ALGEBRA. DEFINITIONS AND NOTATION. 1. ALGEBRA is the art of computing by symbols. It is sometimes also called ANALYSIS; and is a general kind of arithmetic, or universal way of computation. 2. In Algebra, the given, or known, quantities are usually denoted by the first letters of the alphabet, as a, b, c, d, &c. and the unknown, or required quantities, by the last letters, as x, y, z. NOTE. The signs, or characters, explained at the be ginning of Arithmetic, have the same signification in Algebra. And a point is sometimes used for X: thus a+baba+bxa-b. ۱۰ 3. Those quantities, before which the sign + is placed, are called positive, or affirmative; and those, before which the sign is placed, negative. And it is to be observed, that the sign of a negative quantity is never omitted, nor the sign of an affirmative one, ! 1 one, except it be a single quantity, or the first in a series of quantities, then the sign + is frequently omitted: thus a signifies the same as +a, and the series a+bc+d the same as +a+bc+d; so that, if any single quantity, or if the first term in any number of terms, have not a sign before it, then it is always understood to be affirmative. 4. Like signs are either all positive, or all negative; but signs are unlike, when some are positive and others negative. 5. Single, or simple, quantities consist of one term only, as a, b, x. 4 : In multiplying simple quantities, we frequently omit the sign X, and join the letters; thus, ab signifies the same as axb; and abc, the same as aXbXc. And these products, viz. axb, or ab, and abc, are called single or simple quantities, as well as the factors, viz. a, b, c, from which they are produced, and the same is to be observed of the products, arising from the multiplication of any number of simple quantities. 6. If an algebraical quantity consist of two terms, it is called a binomial, as a+b; if of three terms, a trinomial, as a+b+c; if of four terms, a quadrinomial, as a+b+c +d; and if there be more terms, it is called a multinomial, or polynomial; all which are compound quantities. When a compound quantity is to be expressed as multiplied by a simple one, then we place the sign of multiplication between them, and draw a line over the compound quantity only; but when compound quantities are to be represented as multiplied together, then we draw a line over each of them, and connect them with a proper sign. Thus, a+bxc denotes that the compound quantity a+b is multiplied by the simple quantity c ; so that if a were 10, b6, and c 4, then would a+bxc be 10+6X4, or 16 in to 4, which is 64; and a+bxc+d expresses the product of 一番 of the compound quantities atb and c+d multiplied together. 7. When we would express, that one quantity, as a, is greater than another, as b, we write ab, or ab; and if we would express, that a is less than b, we write ab or a b. 8. When we would express the difference between two quantities, as a and b, while it is unknown which is the greater of the two, we write them thus, ab, which denotes the difference of a and b. : 9. Powers of the same quantities or factors are the products of their multiplication: thus axa, or da, denotes the square, or second power, of the quantity represented by a; axaxa, or aaa, expresses the cube, or third power; and axaxaxa, or aaaa, denotes the biquadrate, or fourth power of a, &c. And it is to be observed, that the quantity a is the root of all these powers. Suppose a=5, then will aa=axa= 5×5=25= the square of 5; aaa=axaxa=5×5×5= 125 the cube of 5; and aaaa = axaxaxa = 5×5×5 X5=625= the fourth power of 5. 10. Powers are likewise represented by placing above the root, to the right hand, a figure expressing the number of factors, that produce them. Thus, instead of aa, we write a2; instead of aaa, we write a3; instead of aaaa, we write a1, &c. 2 4 11. These figures, which express the number of factors, that produce powers, are called their indices, or exponents : thus, 2 is the index or exponent of a2; 3 is that of x3; 4 is that of x4, &c. But the exponent of the first power, though generally omitted, is unity, or 1; thus a signifies the same as a, namely, 4 2 namely, the first power of a; axa, the same as axa, or a'f, that is, a, and a2xa is the same as a2xa', ora, or a3. 12. In expressing powers of compound quantities, we usually draw a line over the given quantity, and at the end of the line place the exponent of the power. Thus, a+b denotes the square or second power of a+b, consid ered as one quantity; a+b the third power; a+b the fourth power, &c. And it may be observed, that the quantity a+b, called the first power of a+b, is the root of all these powers. Let a 4 and b=2, then will a+b become 4+2, or 6; and a+b1 =4+2=6=6×6=36, the square of 6; also 3 3 a+b=4+2=63=6×6×6=216, the cube of 6. 13. The division of algebraic quantities is very frequently expressed by writing down the divisor under the dividend with a line between them, in the manner of a vulgar a fraction: thus, - represents the quantity arising by di C viding a by c; so that if a be 144 and 6 4, then will : 144, or 36. And to denotes the quantity aris a -be C ing by dividing a+b by a-c; suppose a=12, b=6 and a 14. These literal expressions, namely, and -, are called algebraic fractions; whereof the upper parts are call a+b ac ed |