Page images
PDF
EPUB
[blocks in formation]

Now so long as m>n, t will be positive, and the problem will be solved in the arithmetical sense of the enunciation. For, if m>n the courier from A will travel faster than the courier from B, and will therefore be continuaily gaining on him: the interval which separates them will diminish more and more, until it becomes 0, and then the couriers will be found upon the same point of the line.

In this case, the time t, which elapses, must be added to 12 o'clock, to obtain the time when they are together.

But, if we suppose m<n, then m-n will be negative, and the value of t will be negative. How is this result to be interpreted ?

It is easily explained from the nature of the question, which, considered in its most general sense, demands the time when the couriers are together.

Now, under the second supposition, the courier which is in advance, travels the fastest, and therefore will continue to separate himself from the other courier. At 12 o'clock the distance between them was equal to a: after 12 o'clock it is greater than a, and as the rate of travel has not been changed, it follows that previous to 12 o'clock the distance must have been less than a. At a certain hour, therefore, before 12 the distance between them must have been equal to nothing, or the couriers were together at some point R'. The precise hour is found by subtracting the value of t from 12 o'clock.

This example, therefore, conforms to the general principle, that, if the conditions of a problem are such as to render the unknown quantity essentially negative, it will appear in the result with the minus sign, whenever it has been regarded as positive in the enunciation.

If we wish to find the distances AR, and BR passed over by the two couriers before coming together, we may take the equation

a

m-n

and multiply both members by the rates of travel respectively: this will give

[merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small]

from which we see that the two distances AR, BR, will both be positive when estimated towards the right, and that AR', BR' will both be negative when estimated in the contrary direction.

109. To explain the terms nothing and infinity, let us considethe equation

a

t=

m-n

If in this equation we make m=n, then m-n=0, and the value of t will reduce to

t=

a

In order to interpret this new result, let us go back to the enunciation, and it will be perceived that it is absolutely impossible to satisfy it for any finite value fort; for whatever time we allow to the two couriers they can never come together, since being once separated by an interval a, and travelling equally fast, this interval will always be preserved.

a

Hence, the result, 이 may be regarded as a sign of impossibility for any finite value of t.

Nevertheless, algebraists consider the result

a

as forming a species of value, to which they have given the name

of infinite value, for this reason :

When the difference m-n, without being absolutely nothing, is supposed to be very small, the result

[blocks in formation]

In short, if the difference between the rates is not zero, the couriers will come together at some point of the line, and the time will become greater and greater as this difference is diminished.

Hence, if the difference between the rates is less than any assignable number, the time expressed by

[blocks in formation]

will be greater than any assignable or finite number. Therefore, for brevity, we say when m-n=0, the result

[blocks in formation]

becomes equal to infinity, which we designate by the character ∞. Again, as the value of a fraction increases as its numerator be. A

comes greater with reference to its denominator, the expression A being any finite number, is a proper symbol to represent an infinite quantity; that is, a quantity greater than any assignable quan. tity.

A ;

A quantity less than any given quantity may be expressed by for a fraction diminishes as its denominator becomes greater with

reference to its numerator. Hence, 0 and

[blocks in formation]

A

are synonymous

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.

110. If, in addition to the hypothesis m=n, we suppose that a=0, we have

[blocks in formation]

To interpret this result, let us reconsider the enunciation, where it will be perceived, that if the two couriers travel equally fast, and are once at the same point, they ought always to be together, and consequently the required point is any point whatever of the line

travelled over. Therefore, the expression

symbol of an indeterminate quantity.

0

is in this case, the

If the couriers do not travel equally fast, that is, if m>, or m<n, and a=0, then will t=0.

Indeed, it is evident, that if the couriers travel at different rates, and are together at 12 o'clock, they can never be together afterwards.

The preceding suppositions are the only ones that lead to remark. able results; and they are sufficient to show to beginners the man. ner in which the results of algebra answer to all the circumstances of the enunciation of a problem.

111. We will add another example to show, that the expression

expresses, generally, an indeterminate quantity.

0

[merged small][merged small][ocr errors][ocr errors][merged small]

Now, if we perform the division the quotient will be 1; and if we make x=1, there will result

[blocks in formation]

If we perform the division, the quotient will be 1+x; then

making x=1, the expression becomes

[blocks in formation]

112. We will add another example showing the value of the ex

[blocks in formation]

Take the equation ax=b, involving one unknown quantity, whence

b

x=

a

1st. If, for a particular supposition made with reference to the given quantities of the question, we have a=0, there results

[merged small][ocr errors]

Now in this case the equation becomes 0xx=b, and evidently

« PreviousContinue »