CHAPTER 1. INTRODUCTION. Algebra, like Arithmetic, treats of number. But the meaning of number, and the mode of representing it, are extended in passing from ordinary Arithmetic to Algebra. § 1. GENERAL NUMBER. 1. In ordinary Arithmetic all numbers have particular values and are represented by definite symbols, the Arabic numerals, 1, 2, 3, etc. The symbol 7, for instance, stands for a group of seven units. In Algebra, however, such symbols as a, b, x, y, are used to represent numbers which may have any values whatever, or numbers whose values are, as yet, unknown. Just as we speak of 10 miles, of 95 dollars, etc., in Arithmetic; so in Algebra we speak of a miles, meaning any number of miles or an unknown number of miles; of x dollars, meaning any number or an unknown number of dollars, etc. For the sake of brevity, we shall say the number a, or simply a, meaning thereby the number denoted by the symbol a. 2. The symbols of Arithmetic, 1, 2, 3, etc., are retained in Algebra with their exact arithmetical meanings. The numbers represented by letters are, for the sake of distinction, called Literal or General Numbers. Other symbols than letters might be used to represent general numbers, but letters are more convenient to write and to pronounce. 3. The operations of Addition, Subtraction, Multiplication, and Division are denoted by the same symbols in Algebra as in Arithmetic. 4. The Symbol of Addition, +, read plus, is placed between two numbers to indicate that the number on its right is to be added to the number on its left. E.g., just as 5+3, read five plus three, means that 3 is to be added to 5; so a + b, read a plus b, means that b is to be added to a. 5. The Symbol of Subtraction, -, read minus, is placed between two numbers to indicate that the number on its right is to be subtracted from the number on its left. E.g., just as 5 – 3, read five minus three, means that 3 is to be subtracted from 5; so a b, read a minus b, means that b is to be subtracted from a. In a chain of additions and subtractions the operations are to be performed successively from left to right. E.g., 7+4-3+2=11-3+2=8+2=10. 6. The Symbol of Multiplication, X, read multiplied by, or times, means that the number on its left is to be multiplied by the number on its right. E.g., just as 5 x 3, read five multiplied by three, or three times five, means that 5 is to be multiplied by 3; so a xb, read a multiplied by b, or b times a, means that a is to be multiplied by b. A dot () is frequently used, instead of the symbol ×, to denote multiplication; as ab for a x b. The symbol of multiplication between two literal numbers, or one literal number and an Arabic numeral, is frequently omitted. E.g., the product x × y × z, xyz. The product a x 6, or a 6, is written, a 6. or xyz, is usually written, It will be proved later that a × b = b × a, or ab = ba. On this account the product a 6 is usually written 6 a, the Arabic numeral being placed first. But the symbol of multiplication between two numerals. cannot be omitted without changing the meaning. E.g., if in the indicated multiplication, 3 x 6, or 3.6, the symbol, x, or, were omitted, we should have 36, not 18. |