a cal sense, whereas by the determinateness of symbolic division is, indeed, not meant actual numerical determinateness, but "symbolical determinateness," that is conformity to Law XI, 163, regarded merely as a symbolic statement. For from the present standpoint the fraction is a mere symbol which does not have numerical meaning apart from the equation () b=a, with which, therefore, the property of numerical determinateness has no possible connection. The same is true of the sum or difference, product of two fractions, and of the quotient of one fraction by another. As for symbolic determinateness, it needs no justification when assumed, as in the case of the fraction and the demonstration 1, 2, 3, 4 cited above, of symbols whose definitions do not exclude it. The deduction, for example, that because which depends on this principle of symbolic determinateness, is of exactly the same nature as the inference that Both are which depends on the associative and commutative laws. pure assumptions made of the undefined symbol for the purpose of giving it a definition identical in form with that of the product of two numerical quotients. 75. The Vanishing of a Product. It has already been shown that the sufficient condition for the vanishing of a product is the vanishing of one of its factors (2239, 3; 41, 7; 63, 1). It follows from the determinateness of division that this is also a necessary condition. If a product vanishes, one of its factors must vanish. Thus: 0, where a and b may represent numbers or any of the symbols which have been considered. Let ab = CHAPTER VI APPLICATIONS OF THE FUNDAMENTAL OPERATIONS SIMPLE EQUATIONS 76. The results learned in addition, subtraction, multiplication, and division, can now be applied to the solution of some simple examples and problems. 77. When two algebraic expressions are connected by the sign of equality, the whole expression thus formed is called an equation (24). The expression on the left of the sign of equality is called the first, and that on the right, the second member of the equation. 78. An Identity is an abbreviated term for an identical equation. An identical equation is one in which the first and second members are equal for all numbers which the letters may represent; for example, (x+a) (x-a) = x2 — a2. The symbol is read identical with. 79. An Equation of Condition is one which is true, not for all values of the letters, but only for a certain definite number of values of the letters; thus, can not be true unless x = 6. x+3=9 On account of its frequent use, an equation of condition involving only the first power of the unknown quantity is called a simple equation. Here the question always is: "What number must x be in order that," say, x+a=b? 80. The Unknown Quantity in an equation is a letter to which a particular value or values must be given in order that the equation may be true. Such a particular value of the unknown quantity is said to satisfy the equation, and is called a root of the equation; thus, 7 is the value which must be given to x in order that the equation X- -2=5 may be satisfied. To solve an equation is to determine the particular value of the unknown quantity for which the equation is satisfied or is an identity. Up to this point four operations have been dealt with namely, addition, subtraction, multiplication, and division. 81. In the discussions which follow, certain propositions are needed which are obvious axioms in Arithmetic and which are still true when the extended meanings in Algebra are given to their terms and symbols; thus, 1. If equal quantities are added to equal quantities, their sums will be equal. (Law VII, (38) 2. If equal quantities are taken from equal quantities, the remainders will be equal. Thus, if x+3= 8, then taking 3 from each of these equal quantities will leave x = 5. (Law VII, (38) 3. If equal quantities are multiplied by the same or equal quantities, the products will be equal. Thus, if 52 +3, then 5×9 = 45; and if a = b, then a" = b", √ã = vb. (2+3) 9: = 18+27 = (Law XI, 62) 4. If equal quantities are divided by the same or equal quantities, their quotients will be equal. (Law XI, 62) 5. If the same quantity is added to and then subtracted from another, the value of the latter will not be altered. (See (38, 3) 6. If a quantity is both multiplied and divided by the same quantity, its value will not be altered. (63, Definition IX) 7. Quantities which are equal to the same quantity are equal to each other. 8. General Axiom.-If the same operation is performed on two equal quantities, the results will be equal. = REMARK.-The student should note that these axioms are true whether the quantities are positive or negative, and when the four fundamental operations have their extended meanings. For example, if l=m, n = p, then in mp, which is evident if the quantities are all positive quantities. Suppose that n is a negative quantity, say -a; then p is a negative quantity, since n = p; and we shall represent p by -b. We have: 82. The axioms can be used to establish some simple rules for solving the simple equations of the first degree. 1. Any quantity may be transferred from one member of an equation to the other by changing its sign. Thus, suppose Here, 3 has been removed from the first member of equation (1) and in its stead +3 appears in the second member of (2). In equation (1), +b has been removed from the second member and in its stead b appears in the first member of (2). 2. If the sign in every term of an equation is changed, the equality still holds. This rule can be proved by the preceding section. It will be noticed that the signs of the corresponding terms in equations (1) and (2) are opposite. 83. The unknown quantities of an equation are usually represented by the last letters of the alphabet x, y, z, u, etc., and the known, by the digits or the first letters of the alphabet, thus, 1, 3, 5, 6, 7, a, b, c, d, e, etc. The following is a rule for the solution of a simple equation: Transpose all the terms which involve the unknown quantities to the first and the known quantities to the second member of the equation; divide both sides by the coefficient, or by the sum of the coefficients of the unknown quantity. The principles just established justify this rule. 8x 29 263x. |