The enunciation of the question may be changed in two ways so as to answer the conditions of this equation. First, we may suppose, that the couriers, setting out from A and B, instead of going towards D, go in the opposite direction, the one from A at 4 miles per hour, and the other from B at 8 miles per hour; at what distance from the points A and B is the point E, where they come together? Or we may suppose that two couriers setting out from the same place E, one travelling at the rate of 4 miles, and the other per hour, have arrived at the same time at the points A and B, which are m miles asunder. What distance are the points A and B from E? How is this result to be interpreted? Observe that in this case a and b being equal, the two cou riers travel equally fast, it is therefore impossible that one should ever overtake the other, however far they may travel in either direction, and no change in the conditions can make it possible. Zero being divisor, then, is a sign of impossibility. We may observe that when there is any difference, however small, between a and b, the values of x and y will be real, and the couriers will come together in one direction or the other and the smaller the difference, the greater will be the distance travelled before they come together; that is, the greater will be the values of x and y. Suppose a = 5 and 64, a-b=1, Again, Suppose a 5, and b 45, ab 5, Again, Suppose a 5, and b = 498, a-b02, Again, Suppose a 5 and b = 4.998, a-b002, α Here observe, that as the difference between a and b becomes very small, the values of x and y become very large, and the difference between them is always m. Hence, since the smaller the divisor the larger the quotient, we may conclude, that when the divisor is actually zero, the quotient must be infinite. From this consideration, mathematicians have called the expression, that is, a quantity divided by zero, a symbol of infinity. They therefore say, that, both couriers travelling equally fast, the distance, travelled before they come together, is infinite. But as infinity is an impossible quantity, I prefer the term impossible, as being a term more easily comprehended, I shall therefore call a symbol of impossibility. 0 If a quantity be divided by an infinite or impossible quantity, the quotient will be zero. If be divided by & b a it be comes Multiply both numerator and denominator by 0, it a becomes 0 0 x b = 0. a In fact, since the larger the divisor, the smaller the quotient, the dividend remaining the same, it follows that if the divisor surpasses any assignable quantity, the quotient must be smaller than and assignable quantity, or nothing. One case more deserves our notice. It is when a = b and m0; in which case we have If we return to the equations themselves, they become This last equation has both its members alike, and is sometimes called an identical equation. The values of the unknown quantities cannot be determined from it. In fact, since m is zero, both couriers set out from the same point. And since they both travel at the same rate, they are always together. Therefore there is no point where they can be said to come together. The expression is here an expression of an indeter minate quantity. 0 0 There are some cases where an expression of this kind is not a sign of an indeterminate quantity, but in these cases it arises from a factor being common to the numerator and denominator, which by some suppositions becomes zero, and renders the fraction of the form of; but being freed from that factor, it has a determinate value. 0 0 rator and denominator contain the factor ab, which becomes zero when a and b are equal. Dividing by ab, the expression becomes a (a + b), which is equal to 2 a when a = b. It is necessary then, when we find an expression of the form before pronouncing it an indeterminate quantity, to see if there is not a factor, common to the numerator and denominator, which, becoming zero, renders the expression of this form. The example of the couriers furnishes some other curious cases, for which we must refer the learner to Lacroix's or Bourdon's Algebra. Let the learner examine the following examples in a similar manner. In Art. IX. examples 15 and 16, the following formulas, relating to interest, were obtained. How are r and t to be interpreted, when p is greater than a; and how when a and p are equal? In Art. XXII. examples 12th and 13th, the following formulas were obtained. In what cases will the results become negative, and how are the negative results to be interpreted? It is required to divide a given number a into two such parts, that if r times one part be added to s times the other part, the sum will be a given number b. Ans. The part to be multiplied by r is b and the part to be multiplied by s is ar—b -as In what cases will one or both of these results be negative ? Can both be negative at the same time? How are the negative results to be interpreted? In what cases will either of them become zero? Can both become zero at the same time? What is to be understood when one or both become zero? In what cases, will one or both become infinite or impossible? Can either of them ever be of the form? XXVII. Equations of the Second Degree. 1. A boy being asked how many chickens he had, answered, that if the number were multiplied by four times itself, the product would be 256. How many had he? This equation is essentially different from any which we have hitherto seen. It is called an equation of the second degree, because it contains a2, or the second power of the unknown quantity. In order to find the value of x, it is necessary to find what number, multiplied by itself, will produce 64. We know immediately by the table of Pythagoras that 8 x 8 64. Therefore Note. The results of these equations may be proved like those of the first degree |