Particular mode of completing a Square (Art. 99) Special Artifices in resolving Quadratics (Art. 106) Quadratic Equations containing two or more unknown quantities ELEMENTS OF ALGEBRA. INTRODUCTION. DEFINITIONS AND AXIOMS. ALGEBRA is a general kind of arithmetic, an universal analysis, or science of computation by symbols. Quantity or magnitude is a general term applied to everything which admits of increase, diminution, and measurement. The measurement of quantity is accomplished by means of an assumed unit or standard of measure; and the unit must be the same, in kind, as the quantity measured. In measuring length, we apply length, as an inch, a yard, or a mile, &c.; measuring area, we apply area, as a square inch, foot, or acre; in measuring money, a dollar, pound, &c., may be taken for the unit. Numbers represent the repetition of things, and when no application is made, the number is said to be abstract. Thus 5, 13, 200, &c., are numbers, but $5, 13 yards, 200 acres, are quantities. In algebraical expressions, some quantities may be known, others unknown; the known quantities are represented by the first or leading letters of the alphabet, a, b, c, d, &c., and the unknown quantities by the final letters, z, y, x, u, &c. THE SIGNS. (1) The perpendicular cross, thus +, called plus, denotes addition, or a positive value, state, or condition. (2) The horizontal dash, thus —, called minus, denotes subtraction, or a negative value, state, or condition. (3) The diamond cross, thus X, or a point between two quantities, denotes that they are to be multiplied together. (4) A horizontal line with a point above and below, thus ÷, denotes division. Also, two quantities, one above another, as numerator and denominator, thus indicates that a is divided by b. (5) Double horizontal lines, thus, represent equality. Points between terms, thus a b c d, represent proportion, and are read as a is to b so is c to d. 3 4 5 (6) The following sign represents root; alone it signifies. square root. With small figures attached, thus √ √ √, &c., indicates the third, fourth, fifth, &c., root. Roots may also be represented by fractions written over a quantity, as aaa, &c., which indicate the square root, the third root, and fourth root of a.* (7) This symbol, ab, signifies that a is greater than b. ab, signifies that a is less than b. This 66 9 (8) A vinculum or bar or parenthesis () is used to connect several quantities together. Thus a+b+cxx or (a+b+c)x, denotes that a plus b plus c is to be multiplied by x. The bar may be placed vertically, thus, which is the same as (a-d+e) y, or the same as ay-dy-ey without the aly -d +el vinculum. (9) Simple quantities consist of a single term, as a, b, ab, 3x, &c. Compound quantities consist of two or more terms connected by their proper signs, as a+x, 3b+2y, 7ab-3xy+c, &c. A binomial consists of two terms; a trinomial of three; and a polynomial of many, or any number of terms above two. (10) The numeral which stands before a quantity is called its coefficient; thus 3x, 3 is the coefficient of x, and indicates that three x's are taken. Coefficients may be literal, simple, or compound, as well as numeral; thus abx, (a+b) x; (c-d+2) x. Here ab, (a+b) and (c-d+2) may be considered coefficients of x. (11) A measure of any quantity is that by which it can be divided without a remainder. 2 is a measure of 4, or any even * The adoption and utility of this last mode of notation, which ought to be exclusively used, will be explained in a subsequent part of this work. number. or 12ax. 5a is the measure of 20a. 3x is the measure of 12x, A multiple of any quantity is that which is some exact number of times that quantity; thus 12 is a multiple of 3, or of 4, or of 6, and 30ab is a multiple of 3ab, of 5ab, &c. AXIOMS. Axioms are self-evident truths, and of course are above demonstration; no explanation can render them more clear. The following are those applicable to algebra, and are the principles on which the truth of all algebraical operations finally rests: Axiom 1. If the same quantity or equal quantities be added to equal quantities, their sums will be equal. 2. If the same quantity or equal quantities be subtracted from equal quantities, the remainders will be equal. 3. If equal quantities be multiplied into the same, or equal quantities, the products will be equal. 4. If equal quantities be divided by the same, or by equal quantities, the quotients will be equal. 5. If the same quantity be both added to and subtracted from another, the value of the latter will not be altered. 6. If a quantity be both multiplied and divided by another, the value of the former will not be altered. 7. Quantities which are respectively equal to any other quantity are equal to each other. 8. Like roots of equal quantities are equal. 9. Like powers of the same or equal quantities are equal. EXERCISES ON NOTATION. When definite values are given to the letters employed, we can at once determine the value of their combination in any algebraic expression. (4) A horizontal line with a point above and below, thus, denotes division. Also, two quantities, one above another, as numerator and denominator, thus, indicates that a is divided by b. (5) Double horizontal lines, thus =, represent equality. Points between terms, thus ab :: c: d, represent proportion, and are read as a is to b so is c to d. (6) The following sign represents root; alone it signifies square root. With small figures attached, thus 345, &c., indicates the third, fourth, fifth, &c., root. Roots may also be represented by fractions written over a quantity, as aaa, &c., which indicate the square root, the third root, and fourth root of a.* (7) This symbol, a➤b, signifies that a is greater than b. This 66 ab, signifies that a is less than b. (8) A vinculum or bar, or parenthesis () is used to connect several quantities together. Thus a+b+cxx or (a+b+c)x, denotes that a plus 6 plus c is to be multiplied by x. b aly -d +el vinculum. The bar may be placed vertically, thus, which is the same as (a-d+e) y, or the same as ay-dy+ey without the (9) Simple quantities consist of a single term, as a, b, ab, 3x, &c. Compound quantities consist of two or more terms connected by their proper signs, as a+x, 3b+2y, 7ab—3xy+c,&c. A binomial consists of two terms; a trinomial of three; and a polynomial of many, or any number of terms above two. (10) The numeral which stands before a quantity is called its coefficient; thus 3x, 3 is the coefficient of x, and indicates that three x's are taken. Coefficients may be literal, simple, or compound, as well as numeral; thus abx, (a+b) x; (c-d+2) x. Here ab, (a+b) and (c-d+2) may be considered coefficients of x. (11) A measure of any quantity is that by which it can be divided without a remainder. 2 is a measure of 4, or any even * The adoption and utility of this last mode of notation, which ought to be exclusively used, will be explained in a subsequent part of this work. |