Therefore, the product of two negative quantities is equal to plus the product of the quantities; i. e., in multiplication minus times minus gives plus. 42. Limitation of Numerical Subtraction.-Equations 1-5, 238, show that subtraction conforms to the same general laws as ad. dition, and therefore it might seem perfectly possible to interchange the rôle of direct and inverse operations. This interchange, however, is seen to be impossible upon examination of these equations. The requirement that the minuend must be greater than the subtrahend sets a comparatively narrow limit to the field of subtraction, making its range much narrower than that of addition. This limitation restricts the use of equations 1-5 of 238 to particular classes of values. For example, such a simple inference as 2a-(2a+3b)+5b = 2b does not hold since 2a+3b > 2a. The use of subtraction as so far defined in any reckoning with symbols must be regarded as unwarranted unless the relative values of the symbols are known. Accordingly the question arises, how is this limitation upon subtraction to be removed? This question is answered in 43, 44, 45. 43. Symbolic Equations.-Definition VI, 36, that is, the equation (a - b) + b = a, as has been seen, is sufficient to define subtraction when a > b. Moreover, (a - b)+b=a, according to definition, only when a - b is a number as defined (236, VI, and 42). However, an equation can be defined in a broader sense. An equation is any declaration of the equivalence of a definite combination of symbols; i. e. one of the combinations may be substituted for the other, and accordingly (a - b) + b = a, may be an equation whatever the values of a and b. b, except that it is a Now if no other meaning is attached to a symbol such that associated with b in the expression (a - b) + b it is equal to a, then the equation (ab) + b = a is a definition of the symbol (a - b). This symbol is not numerical, but purely symbolical. The sign can indicate numerical addition only in case the symbols which it connects represent numbers. 44. Principle of Permanence.--The assumption of the permanance of form of the equation (a−b) + b = a, which is the result of the definition of subtraction, gives at once a symbolic definition of subtraction which is to hold for all values of a and b. The symbolic definition is more general than the definition of numerical subtraction, which is the particular case of the symbolic definition when a and b are numbers and a > b. From the point of view of symbolic subtraction, it is irrelevant whether (ab) is a number or not; only such properties can be attributed to (ab), considered by itself, as follow directly from the generalized equation (a − b) + b = a. Similarly, each of the fundamental laws, I-V, 26, 7, VII, 237, as soon as it fails to be interpreted numerically, becomes, on the assumption of the permanence of its form, a mere declaration of the equivalence of certain particular combinations of symbols. Thus, equations 1-5, 38, become definitions of symbolic addition, subtraction, multiplication, and their combinations. The symbols a, b, etc., are purely symbolic and are unrestricted as to meaning. 45. Some illustrations (27-34) of the increased power gained by considering a, b, etc., as symbols merely have been met with already and many will occur later. In 239 and 40, the introduction of zero and the negative number are the immediate consequences of symbolic definition of subtraction. They greatly increase the simplicity, scope, and power of the operations of Algebra. 46. Review. It is profitable at this point to review the nature of the argument which has been developed in this and the preceding chapter. 1. The associative and commutative laws (Laws I, II, 6) of addition and subtraction, and the determinateness of subtraction (Law VII, 237) followed directly from the definitions of the positive integer, and the operations of addition and subtraction. 2. The result of subtracting b from a, namely ab, is uniquely defined by the equation (ab)+ba for all values of a and b. This assumption led to the definitions of the two symbols 0 and -d, zero and the negative number (39, 40). 3. From the assumption of the permanence of the Laws I-V and VII, were derived the definitions of addition, subtraction, and multiplication of the symbols 0 and d (see 41), and as has been shown in 41, these assumptions were sufficient to determine the meaning of these operations without ambiguity. 4. The Laws I–V, VII, and Definition VI were derived from the properties of numbers and the definitions of their fundamental operations; on the contrary, in the case of the symbols 0 and -d, their characteristics and the definitions of their operations were derived from Laws I-V, VII, and Definition VI. 5. With the introduction of the negative, the character of Arithmetic undergoes a decided change, which gives rise to a symbolic Arithmetic or Algebra. Arithmetic is already in a sense symbolic, since equations and inequalities involving letters as symbols for numbers are used in arithmetical investigations. But its equations, symbols, and operations can be interpreted in terms of the realities which give rise to them, namely, the numbers of things in actually existing groups of things. The introduction of the negative cuts off this connection with reality. The negative, (— d), is purely symbolic, because it is a symbol which stands for an operation that can not be effected with groups of things which actually exist. 47. Not only do the symbols and the fundamental operations performed on them lose all reality, but the equation, which is the fundamental instrument in all mathematical calculations, also loses its reality. In its primary definition, the equation is a declaration (4) of the existence of a one-to-one correspondence between two groups of things. With the introduction of the negative, it loses this interpretation and becomes a mere statement regarding two combinations of symbols, that, in any reckoning, one of them may be substituted for the other. 48. Subtraction of Polynomials. EXAMPLE 1. Subtract 7xy - 5 ab+2m2 from 4xy-3 ab +5 n. Changing the sign of each term of the subtrahend (238, 1 and 2,) and adding the result to the minuend, we have 49. It is customary in subtraction, to perform mentally the operation of changing the signs of the subtrahend. The expressions (1 — m) and (p − q) are to be treated as simple numbers in the subtraction. 18. 19. 3a-4b+5c+3d+7e-8ƒ+ g-h−3k-t a+5b+8c- 9d-10e+12ƒ-7 g 75 a-55b199 e 28 d - 23 e-45ƒ-25 g -- 78 h 21. 22. 23. -2x+1y-1 § z −2} u — 1 } v + 4 } p Ja jbjc&d+je-ff + 1 g } a + 1 b − 4 c − } d− } e+} ƒ 0.8a-3.47b-1.73 c+0.05 d-38.7 e- -411x+53 y 5.3 a +0.5b-9} c+ 33 d+7.75e-17} p+2.1 q 32. 15 a-7b+3c-7d-8e+m-7x-2y- 2+4 10a7b-3c+4d+4e-p− x + y +5z-2 33. 73 a-52b-71c+21d-52x+17y+59z+11t 54a-60b+81c+37 d+18x-33y+99 z+ 7 34. 8.37 a 9.49b+8.5c+57.6 d-5.37e-9.07x+0.09 y 3.97 a 9.8 b83c- 3.46 d +2.63 e−0.57 x 8.91 y 44a+9.38b+2.65 c 134d-53.7x0.375 y - 193 z *The student is expected to handle the fractions in Nos. 20-23 from his knowledge of them in Arithmetic. |