ELEMENTARY HIGHER AND ALGEBRA. SECTION I. DEFINITIONS AND NOTATION. (1.) ALGEBRA is that branch of mathematics in which calculations are made by using any arbitrary characters to stand for the quantities or things considered. The characters used are generally letters of the English or Greek alphabets, and the operations to be performed on the quantities are indicated by signs. (2.) The method of representing quantities may be explained as follows, viz., Let any assumed portion of a quantity (or a quantity of the same kind) be denoted by unity (or 1), then if the quantity contains the assumed unit twice, the quantity will be expressed by 2, or if the quantity contains the unit three times, 3 will express the quantity, and so on to any extent; and if for generality we use a to denote the number of times which the quantity contains the unit, then a will stand for the quantity. Similarly, if b denotes the number of times that another quantity contains the same unit, then b will stand for the other quantity, and so on. All those quantities that can be expressed in terms of the same unit of tity are of the same kind, while quantities that can not be expressed in terms of the same unit of quantity are of different kinds. quan (3.) If the quantity contains the (assumed) unit an exact number of times, the representation of the quantity is inte gral. If the quantity does not contain the unit an exact number of times, then if we suppose the unit to consist of c equal parts, and that the quantity contains one of these parts an exact number of times denoted by b, then if one of these parts is denoted by unity, b will stand for the quantity; but since the first unit contains the second (unit) e times, we must divide b by c, in order to express the quantity in terms of the first unit, as we clearly ought always to do. According to this mode of representation, we shall have what is called a fractional expression of the quantity-wholly fractional if b, called the numerator, is less than e, which is called the denominator—this being a proper fraction. If, however, b is not less than c, the fraction is said to be improper, since it is (clearly) partly integral. (4.) When a finite quantity is represented by a whole number, or by a fraction whose numerator and denominator are finite whole numbers, it is said to be rational; but if the quantity can not be expressed in one of these ways in terms of the assumed unit, it is said to be irrational. (5.) Quantities whose representations in terms of the unit are known, are denoted by some of the first letters, a, b, c, d, etc., of the alphabet; while quantities whose representations in terms of the unit (called unknown quantities) are not known, are denoted by some of the last letters, x, y, z, etc. = (6.) The sign (equal to), when written between any expressions for (numbers or) quantities, denotes that they are equal to each other; and the expressions thus written are called an equation. (7.) The perpendicular cross, + (plus or more), denotes addition; and the horizontal line, (minus or less), denotes subtraction. Thus, 53 denotes that 3 is to be added to 5, or that 5 + 3 = 8, and a + b, or a plus b, means that b is to be added to a. Also, 94, or 9 minus 4, signifies that 4, the number following the sign, must be subtracted from 9, which precedes it; and a c means that c must be subtracted from a. It is to be observed that any quantity which has the sign -- + before it, is called a positive or affirmative quantity; and that any quantity which has no sign before it, is understood to mean the same as if it had the sign + before it, unless the contrary is expressly stated. Also, any quantity preceded by the sign, is called a minus or negative quantity, or (which is the same) it is a quantity to be subtracted; noticing that the sign of subtraction is never omitted, unless the omission is mentioned. Again, it is evident that in subtraction we must suppose the quantity to be subtracted not to be greater than the quantity from which it is to be subtracted. Hence, if we meet with any isolated negative quantity, asa, in any calculation, we may suppose that there is some positive quantity, either expressed or understood, which is not less than the negative quantity, from which the negative quantity is to be subtracted; consequently, if we use — a in our reasonings in the same way that we should do in ba (supposing that a is not greater than b), and obtain a positive result, no absurdity follows from considering — a as a real quantity; but if a negative result is obtained, such that it does not admit of a rational interpretation, it shows that a is not a real quantity, and that the question is absurd, because it requires us to use a as a real quantity, when it is not one. (8.) The oblique cross, × (into), denotes multiplication; the signor (divided by or simply by), when placed between two quantities, denotes that the quantity which precedes it is to be divided by that which follows it. Thus, 6 x 4 means that 6 is to be multiplied by 4, or that 6 x 4 = 24; also, a × b means that the quantity represented by a is to be multiplied by the quantity represented by b; or, more correctly, that the quantity a is to be taken as often as b contains the unit of quantity; for, by the nature of multiplication, if a denotes quantity, b must denote the number of times that a is to be taken, the same as if b were an abstract number; we call a the multiplicand and b the multiplier. It may be observed that multiplication is often expressed by a dot or period placed between the multiplicand and multiplier, and frequently without any sign placed between them, particularly when the quantities to be multiplied are denoted by single letters of the alphabet. Thus, a . b, ab, each signify that a is to be multiplied by b, and of course expresses the same thing as a × b. Again, the expressions ab, a b, mean that a is to be divided by b; we also signify that a is to be divided by b, by the expression, so that a÷b, a: b, and, each mean the same thing; viz., that a is to be divided by b. We observe, further, that the quantity obtained by multiplying any quantities together is called the product of the quantities, and that the quantities which are multiplied together are called factors of the product. If the factors are letters, then they are said to be literal factors; but if they are numbers, the factors are said to be numerical. Each literal factor is said to be of one dimension, and each numerical factor of no dimension; so that a product which involves only one literal factor is said to be of one dimension, a product which has two literal factors is said to be of two dimensions, and so on. If, however, the product has no literal factor, it is said to be of no dimensions. Finally, it may be observed, that any number of letters, when written side by side, after the manner of letters in a word, without any sign between them, always denote that the quantities represented by the letters are to be multiplied by each other, and since the product is independent of the order of the factors, it is best to write them alphabetically, as abcxy, which signifies that the quantities denoted by a, b, c, x, y, are to be multiplied together. (9.) Any quantity is called the first power of itself, or is said to be of the first power; the product obtained from the multiplication of any quantity by itself is called the second power, or square of the quantity; and again, if we multiply the square by the quantity, we obtain a product which is called the third power, or cube of the quantity, and so on. Thus a, aa, aaa, aaaa, are called the first, second, third, and fourth powers of a, and we denote them by a', a2, a3, a1, the small figures being called exponents or indices, which are used merely for brevity. Hence if a quantity has no exponent expressed, it (the exponent) is always understood to be one, and it is plain that a is the same thing as a1, for the ex ponent 1 does not in the least affect the value of the quantity, since it means the same thing as a × 1, or a. (10.) If any quantity is resolved into two equal factors, then either factor is called the second or square root of the quantity; if any quantity is resolved into three equal factors, either factor is called the third or cube root of the quantity, and so on. Hence the second or square root of a quantity is that quantity whose second power or square produces the (original) quantity; and the third or cube root of a quautity is that whose third power or cube produces the (original) quantity, and so on. To express roots, we make use of the character, called the radical or surd sign, written before the quantity whose root is to be taken, with the small figure 2 (as an index) above the sign for the square root, and with the small figure 3 (as an index) above the sign for the third root, and so on. It is customary to omit writing 2 above the sign when the square root is to be taken; thus, Va, Va, Va, denote severally, that the square, cube, and fourth roots of a are to be taken; and a is written to show that the square root of a is to be taken (instead of expressing it by Va), according to the usual practice. Roots are also expressed by the use of vulgar fractions as indices (or exponents) of the quantities whose roots are to be taken, the numerators of the fractions being always one, and their denominators the numbers which denote the roots that are to be taken; thus, the index denotes the square root, the cube root, and so on. denote the second, third, and 14 Hence, a3, Hence, a, a, a, severally fourth roots of a. (11.) The reciprocal of any quantity is unity divided by the quantity; thus, is the reciprocal of 2, that of 3, and the reciprocal of a. It may be observed that for 8 13 1 1 1 1 1 1 a' a2' as' a'' a' J we often write a1, a-3, a-3, a-1, a -, a-, etc. 2 1 etc., (12.) Any quantity that is expressed by letters (with figures when necessary) and signs, is called an algebraic quantity. |