ALGEBRA. CHAPTER I. INTRODUCTORY LESSONS. 1. Signs and Symbols in Algebra. In algebra a letter is frequently used to denote a number. When so used in a connected series of operations, a given problem for example, the letter must be regarded as standing for the same number in all the operations of the series. Several distinct letters may be used to denote several distinct numbers in the same problem. In these Introductory Lessons the signs +, -, X, ÷, and =, have the usual significations given to them in arithmetic. Their formal definitions are given in Arts. 12, 13, 14, 15, and 21, of the next chapter. When letters are used to represent numbers, the sign of multiplication is frequently omitted. Thus ab means a x b. 2. The Method of Algebra. The statement of an equality, made by placing the sign between two numbers, or two sets of numbers, indicating that the two numbers, or two sets of numbers, are equal to one another, is called an equation. Thus 2 x = 5 and x + 2 = a + 3 are equations. Problems stated in the form of equations are said to be stated algebraically. In this introductory chapter some of the uses of algebra will be explained by stating and solving, with the help of equations, certain classes of problems already familiar to the learner who has studied arithmetic. In applying the method of algebra, the first step is to translate the ordinary language in which the problem is expressed into a concise symbolical statement, and the form used for this purpose is the equation. The solution of this equation then solves the problem itself. Algebra has no one universal rule that will put every problem expressed in ordinary language into the algebraic form, but rather it devises ways and means for solving the various classes of equations that are produced by dif ferent practical problems. Our first concern is to state the given problem in the form of an equation, and the way to do this must be sought in the language of the problem itself. A few simple examples will best explain how. Ex. 1. If $10 were added to twice the money would be $150. How much money have I? have, the result The specifications of this problem may be expressed more concisely thus: Twice the money I have, added to $10, equals $150, or again, using the signs + and in the ordinary arithmetical sense, = Twice the money I have + $10 = $150. The problem is here stated in the form of an equation, which as yet, however, is not purely algebraic. This analysis will not be in any way disturbed by the replacement of 10 and 150 by any other numbers we choose to insert, and we may analyze with equal facility the following problem, in which a and b stand for any numbers, although we impose, for the present, the condition that b shall be greater than a. Ex. 2. If a dollars were added to twice the money I have, the result would be b dollars. How much money have I ? Stated in the briefer form, this problem says: Twice the number of dollars I have + $a = $b. Ex. 3. The sum of two numbers is 50, and their difference is 20. What are the numbers? The first condition of this problem asserts that The larger number + the smaller number = 50; and since we obviously get the larger number by adding their difference to the smaller, the second condition asserts that 20+ the smaller number = the larger number. Hence, replacing the larger number' in the former of these two statements by its equal 20 + the smaller number,' we obtain 20+ the smaller number + the smaller number or, in briefer form, 20+ twice the smaller number= 50. 50, From this statement the smaller number is at once seen to be 15. Observe that here, as in Ex. 1, the numbers 20 and 50 may be replaced by letters representing any other numbers, without in any way affecting the analysis of the problem, or the form of its solution. These examples point to the fact (they do not prove it) that no matter what the relation of the quantities involved in a determinate problem may be, it can be expressed in the form of an equation. |