be otherwise expressed thus, a(c + d)(e -/); or thus, a x (c + d) x (e -f). A number prefixed to a letter is called a numerical co efficient, and denotes how often that quantity is to be taken. Thus, 3a signifies that a is to be taken three times. When no number is prefixed, the coefficient is understood to be unity. 5. The quotient arising from the division of one quan tity by another is often expressed by placing the dividend 12 above a line, and the divisor below it. Thus, denotes o the quotient arising from the division of 1 2 by 3, or 4 ; - denotes the quotient arising from the division of b by a. 6. The equality of two quantities is expressed by putting the sign between them. Thus, a + b = c-d denotes that the sum of a and b is equal to the excess of c above d. 7. Simple quantities, or the terms of compound quanti ties, are said to be like, which consist of the same letter or letters taken together in the same way. Thus, + ab and 5ab are like quantities, but + ab and + abb are unlike. There are some other characters, such as > for greater than, < for less than, . . for therefore, which will be explained when we have occasion to use them; and in what follows we shall suppose that the operations and no tation of common arithmetic are sufficiently understood. 8. As the science extends itself beyond its original Extensh boundaries, it begins gradually to appear that the limits of defini imposed by these definitions have been transgressed, so tlons - that almost insensibly the symbols have acquired for them selves significations much more comprehensive than those originally attached to them. Thus, were + a to signify a gain of a, a would signify a loss of the same sum ; were + a to signify motion forwards through a feet, a would signify motion backwards through the same space. The extended definitions of + and may now be such as the following : + and are collective symbols of operations the reverse of each other. From similar con siderations to those by which the signification of + and - has been extended, we extend that of x and -7- to something like the following : x and -f- are cumulative symbols of operations the inverse of each other. We may now exhibit the most general definition of the four sym bols in the following form : + and are symbols of operations prefixed to algebraical symbols of quantity, and are such that +a a = + or 0, where + means simply or very nearly increased by ; 0, diminished by 0. x and -f- are symbols of operations prefixed to alge braical symbols of quantity, and are such that x a -j- a = x 1 or -7-1, where x 1 means simply or very nearly multi* plied by 1 ; -7- 1, divided by 1. 9. The laws by which the symbols are combined are the Laws of same as in arithmetic. It is desirable, however, to exhibit cpmMna them. They are three, LAW I. Quantities affected by the signs + and are in no way influenced by the quantities to which they are united by these signs. LAW II. The Distributive Law. Additions and subtrac tions may be performed in any order. LAW III. The Commutative Law. Multiplications and divisions may be performed in any order. We may remark that these laws are assumed for algebra f so that the science is limited by their applicability. Algebra has been extended into the science of quaternions by freeing it from part of the limitation imposed by the third of these laws. In this new science ab is not the same thing as ba. We add a few examples of the substitution of numbers Examplt for letters. (Ex. 3 and 4 involve processes that will be explained later.) Ex. 1. If = 1, 6 = 2, c = 3, find the value of (a + b + c). (a + 2b-c).(b + 2c-a). It is (1 + 2 + 3) . (1 + 4 - 3) . (2 + 6 - 1) = 84. Ex. 2. If a = A, 6 = 1, c = l, x = Q find the value of Ex. 3. With the same data as in example 2, find the , W w V W value of 5 X X" The first term is infinite, and the second is infinitely greater than the first, because x- = xxx . . the answer is CO . Ex. 1 i 4. If # = - = - = 0; find the value of
y *
