the Graphical Representation of Functions, Arts. 682 to 688; Art. 689; the solution of Cubic and Biquadratic Equations, Arts. 706 to 716; Art. 718. In Appendix I. will be found graphical demonstrations of the fundamental laws of Algebra for pure imaginary and complex numbers; and in Appendix II., Cauchy's proof that every equation has a root. BOSTON, 1890. WEBSTER WELLS. ANSWERS TO THE EXAMPLES. ALGEBRA. I. DEFINITIONS AND NOTATION. 1. Algebra is that branch of mathematics in which the relations of numbers are investigated by representing them by symbols which may have any values whatever. Note. It is customary in Algebra to use the word "quantity" as ynonymous with "number"; and this meaning will be attached to he word throughout the present work. 2. The Symbols used in Algebra are of four kinds : SYMBOLS OF NUMBER. 3 The symbols of number usually employed are the Arac numerals, and the letters of the alphabet. Tl numerals are used to denote known or determinate numlrs, and the letters any numbers whatever, known or unknyn. Not For the sake of brevity, we shall often speak of "the number a," hen we mean "the number represented by the symbol a." 4. Nmbers occupying similar relations in the same problem areften represented by the same letter, distinguished by diffent accents; as a', a", a'', read "a prime," "a second," "chird," etc. 1 They may also be distinguished by different subscript numbers; as a1, a2, a3, read "a one," "a two," "a three," etc. SYMBOLS OF OPERATION. 5. The Sign of Addition, +, is read "plus." Thus, a + b signifies that the number b is to be added to the number a; a+b+csignifies that b is to be added to a, and then c added to the result; and so on. If two or more numbers are added together, the result is called their sum. 6. The Sign of Subtraction, -, is read "minus." Thus, a - b signifies that the number b is to be subtracted from the number a. Note. The sign ~ is used to denote the difference of the numbes between which it is placed; thus, a~b signifies the difference of the numbers a and b. 7. The Sign of Multiplication, X, is read "times," "int," or "multiplied by." Thus, a × b signifies that the number a is to be mutiplied by the number b; axbx c signifies that a is to be multiplied by b, and the result multiplied by c; and son. The multiplicand is the number to be multiplied. The multiplier is the number by which the multipland is to be multiplied. If two or more numbers are multiplied together, theesult is called their product. The sign of multiplication is usually omitted in Agebra except between two numbers expressed in Arabic nuerals; the multiplication of numbers is therefore indicate by the absence of any sign between them. Thus, 2 ab signifies 2 × a × b. A point is often used instead of the sign x; the, 2.3.4 is the same as 2 x 3 x 4. |