A FIRST BOOK OF ALGEBRA CHAPTER I INTRODUCTION THE LANGUAGE OF ALGEBRA. THE EQUATION 1. Algebra, like arithmetic, deals with number. Algebra, however, goes farther than arithmetic, and enables one to solve with readiness problems which cannot be solved by arithmetic. Algebra makes use of the symbols of arithmetic. Thus, 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, are used in algebra to represent numbers in exactly the same way that they are used in arithmetic. In algebra the sign of addition is +; the sign of subtraction is; the sign of multiplication is x, or a period; the sign of division is, /, or the fractional notation. In arithmetic it is customary to use abbreviations for concrete quantities. Thus, instead of the word cent or cents, the letter c. is used. For example, 54 cents is written 54. Instead of the word acre or acres, the letter A. is used. Thus, 5 acres is written 5 A. word ton or tons, the letter T. is used. Instead of the is written 10 T. In the metric system of measures the linear unit is the meter. The abbreviation for meter is m. Thus, 12 meters is written 12 m. A meter is, approxi mately, 39.37 inches. Hence, 12 m. is equivalent to 12 x 39.37 inches. 2. In algebra, symbols are carried farther than in arithmetic. In algebra, a letter of the alphabet may stand for any number whatever. When a number is written before a letter, it indicates multiplication. Thus, 4 a means 4 times a. If a stands for 5, then 4 a stands for 4 times 5. If a stands for 90, then 4 a stands for 4 times 90. Letters written in succession indicate multiplication. Thus, ab means a times b. abc means the continued product of the numbers a, b, and c, that is, a times b, and this result times c. If a = 4, b = 7, and c = abc-4.7.8, or 4 x 7 x 8. 8, then 3. Each of the letters in an indicated product is called a factor of the product. Any one factor is called a coefficient of the number expressed by the remaining factors. Thus, in the indicated product, 9x, 9 is the coefficient of X. In the indicated product, 4 ab, 4 is the coefficient of ab, and 4 a is the coefficient of b. When no coefficient is written, 1 is understood. In the expression x + 3, 1 is the coefficient of x. 4. Algebra uses a language of its own. The sum of two numbers, 7 and 4, is written 7+4. The sum of two The difference of numbers, x and y, is written x+y. The difference b. The quotient of two numbers, a and b, is written a of 56 divided by 11 is written 5. The quotient of a divided by b is written. b a2 is a short method of writing a times a, or aa. a3 is a short method of writing a times a times a, or aaa. at is a short method of writing a times a times a times a, or aaaa. The number indicating how many times a number or letter is taken as a factor is called an exponent. Thus, in a2, 4 is the exponent. It is important that the student distinguish carefully between coefficient and exponent. 4 a means 4 times a. If a 10, 4a= 40, and a4 = 104 10,000. = EXERCISE I 1. If x=7, find the value of 2x, 7x, 11x, 19 x 25x. 2. If x = 11, find the value of 8x, 12 x, 15 x, 24 x, 32 x. 3. If x 8, find the value of 5x, = 9x, 7x, 14 x. 16 a, 20 a, 30 a. a2, 3 a, a3, 5 a. 362, 63, 462, 7b2. 7. If x=5, find the value of x2, x2+x, 2 x2+3 x, x2—x. 8. If a = 4, b = 3, find the value of a+b, a−b, 2 a+b, 3 a-b. 9. If a 4, 67, find the value of ab, 5 ab, = 7 a a2 a3 b'b' b 10. If a = 12, b = 3, find the value of ab2, = 11. If a 6, 61, find the value of a2 + b2, a2 - ab. = 12. If a 7, c=3, find the value of 2 ac, a2+ ac, a2 - ac, ac2. |