Ans. When a nb, which signifies that the age of the father is now n times that of the son. In what case would the values of the unknown quantities in Prob. 38, page 96, become zero, and what would these values 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. 166. 4th. We may obtain for the unknown quantity values of the form of 4. In what case does the value of the unknown quantity in Prob. 20, page 94, reduce to 4, and how shall we interpret this result? 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 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, and for the following reason: If the denominator of a fraction is made to diminish, while the numerator remains unchanged, the value of the fraction must increase. 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 to be less than any assignable quantity, the time α represented by will be greater than any assignable quantity. m-n Hence we infer that every expression of the form 4 found for the unknown quantity indicates the impossibility of satis fying the problem, at least in finite numbers. In what case would the value of the unknown quantity in Prob. 10, page 92, reduce to the form, and how shall we interpret this result? 167. The symbol 0, called zero, is sometimes used to denote the absence of value, and sometimes to denote a quantity less than any assignable value. The symbol ∞, called infinity, is used to denote a quantity greater than any assignable value. A line produced beyond any assignable limit is said to be of infinite length; and time extended beyond any assignable limit is called infinite duration. m We have seen that when the denominator of the fraction -n a becomes less than any assignable quantity, the value of the fraction becomes greater than any assignable quantity. Hence we conclude that %= that is, a finite quantity divided by zero is an expression for înfinity. Also, if the denominator of a fraction be made to increase while the numerator remains unchanged, the value of the fraction must diminish; and when the denominator becomes greater than any assignable quantity, the value of the fraction must become less than any assignable quantity. Hence we conclude that that is, a finite quantity divided by infinity is an expression for zero. 168. 5th. We may obtain for the unknown quantity values of the form of o result? In what case does the value of the unknown quantity in Prob. 20, page 94, reduce too, and how shall we interpret this 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. The problem, then, is entirely indeterminate, or admits of an infinite number of solutions; and the expression may represent any finite quantity. We infer, therefore, that an expression of the form of 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 Prob. 44, page 97, reduce to, and how would they satisfy the conditions of the problem? Ans. When a=b=c, which indicates that the coins are all of the same value. B 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 Prob. 38, page 96, reduce to , and how shall we interpret this result? 169. The expression may be conceived to result from a fraction whose numerator and denominator both diminish simultaneously, but in such a manner as to preserve the same relative value. If both numerator and denominator of a fraction are divided by the same quantity, its value remains unF changed. Hence, if represent any fraction, we may conceive both numerator and denominator to be divided by 10, 100, 1000, etc., until each becomes less than any assignable quantity, or 0. The fraction then reduces to the form of, but the value of the fraction has throughout remained unchanged. For example, we may suppose the numerator to represent the circumference of a circle, and the denominator to represent its diameter. The value of the fraction in this case is known to be 3.1416. If now we suppose the circle to diminish until it becomes a mere point, the circumference and diameter both become zero, but the value of the fraction has throughout remained the same. Hence, in this case, we have =3.1416. Again, suppose the numerator to represent the area of a circle, and the denominator the area of the circumscribed square; then the value of the fraction becomes .7854. But this value remains unchanged, although the circle may be supposed to diminish until it becomes a mere point. Hence, in this case, finite quantity. may represent any So, also, we may conceive both numerator and denominator of a fraction to be multiplied by 10, 100, 1000, etc., until each becomes greater than any assignable quantity; the fraction then reduces to the form of. Hence we conclude that the ∞ ∞ ∞ symbol may also represent any finite quantity. INEQUALITIES. 170. An inequality is an expression denoting that one quantity is greater or less than another. Thus 3x > 2ab denotes that three times the quantity x is greater than twice the prod uct of the quantities a and b. 171. In treating of inequalities, the terms greater and less must be understood in their algebraic sense; that is, a negative quantity standing alone is regarded as less than zero; and of two negative quantities, that which is numerically the greatest is considered as the least; for if from the same number we subtract successively numbers larger and larger, the remainders must continually diminish. Take any number, 5 for example, and from it subtract successively 1, 2, 3, 4, 5, 6, 7, 8, 9, etc., we obtain 5—1, 5–2, 5—3, 5—4, 5—5, 5—6, 5–7, 5–8, 5–9, etc.; or, reducing, we have 4, 3, 2, 1, 0, −1, −2, −3, −4, etc. Hence we see that -1 should be regarded as less than zero; -2 less than -1; -3 less than -2, etc. 172. Two inequalities are said to subsist in the same sense when the greater quantity stands at the left in both, or at the right in both; and in a contrary sense when the greater quantity stands at the right in one and at the left in the other. Thus 9>7 and 7>6, or 5<8 and 3<4, are inequalities which subsist in the same sense; but the inequalities 10>6 and 3<7 subsist in a contrary sense. 173. Properties of Inequalities.—1st. If the same quantity be added to or subtracted from each member of an inequality, the resulting inequality will always subsist in the same sense. Adding 6 to each member, we have −3+6<−2+6, or 3<4; and subtracting 6 from each member, -3-6-2-6, or -9<-8. |