1 who has $15000 more than the first, and gets 2 per cent. more, has a greater income by $1500. What are the three capitals, and the rates of interest? Ans. The capitals are $30000, $40000, and $45000. The rates are 4, 5, Problems which give rise to Negative Results. Theory of Negative Quantities. 58. The use of algebraical signs in the resolution of problems often leads to results which at first view are embarrassing; on reflection, however, it will appear, not only that they are capable of explanation, but that by their means the language of algebra may be still further generalized. Let us take the following problem. To find To find a number which, added to a number b, gives for their sum a number a. Let x be the number required, then evidently b+ x = a, or x = a— - b. This expression or formula gives the value of x, in all particular cases of the problem. Let now a 24, b = 31; then x = 24 31. Since 31 = 24 +7, this expression may be put under the form x = 24 24-7, or, by reduction, x=-7. This value of x is what is called a negative solution. What is the meaning of it? If we return to the statement of the problem, we see that it is impossible that 31 added to any number can give 24, a number less than 31. Therefore no number can solve the question in this case. Nevertheless, if, in the equation of the problem, 31 x 24, we put, instead of the term +x, the negative value 7, it becomes 31 — 7 = 24, a true equation, which amounts to saying that 31 diminished by 7 is 24. x = The negative solution 7 indicates the impossibility of solving the problem in the sense in which it was proposed; but if we consider the solution independently of its sign, that is, x = 7, we may see that it is the solution of the following problem, To find a number which, subtracted from 31, gives 24; which only differs from the first, viz. To find a number which, added to 31, gives 24, in this, that the words added to are supplied by the words subtracted from. The new question, when solved directly, gives the equation Let us take the following problem. A father is a years, his son b years old. In how many years will the son's age be onefourth of his father's? Let x be the number of years, then a +x and b + x represent the ages of the father and son at the end of this number of years; then the equation is, The father being 54 years old, and the son 9, in six years the father will be 60, and the son 15; now, 15 is the fourth of 60; therefore 6 is the answer to the problem. This expression may be reduced to x5 by the ordinary rules. How is this negative result x5 to be explained? If we return to the equation of the problem, we shall find that in this case it becomes 45 4 This contains a manifest contradiction; for the second member is +1; and each of these two parts is less than the correspond 4 ing part of the other member. But if we substitute 5 for x, it an exact equation, which indicates that if, instead of adding to the away 5 years, the age of the son will be ages of the two, we take Thus the solution which has been found, considered independently of its sign, is the solution of the following problem. A father is 45 years old, and his son 15; when was the age of the son one-fourth of that of the father? The equation of this new problem is The least consideration of the problem will show that as the ratio of the ages of the two is 15 become one-fourth of that of the father, but has been so already; because, as has been proved (6), by adding to both terms of a fraction the same number, the fraction is increased in value. On the contrary, it is diminished in value by the subtraction of the same quantity from both terms. 59. We are led by these analogies to the establishment of the following principle. (1.) When the unknown quantity is found to have a negative value, it is indicative of some incorrectness in the manner of stating the question, or, at least, in the equation which is the algebraical translation of it. (See the Remark at the end of this article.) (2.) This value, independently of its sign, may be regarded as the solution of a problem, which differs from the proposed problem in this, that certain quantities, which were additive in the first, are subtractive in the second, and the reverse. Demonstration. The first part of this principle may be easily demonstrated. The finding a negative value for a must arise from our being led, by the nature of the equation, to subtract the greater of two numbers from the less, which is impossible. Thus the values ≈ — — -7, x == - 5, (58) arose from the equations Now, if no absolute number,* when substituted for x, can verify the equation of the problem, after the transformations (43, 45) have been made, it follows that the original equation itself cannot be verified by the substitution of any absolute number for x; for the correctness of the transformations has been shown for all equations which are capable of being verified. An absolute number is one considered without reference to its sign, as in arithmetic. Bour. Alg. 10 Sometimes the impossibility of solving the problem in the way in which it has been put, is evident on the mere inspection either of the enunciation or the equation; the two preceding problems are examples of this. Sometimes it is difficult to discover this impossibility; but it is always made evident in the course of the solution. We now go on to the second part of the principle. x, or Observe, first, that if, in the equation, — be substituted for +x, all the terms containing x, which were additive, are now subtractive, and the reverse. If there be, for example, the term + a x, when x is put instead of x, it becomes + a × In like manner, if we have the term -b it becomes X, or+bx. If this new equation be translated into ordinary language, a new enunciation is obtained, which differs from the first only in having some quantities subtractive which before were additive, and the reverse. It remains to show that the substitution of x in the place of x gives xp, the former result being x-p. (p is here considered as an absolute number). Now, whatever the original equation of the problem may be, we can always reduce it by known transformations to the form ax+b (a and b being absolute numbers). of x in the original equation, the same transformations will reduce Ꮖ It will hence be seen how we are to interpret negative results. When the sign is taken away, they may be regarded as the solutions, not of the questions proposed, but of questions of the same nature, certain conditions of which have been modified; and the surest method of obtaining the new question is to change x into x in the equation of the problem, and translate the result into ordinary language. 60. Remark, The principle which has been established is rigorously true for the equations only, and not always for the enunciations of problems; that is, the enunciation of a problem may be correct, although the resolution of the equation gives. a negative value. The cause of this is, that the algebraist, in the application of his methods to the resolution of a problem, is apt frequently to interpret certain conditions in a sense exactly opposite to that in which they ought to be taken; in this case, the negative solution corrects the effects of the wrong view which he has taken of these conditions. Thus, the equation is false, although the problem is capable of being resolved; and it is only when the equation is a faithful translation of the enunciation and of the meaning of all its conditions, that the principle is applicable to the enunciation. We shall see examples of this in what follows; but it is mostly in the application of Algebra to Geometry that the principle is applicable, not to the enunciations, but to the equations. 61. In the preceding demonstration, we have been led to multiply a by-x, to divide -b by +a, and -b by—a, and the results were obtained by applying to simple quantities the rule of the signs established for the multiplication and division of polynomials. It may appear at first view necessary to demonstrate these rules with reference to insulated simple quantities, and this is what most authors have attempted to do. But the demonstrations which they have given have only the appearance of rigour, and leave much uncertainty in the mind. We say then, that the rule for the signs, established for polynomial quantities, is extended to simple quantities, in order to interpret the peculiar results to which algebraical operations lead. Those who do not admit this extension deprive themselves of one of the principal advantages of algebraical language, which consists in comprehending, under one formula, the solutions of several questions of the same nature, whose enunciations differ only in the way of stating certain conditions, that is, in certain quantities which are additive in the first, being subtractive in the second, and the reverse. The extension to simple quantities of the rules established for polynomials, may appear desirable from the following considerations. The demonstration of article 17 for the multiplication of the binomials ab and c-d, evidently supposes ab and cd. For, if the contrary be the case, the course of reasoning loses all meaning; nevertheless, having once established the rule for the |