and B $150, A had twice as much money as B; how much did each have at first? Ans. A was in debt $300, and B $100. 5. A man worked 7 days, and had his son with him 3 days, and received for wages 22 shillings, and the board of his son and himself while at work. He afterward worked 5 days, and had his son with him one day, and received 18 shillings. What were his daily wages, and what the daily wages of his son? Ans. The father received 4 shillings per day, and paid 2 shillings for his son's board. 6. A man worked for a person 10 days, having his wife with him 8 days, and his son 6 days, and he received $10.30 as compensation for all three; at another time he wrought 12 days, his wife 10 days, and son 4 days, and he received $13.20; at another time he wrought 15 days, his wife 10 days, and his son 12 days, at the same rates as before, and he received $13.85. What were the daily wages of each? Ans. He received $.75 for himself, $.50 for his wife, and paid $.20 for his son's board. 7. A man wrought 10 days for his neighbor, his wife 4 days, and son 3 days, and received $11.50; at another time he served 9 days, his wife 8 days, and his son 6 days, at the same rates as before, and received $12.00; a third time he served 7 days, his wife 6 days, and his son 4 days, at the same rates as before, and he received $9.00. What were the daily wages of each? Ans. Husband's wages, $1.00; Wife's, 0; Son's, $.50. 8. What fraction is that which becomes when 1 is added to its numerator, and when 1 is added to its denominator ? Ans. In an arithmetical sense, there is no such fraction. The algebraic expression, 18, will give the required results. How shall the enunciation be modified, to form an analogous question involving no absurdity? 9. Four merchants, A, B, C, D, find by their balance sheets that if they unite in a firm, receiving the assets and assuming the liabilities of each, they will have a joint net capital of $5780. If A, B, and C unite on the same conditions, their joint capital will be $7950; if B, C, and D unite, their joint capital will be $2220; and if C, D, and A unite, their joint capital will be $7320. the net capital or the net insolvency of each. Required 10. Two men were traveling on the same road towards Boston, A at the rate of a miles per hour, and B at the rate of b miles per hour. At 6 o'clock A was at a point m miles from Boston, and at 10 o'clock B was at a point n miles from Boston. Find the time when A passed B upon the road. 11. What time of day will be indicated by the preceding formula, if m = 36, n = 28, a = 5, and b = 3? Ans. 4 o'clock. 12. There are two numbers whose difference is a; and if 3 times the greater be added to 5 times the less, the sum will be b. What are the numbers? Ans. Greater, b+5a ; Less, b-3a 8 How shall this result be interpreted if a = 24 and b = 48? NOTHING AND INFINITY. 183. The limits between which all absolute values are comprised, are nothing and infinity; and the symbols by which these limits are denoted, are 0 and co. 184. In certain algebraic investigations it is convenient to employ these symbols in connection with each other and the ordinary symbols of quantity. They may thus sustain the relations of divisor, dividend, quotient, or factor. Such relations, however, can not really exist except between symbols of quantity. Hence, in Algebra, 0 does not always signify merely absence of value; nor does ∞ represent infinity, in the highest sense of the word. The more complete definition of these symbols may be given as follows: 185. The symbol 0, called nothing, or zero, may be used to denote the absence of value, or to represent a quantity less than any assignable value. 186. The symbol oo, called infinity, is used to represent a quan tity greater than any assignable value. A A INTERPRETATION OF THE FORMS AND 号 187. In order to understand the signification of the expressions, we may consider the symbols 0 and co as resulting from an arbitrary or varying quantity, made to diminish until it becomes indefinitely small, or to increase until it becomes indefinitely great. 188. Let a b represent a fraction, a and b being arbitrary quan tities. And let it be remembered that the value of a fraction depends simply upon the relative values of the numerator and denominator. 1.-If the denominator b is made to diminish, becoming less and less continually, while the numerator a remains unchanged, the value of the fraction must increase, becoming greater and greater continually, (119, II); and thus when the denominator b becomes less than any assignable quantity, or 0, the value of the fraction must become greater than any assignable quantity, or co. Hence, we conclude that A finite quantity divided by zero is an expression for infinity. 2. If the denominator b is made to increase, becoming greater and greater continually, while the numerator a remains unchanged, the value of the fraction must diminish, becoming less and less continually, (119, II); and when the denominator b becomes greater than any assignable quantity, or oo, the value of the fraction must become less than any assignable quantity, or 0. Hence, A finite quantity divided by infinity is an expression for zero or nothing. 3. If the numerator a is made to diminish, becoming less and less continually, while the denominator b remains unchanged, the value of the fraction must diminish continually, (119, I); and when a becomes less than any assignable quantity, or 0, the value of the fraction also must become 0. Hence, Zero divided by a finite quantity is an expression for nothing or zero. 4.-If both a and b are made to diminish simultaneously, but in such a manner as to preserve their relative value, then the value of the fraction will remain unchanged, however small the terms become, (119, III); and when both a and b become less than any 0 assignable quantity, or 0, we shall have the expression represent a 0 ing the value of. And since this value may be any quantity whatever, we conclude that That is, 0 represents an indeterminate quantity. Zero divided by zero is a symbol of indetermination. NOTE.-If it should be difficult for any one to conceive how both terms of a fraction may, by being diminished, become nothing at the same time, and yet preserve the same relative value to the last, it may be useful to consider the following illustrations: с Take the fractiona, in which d represents the diameter of a circle, and c the circumference. Now the diameter and circumference of a circle have the same ratio to each other, whatever the dimensions of the circle. Hence, if the circle be made to diminish until it shall become a point, or vanish, both terms of the fraction, will diminish, and become O at the same instant, the value of the fraction remaining the same through с out, and reducing to the form, ' at the instant the circle vanishes. Now the ratio of the diameter to the circumference of a circle is known to be 3.1416-; hence, in the present case, we shall have = 3.1416. Again, let 8 represent the side of a square and d the diagonal. Then we have the well known ratio If the square is supposed to diminish by insensible degrees, both d and I will vanish at the same instant, and we shall have finally, 0 PROBLEM OF THE COURIERS. 189. The anomalous forms which have been explained in the last article will now be viewed in connection with a general problem, involving certain relations of motion, time and distance. The discussion will also confirm our interpretation of negative results. PROBLEM.-TWO couriers, A and B, were traveling along the same road and in the same direction, namely, from C' toward C; the former going at the rate of a miles per hour, and the latter at the rate of 6 miles per hour. At 12 o'clock, A was at a certain point P, and B was d miles in advance of A, in the direction of C. It is required to find when and where the couriers were together. C' Р d C This problem is entirely general, and we do not know from the enunciation whether the couriers were together after, or before 12 o'clock; nor whether the place of meeting was to the right, or to the left of P. But in order to effect a statement of the problem, we will suppose the required time to be after 12 o'clock. Then we must regard time after 12 o'clock as positive, and time before 12 o'clock as negative; also, distance reckoned from P toward Cas positive, and distance reckoned from P toward C" as negative. Accordingly, Let t = the number of hours after 12 o'clock; the distance from P to the point of meeting. And since A traveled at the rate of a miles per hour, and B at the rate of 6 miles per hour, we have But since A and B were d miles apart, at 12 o'clock, we have We may now discuss this problem with reference to the time f. and the distance, which are the two unknown elements |