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187. In order to understand the signification of the expressions

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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.

b

188. Let represent a fraction, a and b being arbitrary quantities. 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 ∞. Hence, we conclude that

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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 ∞, the value of the fraction must become less than any assignable quantity, or 0. Hence, That is,

a

= 0.

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,

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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 assignable quantity, or 0, we shall have the expression

α

0

representing the value of And since this value may be any

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quantity whatever, we conclude that

nate quantity. That is,

0

represents an indetermi

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:

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Take the fraction 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 0 at the same in

0

0'

с

ď

stant, the value of the fraction remaining the same throughout, 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

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Again, let 8 represent the side of a square and d the diagonal. Then we have the well-known ratio

d

= √2.

8

If the square is supposed to diminish by insensible degrees, both d and s 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'

P

P

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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 C as positive, and distance reckoned from P toward C' as negative. Accordingly,

Let

t = the number of hours after 12 o'clock;

x 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 b miles per hour, we have

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But since A and B were d miles apart, at 12 o'clock, we have

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We may now discuss this problem with reference to the time t, and the distance x, which are the two unknown elements.

I. Suppose a >b.

Under this hypothesis the values of both t and x will be positive, because the common denominator, ab, is positive. Now since t is positive, we conclude that the two couriers came together after 12 o'clock; and as x is positive, we infer that the point of meeting is somewhere to the right of P.

These conclusions agree with each other, and are consistent with the conditions of the problem. For, the supposition that a is greater than b implies that A was traveling faster than B. A would therefore gain upon B, and overtake him sometime after 12 o'clock, and at a point situated in the direction, of C.

II. Suppose a <b.

Then in equations (1) and (2) the denominator, ab, is negative, and consequently both t and x will be negative.

This implies that t and x must be taken in a sense contrary to that in which they were employed under the hypothesis (I), where they were positive; that is, the time when the couriers were together was before 12 o'clock, and the place of meeting was situated to the left of P.

This interpretation, also, agrees with the conditions of the problem, under the present hypothesis. For, if a is less than b, then B was traveling faster than A; and as B was in advance of A at 12 o'clock, he must have passed A before that time, somewhere to the left of P, in the direction of C'.

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Under this hypothesis we shall have ab=0, and

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Now, according to these results, t, the time to elapse before the couriers are together, is greater than any assignable quantity, or infinity; therefore they can never be together. And likewise x, the distance from P to the supposed point of meeting, is greater than any assignable quantity, or infinity; hence there can be no such point, however distant from P.

This interpretation is in accordance with the conditions of the problem, under the present hypothesis. For, at 12 o'clock the two

couriers were d miles apart; and if ab, they were traveling at equal rates, neither approaching nor separating. Hence, they could always continue in motion, and remove to any distance from P, without meeting.

IV. Suppose d= 0, and a > b or a <b.

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That is, both the time and distance are nothing. These results must be interpreted to mean that the couriers were together at 12 o'clock, at the point P, and at no other time or place.

And this interpretation is also confirmed by the conditions of the problem. For, if d= 0, then at 12 o'clock B must have been with A, at the point P. And if a > b or a <b, the couriers were traveling at different rates, and must be either approaching or receding from each other at all times except at the moment of passing; hence, they could be together only at a single point. V. Suppose d= 0, and a = b.

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Here the values of both t and x are represented by the symbol of indetermination, which signifies that the time and the distance may be anything whatever; and we infer that the couriers must be together at all times, and at any distance from P.

And this conclusion is evidently confirmed by the conditions of the problem. For, if d = 0, the couriers were together at 12 o'clock; and if ab, they were traveling at equal rates, and would never separate.

190. To the foregoing interpretations, there is an apparent exception in the case of the expression. For, a fraction which

is not indeterminate will reduce to this form, if its terms contain a common factor that becomes zero under the hypothesis. Thus, in the solution of a problem, suppose

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