(13.) When an algebraic quantity is not connected with any other by either of the signs, +, —, it is called a monomial or a term. (14.) A compound quantity is that which has two or more terms, as a + b, ac + bd — ef, etc. A compound quantity which has but two terms is called a binomial, as x+y, ab; the last is sometimes called a residual, since the connecting sign is. A compound quantity that consists of three terms is called a trinomial, one of four terms is a quadrinomial, and one of more than four terms is called a polynomial. Some authors call any algebraic quantity that consists of more than one term a polynomial, or a multinomial. 2 (15.) In order to show that any operation is to be performed on a polynomial, we connect its terms by a right line drawn across the top of it, which right line is called a vinculum or bar; thus, abcd +xy, shows that the polynomial is to be squared; a xp+q-r shows that a is to be multiplied by the polynomial p+q-r; again, m + n − vw + xy x 183 α bc shows that the square root of m + n vw + xy is to be taken, and that the cube root of a bc is also to be taken, and the sign of multiplication shows that the product of these two roots is to be found. It may be remarked that, instead of using the bar, we sometimes put the polynomial within a parenthesis or braces; thus, (a + b) expresses the same thing as a+b; again, the same thing is denoted by {ab} or by [a +b]; the character () being called a parenthesis, and the characters { }, [ ], braces. It may be noted further that the bar is sometimes placed vertically; thus, p, p, mean the same thing as m + n × p. (16.) The coefficient of a quantity is a number or letter which shows how often the quantity is to be taken, and is prefixed to the quantity, between it and its sign. If no coefficient is expressed, it is understood to be 1, or unity. Thus, in 5ab, 5 is the coefficient; and in -Va+b2, is the coefficient of — √a2 + b2. In axy we may, if we please, consider a as the coefficient of xy, and of ab 1 is the coefficient. (17.) Quantities that do not differ from each other, or those which do not differ from each other except in their coefficients and signs, are said to be similar or like quantities. Thus, 6ab, 6ab, 3ab, — 4ab are like quantities; also, 5 √ a2 + b2 -9a+b2 are like quantities. Unlike or dissimilar quantities are those which differ from each other in other respects than their signs and coefficients. Thus, ab, xy, av √ a2 + b2, Va+b2, are dissimilar or unlike quantities. (18.) Quantities that are all plus or all minus are said to have like signs; but if some have the sign plus, and others the sign minus, they are said to have unlike signs. (19.) We shall here add some signs that are occasionally used. The sign ~, when placed between two quantities, is used to express their difference, when it is not known which is the greater; thus, ab signifies the difference of a and b when it is not known which of the two quantities denoted by a and b is the greater. The sign> or <, when placed between two quantities, is used to show that the quantity toward which the opening is turned is larger than the other quantity; thus, a > b is read, a is greater than b; and cd is read, c is less than d; in accordance with the meaning of the sign. The sign is used to signify that the quantity which precedes it is infinite, or that it is of unlimited value; thus, a co is read, a is infinite. The sign... is used for the word therefore, or consequently; and the sign. is used for the word because. Sometimes we use the sign: for, especially in the doctrine of proportion. (20.) To what has been done, it may not be improper to add the following, viz.: A definition is the explanation of the meaning of any term or word. A proposition is something proposed for consideration, and is either self-evident, to be proved, or done. When the proposition is to be proved, it is called a Theorem, and the proof is called a Demonstration. If the proposition requires something to be done, it is called a Problem, and the process of doing the thing required is called the Solution of the Problem. A self-evident proposition is called an Axiom. A self-evident problem is called a Postulate. A proposition that is premised to aid in the demonstration of a Theorem, or the solution of a Problem, is called a Lemma. A Corollary or Consectory is an obvious inference drawn from one or more preceding propositions. A Scholium is a remark made on one or more preceding propositions, for the purpose of explaining their connection, use, etc. I. The same or equivalent operations may be performed on equals. II. Any quantity may be considered as composed of any number of parts, which are equal or unequal, according to the nature of the case, provided they are taken sufficiently small. I. Any quantity is greater than any one of its parts, and is equal to the sum of all its parts. The parts may be interchanged, and that in any manner, without altering the value of the quantity. II. If the same or equivalent operations are performed on equals, the results are equal. Thus, equals added to equals, or subtracted from equals, give equals. Also, equals multiplied or divided by equals, give equal products, or equal quotients. Equal powers or roots of equals give equals. III. Equals added to or subtracted from unequals, give unequals. IV. Things equal to the same or equal things, are equal to each other. V. Two or more algebraic expressions for the same or equal things, are equal or equivalent to each other. VI. Equal quantities, when added with opposite signs, that is, one having the sign (plus) + and the other the sign destroy each other, and give nothing, or zero (0), for the result. VII. If the difference between two quantities is less than any assignable quantity, the quantities are to be considered as equal to each other; the quantities being, of course, of the same kind. SECTION II. ADDITION. (1.) ADDITION Consists in uniting quantities of the same kind, according to their signs, into one sum or whole, and in finding a proper expression for the result. The rule of addition may be separated into three cases. CASE I. (2.) When the quantities are similar and have like signs. RULE. Add the coefficients of the several quantities together, and to the sum annex the common literal quantity, and prefix the common sign. For convenience, the quantities may be placed in a vertical column, so that their coefficients shall stand under each other, after the manner of placing figures for addition in Arithmetic. Then the coefficients may be added in the same way that numbers are added in Arithmetic. Thus, the sum of a, 3a, 11a, 20a, 19a, 2a, is found by placing the quantities under each other, which gives α За 11a 20a 19a 2a whole sum = 56a, which is evidently correct, since a is plainly to be taken as often as there are units in the sum of the coefficients; observing that the coefficient of a is 1. In like manner, to find the sum of 2a,- 5a,- 6a,- 7a, we have for evidently we must take a as often as there are units in the sum of the coefficients. It is to be noted that when no sign is expressed before a quantity, we are always to understand that its sign is supposed to be ; also, when a quantity has no coefficient prefixed, it is always understood to be 1. CASE II. (3.) When the quantities are similar, but have unlike signs. |