ALGEBRA

An important branch of the math sciences, and may be defined to be the method oi ting indeterminate quantities. It is a sort of universal arithmetic, founded on the same principles as common arithmetic, and proceeding by rules and operations precisely similar. But it is not confined merely to questions relating to numbers, being applied generally to investigate the relations that subsist among quantities of all kinds, whether arithmetical or geometrical. The reasoning is carried on by general symbols; and it is to the complete system of notation, which has been introduced by its succe ive cultivators and improvers, that it owes its immense superiority over the ancient analysis. The symbols employed in algebra are of two kinds: those which denote quantities; and those which denote the affections or relations, or properties of quantities, and operations to be performed on them. For representing quantities or magnitudes, the letters of the alphabet are employed. Thus, in the solution of an arithmetical problem,a number maybe represented by the letter a; in geometry, a may represent a line or an angle; in mechanics,aforce. The relations of quantities are expre ed by other conventional symbols. The relation of equality is expre ed by the sign =; thus, to expre that the quantity represented by a is equal to the quantity represented by b, we write a = b. The symbol > or b signifies that a is greater than b, and a < b denotes thai a is le than 6. The two primary operations of which quantities are susceptible, are addition and subtraction, and these are respectively indicated by the symbols + plus, and — minus. For example, a -f b denotes the sum of the two quantities a and b, or that a is to be increased by b; and a — b denotes the difference between a and b, or that a is to be diminished by b. Multiplication is indicated by the symbol Xi °r by simply placing the letters beside each other without an intervening symbol. Thus, in numbers, "X b or a b denote the same thing, namely, the product arising from the multiplication of the number u into b. In geometry, two letters joined together, as a b, denote a rectangular parallelogram, one of the sides of which is represented by a, and Ihe other by b. Division is indicated by -7-; or more frequently by placing one of the numbers above the other in the form of a fraction; thus: 30-i-lO, or SB.. In addition and subtraction, the quantities connected by the appropriate symbols must be homogeneous, or of the same kind; for it is only such quantities that admit of addition or subtraction. Of two quantities connected by the symbol of multiplication, one must nece arily be an abstract number, for a quantity can only be multiplied by a number, or, which is the same thing, added to itself once or twice, or some other number of times. When division is to be performed, the divisor may either be a quantity of the same kind as the dividend, or it may be an abstract number; in the former case, the quotient is an abstract number; in the latter, it is a quantity of the same kind as the dividend. In the multiplication of quantifies, the frequent repetition of the same symbol would become inconvenient; it is usual, therefore, to write Ihe root only once, and to place over it, on the right, the exponent or number indicating the power: thus, a'-* denotes the same thing as a a, or the square of a; aS is the same as aaa, or the cube of a, and a» denotes the nth power of a, or a multiplied by n limes into itself. By analogy, a denotes the square root of a;Symbol.)a* the cube root of a, and so on. (.See Notation, Algebra is in its nature e entially distinct from arithmetic.In arithmetic absolute numbers are given, from which other absolute numbers are required to be determined. But in algebra the symbols that are employed are perfectly general, and may represent any numbers whatever; and the expre ions which result from combining them according to the conditions of the problem, indicate the solution not of a particular question, but of all questions whatever, in which numbers are subjected to the same series of operations. In this manner the general properties of numbers are discovered. For example, the expre ion(a+ b)(a — b), which signifies that the sum of Ihe two numbers a and b is to be multiplied by their difference, becomes, on performing the multiplication, a2 — bi; whence we infer this general or universal truth, namely, that the product of the sum and the difference of anv f.—- [s. 44]