To find a number which, subtracted from 8, gives a differ ence of 5; an enunciation which differs from the former only in this, that we put subtract for add, and difference for sum. If we wish to solve this new question directly, we shall have Whence a-nb 8-x=5. x=8-5, or x=3. (123.) For another example, take Problem 50, page 77 The age of the father being represented by a, and that of the son by b; then will represent the number of years beN- - 1 of the father will be n times that of the son. fore the age Thus, suppose Then That is, the father having lived 54 years and the son 9, in 6 years more the father will be 60 years old and the son 15. But 60 is 4 times 15; hence this value, x=6, satisfies the enunciation of the problem. Again, suppose a=45, b=15, and n=4. Here again we obtain a negative solution. How are we to interpret it? By referring to the problem, we see that the age of the son is already more than one fourth that of the father, so that the time required is already past by five years. The value of x just obtained, taken with a contrary sign, satisfies the following enunciation: A father is 45 years old, his son 15; how many years since the age of the father was four times that of his son? The equation corresponding to this new enunciation is 45-x 15-x= 4 Whence 60-4x=45-x; and x=5. (124.) Reasoning from analogy, we deduce the following general principles: 1. Every negative value found for the unknown quantity in a problem of the first degree, indicates an absurdity in the conditions of the problem, or at least in its algebraic statement. 2. This value, taken with a contrary sign, may be regarded as the answer to a problem, whose enunciation only differs from that of the proposed problem in this, that certain quantities which were ADDED should have been SUBTRACTED, and reciprocally. (125.) In what case would the value of the unknown quantity in Prob. 20, page 72, be negative? Ans. When n>m. Thus, let Then m=20, n=25, and a 50 miles. x= 60 60 To interpret this result, observe that it is impossible that the second train, which moves the slowest, should overtake the first. At the time of starting, the distance between them was 60 miles, and every subsequent hour the distance increases. If, however, we suppose the two trains to have been moving uniformly along an endless road, it is obvious that at some former time they must have been together. This negative solution then shows an absurdity in the conditions of the problem. The problem should have been stated thus: Two trains of cars, 60 miles apart, are moving in the same direction, the forward one 25 miles per hour, the other 20. How long since they were together? To solve this problem, let x = the required number of hours the distance traveled by the first train, Then 25x 20x = And since they are now 60 miles apart, 25x=20x+60. second train. We thus obtain a positive value of x. In order to include both of these cases in the same enuncia tion, the question should have been asked, Required the time of their being together, leaving it uncertain whether the time was past or future. In what case would the value of one of the unknown quan tities in Problem 34, page 74, be negative? Why should it be negative and how could the enunciation be corrected for this case? In what case would the value of one of the unknown quantities in Problem 4, page 67, be negative? 0 (126.) III. Values of the form of zero, or Α In what case would the value of the unknown quantity in Problem 20, page 72, become zero, and what would this value signify? Ans. This value becomes zero when a=0, which signifies that the two trains are together at the outset. In what case would the value of the unknown quantity in Problem 50, page 77, become zero, and what would this value signify? Ans. When a=nb, which signifies that the age of the fa ther is now n times that of the son. In what case would the values of the unknown quantities in Problem 38, page 75, become zero, and what would this signify? When a problem gives zero for the value of the unknown quantity, this value is sometimes applicable to the problem, and sometimes it indicates an impossibility in the proposed question. (127.) IV. Values of the form of 4. A In what case does the value of the unknown quantity m Problem 20, page 72, reduce to? and how shall we interpret this result? 0 Ans. When m=n. On referring to the enunciation of the problem, we see that it is absolutely impossible to satisfy it; that is, there can be no point of meeting, for the two trains being separated by the distance a, and moving equally fast, will always continue at a the same distance from each other. The result may then be regarded as indicating an impossibility. α The symbol is sometimes employed to represent infinity ; 0 and for the following reason: When the difference m-n, without being absolutely nothing, α is very small, the quotient is very large. m-n Hence, if the difference in the rates of motion is not zero, the two trains must meet, and the time will become greater and greater as this difference is diminished. If, then, we suppose this difference less than any assignable quantity, the time a m-n represented by will be greater than any assignable quantity, or infinite. A Hence we infer, that every expression of the form found O' for the unknown quantity, indicates the impossibility of satisfying the problem, at least in finite numbers. In what case would the value of the unknown quantity in Problem 10, page 70, reduce to the form 4? and how shall we interpret this result? (128.) V. Values of the form of In what case does the value of the unknown quantity in Problem 20, page 72, reduce to? and how shall we interpret this result? Ans. When a=0, and m=n. To interpret this result, let us recur to the enunciation, and observe that, since a is zero, both trains start from the same point; and since they both travel at the same rate, they will always remain together, and therefore the required point of meeting will be any where in the road traveled over. G Th problem, then, is entirely indeterminate, or admits of an infinite 0 number of solutions, and the expression may represent any finite quantity. 0 We infer, therefore, that an expression of the form found for the unknown quantity, generally indicates that it may have any value whatever. In some cases, however, this value is subject to limitations. In what case would the values of the unknown quantities in ? and how would they satisfy Problem 44, page 76, reduce to the conditions of the problem? 0 0 Ans, When a=b=c, B which indicates that the coins are all of the same value. might therefore be paid in either kind of coin; but there is a limitation, viz., that the value of the coins must be one dollar. In what case do the values of the unknown quantities in Problem 38, page 75, reduce to? and how shall we interpret this result? OF ZERO AND INFINITY. (129.) From Art. 127, it is seen that in Algebra we sometimes have occasion to consider infinite quantities. It is necessary, therefore, to establish some general principles respecting them. An infinite quantity is one which exceeds any assignable limit. It is often expressed by the character. Thus, a line produced beyond any assignable limit is said to be of infinite length. A surface indefinitely extended, and also a solid of indefinite extent in any one of its three dimensions, are examples of infinity. An infinite quantity does not mean an infinite number of terms. Thus, the fraction reduced to a decimal, is .333333 &c., without end, but the value of this series is less than unity. Infinite quantities are not all equal among themselves. |