If we wish to include both of these cases in the same enunciation, the question should be, Required the time of their being together, leaving it uncertain whether the time was past or future.

EXAMPLES.

1. What number is that whose fourth part exceeds its third part by 16? Ans. -192. How should the enunciation be modified in order that the result may be positive?

2. The sum of two numbers is 2, and their difference 8. What are those numbers? Ans. 3 and +5.

How should the enunciation be modified in order that both results may be positive?

3. What fraction is that from the numerator of which if 4 be subtracted the value is one half, but if 7 be subtracted from the denominator its value is one fifth ?

-5

Ans. 18

How should the enunciation be modified in order that the

problem may be possible in its arithmetical sense?

4. Find two numbers whose difference is 6, such that four times the less may exceed five times the greater by 12.

Ans. -42 and -36.

Change the enunciation of the problem so that these numbers, taken with the contrary sign, may be the answers to the modified problem.

165. 3d. We may obtain for the unknown quantity values of 0

the form of zero, or 2.

Α

In what case would the value of the unknown quantity in Prob. 20, page 94, become zero, and what would this value signify?

Ans. This value becomes zero when a=0, which signifies that the two trains are together at the outset.

In what case would the value of the unknown quantity in Prob. 50, page 98, become zero, and what would this value signify?

Ans. When a=nb, which signifies that the age of the father is now n times that of the son.

In what case would the values of the unknown quantities in Prob. 38, page 96, become zero, and what would these values signify?

When a problem gives zero for the value of the unknown quantity, this value is sometimes applicable to the problem, and sometimes it indicates an impossibility in the proposed question.

166. 4th. We may obtain for the unknown quantity values of the form of 4.

In what case does the value of the unknown quantity in Prob. 20, page 94, reduce to and how shall we interpret

this result?

А

Ans. When m=n.

On referring to the enunciation of the problem, we see that it is absolutely impossible to satisfy it; that is, there can be no point of meeting; for the two trains, being separated by the distance a, and moving equally fast, will always continue at the same distance from each other. The result may then be regarded as indicating an impossibility.

α

0

α

The symbol is sometimes employed to represent infinity, and for the following reason:

If the denominator of a fraction is made to diminish, while the numerator remains unchanged, the value of the fraction must increase.

Hence, if the difference in the rates of motion is not zero, the

two trains must meet, and the time will become greater and greater as this difference is diminished. If, then, we suppose this difference to be less than any assignable quantity, the time

α

represented by will be greater than any assignable quantity.

A

Hence we infer that every expression of the form found

for the unknown quantity indicates the impossibility of satisfying the problem, at least in finite numbers.

In what case would the value of the unknown quantity in Prob. 10, page 92, reduce to the form 4, and how shall we in

terpret this result?

A
0'

167. The symbol 0, called zero, is sometimes used to denote the absence of value, and sometimes to denote a quantity less than any assignable value.

The symbol ∞, called infinity, is used to denote a quantity greater than any assignable value. A line produced beyond any assignable limit is said to be of infinite length; and time extended beyond any assignable limit is called infinite duration.

We have seen that when the denominator of the fraction

α becomes less than any assignable quantity, the value of the fraction becomes greater than any assignable quantity. Hence we conclude that 8=0;

а

that is, a finite quantity divided by zero is an expression for infinity.

Also, if the denominator of a fraction be made to increase while the numerator remains unchanged, the value of the fraction must diminish; and when the denominator becomes greater than any assignable quantity, the value of the fraction must become less than any assignable quantity. Hence we conclude that

that is, a finite quantity divided by infinity is an expression for

zero.

168. 5th. We may obtain for the unknown quantity values

of the form of o

In what case does the value of the unknown quantity in

0

Prob. 20, page 94, reduce to %, and how shall we interpret this Ans. When a=0, and m=n.

result?

To interpret this result, let us recur to the enunciation, and observe that, since a is zero, both trains start from the same point; and since they both travel at the same rate, they will always remain together; and, therefore, the required point of meeting will be any where in the road traveled over. The problem, then, is entirely indeterminate, or admits of an infinite number of solutions; and the expression may represent any finite quantity.

We infer, therefore, that an expression of the form of found for the unknown quantity generally indicates that it may have any value whatever. In some cases, however, this value is subject to limitations.

In what case would the values of the unknown quantities in Prob. 44, page 97, reduce to and how would they satisfy the conditions of the problem? Ans. When a=b=c, which indicates that the coins are all of the same value. B might therefore be paid in either kind of coin; but there is a limitation, viz., that the value of the coins must be one dollar.

In what case do the values of the unknown quantities in Prob. 38, page 96, reduce to, and how shall we interpret this result?

169. The expression may be conceived to result from a fraction whose numerator and denominator both diminish simultaneously, but in such a manner as to preserve the same relative value. If both numerator and denominator of a fraction are divided by the same quantity, its value remains un

α

changed. Hence, if represent any fraction, we may conceive

b

both numerator and denominator to be divided by 10, 100, 1000, etc., until each becomes less than any assignable quantity, or 0. The fraction then reduces to the form of o, but the 0' value of the fraction has throughout remained unchanged.

For example, we may suppose the numerator to represent the circumference of a circle, and the denominator to represent its diameter. The value of the fraction in this case is known to be 3.1416. If now we suppose the circle to diminish until it becomes a mere point, the circumference and diameter both become zero, but the value of the fraction has throughout remained the same. Hence, in this case, we have

-3.1416.

Again, suppose the numerator to represent the area of a circle, and the denominator the area of the circumscribed square; then the value of the fraction becomes .7854. But this value remains unchanged, although the circle may be supposed to diminish until it becomes a mere point. Hence, in this case,

Hence we conclude that the symbol may represent any finite quantity.

So, also, we may conceive both numerator and denominator of a fraction to be multiplied by 10, 100, 1000, etc., until each becomes greater than any assignable quantity; the fraction then reduces to the form of Hence we conclude that the also represent any finite quantity.

∞

symbol

may

∞

INEQUALITIES.

170. An inequality is an expression denoting that one quantity is greater or less than another. Thus 3x > 2ab denotes that three times the quantity x is greater than twice the product of the quantities a and b.