XXXIV. Theory of Quadratic Equations Relations of Roots and Coefficients Formation of Equations with Given Roots 256 XXXVI. Arithmetical Progression . XXXVII. Permutations and Combinations XXXIX. Binomial Theorem: Proof for Positive Integral Application when Index is Negative or Frac- Advantages of Common Logarithms. Four-Place Table of Logarithms XLII. Limiting Values and Vanishing Fractions . 345 XLIII. Convergency and Divergency of Series . XLIV. Undetermined Coefficients Expansion of Fractions into Series Formation of Successive Convergents . Recurring Continued Fractions . XLVII. Binomial Theorem : Proof for Any Index XLVIII. Exponential and Logarithmic Series Horner's Method of Synthetic Division Graphical Representation of Functions Solution of Higher Numerical Equations ALGEBRA. CHAPTER I. DEFINITIONS. SUBSTITUTIONS. we say 1. Algebra treats of quantities as in Arithmetic, but with greater generality; for while the quantities used in arithmetical processes are denoted by figures which have one single definite value, algebraical quantities are denoted by symbols which may have any value we choose to assign to them. The symbols employed are letters, usually those of our own alphabet; and, though there is no restriction as to the numerical values a symbol may represent, it is understood that in the same piece of work it keeps the same value throughout. Thus, when “let a=1," we do not mean that a must have the value 1 always, but only in the particular example we are considering. Moreover, we may operate with symbols without assigning to them any particular numerical value at all; indeed it is with such operations that Algebra is chiefly concerned. We begin with the definitions of Algebra, premising that the usual symbols of operation +, -, x, , ( ), will have the same meanings as in Arithmetic. 2. An algebraical expression is a collection of symbols ; it may consist of one or more terms, which are separated from each other by the signs + and Thus 7a+5b – 3c - x + 2y is an expression consisting of five terms. Note. When no sign precedes a term the sign + is understood. 3. Expressions are either simple or compound. A simple expression consists of one term, as 5a. A compound expression consists of two or more terms. Compound expressions may be further distinguished. Thus an expression of two terms, as 30 – 2b, is called a binomial expression; one of three terms, as 2a – 36+c, a trinomial; one of more than three terms a multinomial. 4. When two or more quantities are multiplied together the result is called the product. One important difference between the notation of Arithmetic and Algebra should be here remarked. In Arithmetic the product of 2 and 3 is written 2 x 3, whereas in Algebra the product of a and b may be written in any of the forms a xb, a.b, or ab. The form ab is the most usual. Thus, if a=2, b=3, the product ab=a xb=2x3=6; but in Arithmetic 23 means “twenty-three,” or 2 x 10+3. 5. Each of the quantities multiplied together to form a product is called a factor of the product. Thus 5, a, b are the factors of the product 5ab. 6. When one of the factors of an expression is a numerical quantity, it is called the coefficient of the remaining factors. Thus, in the expression 5ab, 5 is the coefficient. But the word coefficient is also used in a wider sense, and it is sometimes convenient to consider any factor, or factors, of a product as the coefficient of the remaining factors. Thus, in the product 6abc, 6a may be appropriately called the coefficient of bc. A coefficient which is not merely numerical is sometimes called a literal coefficient. Nors. When the coefficient is unity it is usually omitted. Thus we do not write la, but simply a. 7. If a quantity be multiplied by itself any number of times, the product is called a power of that quantity, and is expressed by writing the number of factors to the right of the quantity and above it. Thus axa is called the second power of a, and is written a”; a xa x a...............third power of a, and so on. The number which expresses the power of any quantity is called its index or exponent. Thus 2, 5, 7 are respectively the indices of a?, as, a?. NOTE. a” is usually read “a squared”; q3 is read “ a cubed"; a4 is read “a to the fourth"; and so on. When the index is unity it is omitted. Thus we do not write a?, but simply a. a”; b=5, 8. The beginner must be careful to distinguish between coefficient and index. Example 1. If a=4, 3a = 3 x a=3 x 4=12; but al=a xa xa=4x4x4=64. Example 2. If 462=4 xbxb=4x 5 x 5=100; Şax=ş xa x x= x 6 x 7 = 70. 5.x2 = 5 x xa=5x14=5x1=5. EXAMPLES I. a. If a=7, b ---2, c=1, x=5, y = 3, find the value of 1. 14x. 2. 23. 3. 3ax. 4. a?. 6. 65. 7. 36? 8. 2xa. 9. 6c. 11. Tc, 12. 964, 5. 5ly. 10. 4y 23. a. If a=8, b=5, C= 4, x = 1, y=3, find the value of 13. 3c2. 14. 743 15. 5ab. 16. 9xy. 17. 863 18. 3.0.5. 19. 28. 20. 774 21. Cx. 22. by. 25. 27. br. 28. a. If a=5, b=l, c=6, x=4, find the value of 29.1.x4. 30. Rc3 31. 323. 32. vax. 33. 3. 34. 2C. 35. 8. 36. 7*. 37. Tacx. 38. lbox. 205 39. c. 40. 64 9. When several different quantities are multiplied together a notation similar to that of Art. 7 is adopted. Thus aabbbbcddd is written a-b4cd3. And conversely 7a%cdî has the same meaning as 7 xaxaxaxcxd x d. 1 |