ALGEBRA. INTRODUCTION. - DEFINITIONS AND AXIOMS. ALGEBRA is a general kind of arithmetic, an universal analysis, or science of computation by symbols. 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. চ (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. (6) The following sign represents root ✓, alone it signi fies square root. With small figures annexed thus 3 5 ✔ &c. indicates the third, fourth, fifth, &c. root. Roots may also be represented by fractions written over aat &c. which indicate the square root, a quantity, as a (7) This symbol a>b signifies that a is greater than b. This 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 b plus c is to be multiplied by x. The bar may be placed vertically thus aly -d which is the same as (a-d+e)y, or the same as +eay-dy+ey without the 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 a, and indicates that three a'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 number. 5a is the measure of 20a. 3x is the measure of 12x, or 12ax. 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. 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. 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 rest: 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. 4. If equal quantities be multiplied into the same, or equal quantities, the products 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. Equal roots of equal quantities are equal. 9. Equal powers of the same or equal quantities are equal. EXERCISES ON NOΤΑΤΙΟΝ. When definite values are given to the letters employed, we can at once determine the value of their combination in any algebraic expression. Let a=5 b=20 c=4 d=1 SECTION I. ADDITION. (Art. 1.) Before we can make use of literal or algebraical quantities to aid us in any mathematical investigation, we must not only learn the nature of the quantities expressed, but how to add, subtract, multiply, and divide them, and subsequently learn how to raise them to powers, and extract roots. The pupil has undoubtedly learned in arithmetic, that quantities representing different things cannot be added together; for instance, dollars and yards of cloth cannot be put into one sum, but dollars can be added to dollars, and yards to yards, units can be added to units, tens to tens, &c So in algebra, a can be added to a, making 2a, 3a can be added to 5a making 8a. As a may represent a dollar, then 3a would be 3 dollars, and 5a would be 5 dollars, and the sum would be 8 dollars. Again, a may represent any num ber of dollars as well as one dollar; for example, suppose a to represent 6 dollars, then 3a would be 18 dollars, and 5a would be 30 dollars, and the whole sum would be 48 dollars. Also, 8a is 8 times 6 or 48 dollars; hence any number of a's may be added to any other number of a's by uniting their coefficients, but a cannot be added to b, or 4a to 36, or to any other dissimilar quantity, because it would be adding unlike things, but we can write a+b and 3a+3b, indicating the addition by the sign, making a compound quantity. Let the pupil observe that a broad generality, a wide latitude must be given to the term addition. In algebra, it rather means uniting, condensing, or reducing terms, and in some cases, the sum may appear like difference, owing to the difference of signs. Thus, 4a added to -a is 3a; that is, the quantities united can only make 3a. Again, 76+ 3b-4b, when united, can only give 66, which is in fact the sum of these quantities, as 46 has the minus sign which demands that it should be taken out, hence to add similar quantities we have the following RULE. Add the affirmative coefficients into one sum and the negative ones into another, and take their difference with the sign of the greater, to which prefix the common literal quantity. N. B. Like quantities of whatever kind, whether of powers or roots, may be added together the same as more simple or rational quantities. Thus 3a2 and Saa are 11a2, and 763+363=1063. No matter what the terms may be, if they are only alike in kind. Let the reader observe that 2(a+b)+3(a+b) must be together 5(a+b), that is, 2 times any quantity whatever added to 3 times the same quantity must be 5 times that quantity. Therefore, 4√x+y+3√x+y =7√x+y, for √x+y which represents the square root of x+y, may be considered a single quantity. (Art. 2.) To find the sum of various quantities we have the following RULE. Collect together all those that are alike, by uniting |