becomes equal to infinity, which we designate by the charac ter ∞. Hence we conclude, that a finite quantity divided by 0, gives a quotient greater than any assignable quantity, which we call, INFINITY. 111. Again, let A represent any finite number: then, since the value of a fraction increases as its numerator becomes greater with reference to its denominator, the expression A is a proper symbol to represent an infinite quantity; that is, a quantity greater than any assignable quantity. Since the value of a fraction diminishes as its denominator becomes greater with reference to its numerator, the expression is a proper symbol for a quantity less than any assignable quantity. Hence, We have been thus particular in explaining these ideas of infinity, because there are some questions of such a nature, that infinity may be considered as the true answer to the enunciation. In the case just considered, where m = n, it will be perceived that there is not, properly speaking, any solution in finite and determinate numbers; but the value of the unknown quantity is found to be infinite. 112. If, in addition to the hypothesis m = n, we also suppose that a 0, we have t = 0 0 be satisfied by any value of t. or t x 0 0; a result which will = To interpret this result, let us consider again the enunciation, from which it is perceived, that if the two couriers travel equally fast, and are once at the same point, they ought, ever after, to be together, and consequently the required time is entirely unde termined. Therefore, the expression symbol of an indeterminate quantity. is, in this case, the The preceding suppositions are the only ones that lead to remarkable results; and they are sufficient to show to beginners the manner in which the results of Algebra answer to all the circumstances of the enunciation of a problem. 0 113. It should be observed, that the expression is not a certain symbol of indetermination, but frequently arises from the existence of a common factor in each term of the fraction, which factor becomes nothing, in consequence of a particular hypothesis. For example, suppose the value of the unknown quantity to be If, in this formula, a is made equal to b, there results Now, if we suppress the common factor ab, and then suppose a = b, we shall have Let us suppose, that in another example, we have If, however, we suppress the factor common to the numerator and denominator, in the value of x, we have, Therefore we conclude, that before pronouncing upon the true value of the fraction, 0 it is necessary to ascertain whether the two terms do not contain a common factor. If they do not, we conclude that the fraction is really indeterminate. If they do contain one, suppress it, and then make the particular hypothesis; this will give the true value of the fraction, which will assume one of the three forms A A 0 B' 0' 0' that is, it will be determinate, infinite, or indeterminate. This observation is very useful in the discussion of problems. Of Inequalities. 114. In the discussion of problems, we have often occasion to suppose quantities unequal, and to perform transformations, upon them, analogous to those executed upon equalities. We sometimes do this, to establish the necessary relations between the given quantities, in order that the problem may be susceptible of a direct, or at least, of a real solution. We often do it, to fix the limits between which the particular values of certain given quantities must be found, in order that the enunciation may fulfil a particular condition. Now, although the principles established for equations are, in general, applicable to inequalities, there are nevertheless some exceptions, of which it is necessary to speak, in order to put the beginner upon his guard against some errors that he might commit, in making use of the sign of inequality. These exceptions arise from the introduction of negative expressions into the calculus, as quantities. In order to be clearly understood, we will give examples of the different transformations to which inequalities may be subjected, taking care to point out the exceptions to which these transformations are liable. 115. 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, 2520 and 18> 10, or 6 <8 and 7< 9, are inequalities which subsist in the same sense; and the inequalities 15 13 and 12 14, subsist in a contrary sense. 1. If we add the same quantity to both members of an inequality, or subtract the same quantity from both members, the resulting inequality will subsist in the same sense. Thus, take 8>6; by adding 5, we still have 8565; and subtracting 5, we have 8 5 > 6-5. When the two members of an inequality are both negative, that one is the least, algebraically considered, which contains the greatest number of units. Thus, 25-20; and if 30 be added to both members, we have 5 < 10. This must be understood entirely in an algebraic sense, and arises from the convention before established, to consider all quantities preceded by the minus sign, as subtractive. The principle first enunciated, serves to transpose certain terms from one member of the inequality to the other. Take, for example, the inequality there will result, by transposing, a2 + 2a2 > 362 — b2, or 3a2 > 2b2. 2. If two inequalities subsist in the sume sense, and we add them member to member, the resulting inequality will also subsist in the same sense. Thus, add there results a>b, c>d, e>f: and a+c+e>b + d + f. But this is not always the case, when we subtract, member from member, two inequalities established in the same sense. Let there be the two inequalities 4 <7 and 2 <3, we have But if we have the inequalities 910 and 68, by subtracting we have 96 or 310 8 or 2. We should then avoid this transformation as much as possible, or if we employ it, determine in which sense the resulting inequality exists. 3. If the two members of an inequality be multiplied by a positive number, the resulting inequality will exist in the same sense. 3a (a2 - b2) > 2d (c2 — d2), and the same principle is true for division. But, when the two members of an inequality are multiplied or divided by a negative number, the inequality will subsist in a contrary sense. Take, for example, 8>7; multiplying by 3, we have - 3' < 3 Therefore, when the two members of an inequality are multiplied or divided by a number expressed algebraically, it is necessary to ascertain whether the multiplier or divisor is negative; for, in that case, the inequality will exist in a contrary sense. 4. It is not permitted to change the signs of the two members of an inequality, unless we establish the resulting inequality in a contrary sense; for this transformation is evidently the same as multiplying the two members by — 1. 5. Both members of an inequality between positive numbers can be squared, and the inequality will exist in the same sense. Thus, from 5>3, we deduce, 25 > 9; from a + b> c, we find (a + b)2 > c2. |