S. Three persons, A, B, and C, talking of their money, says A to B and C, give me half of your money and I shall have a sum d; says B to A and C, give me one third of your money and I shall have d; says C to A and B, give me one fourth of your money, and I shall have d. How much had each?

XXIV. Negative Quantities.

It sometimes happens in the course of a calculation, through some misconception of the conditions of the question, that a quantity is added which ought to have been subtracted, or a quantity subtracted which ought to have been added. In this case, algebra will detect the error, and show how to correct it.

The length of a certain field is a, and its breadth b; how much must be added to its length, that its content may be c? Let x = the quantity to be added to the length. Then a + x = the length after adding î.

Suppose the length to be 8 rods, and the breadth 5; how much must be added to the length, that the field may contain 60 square rods ?

Ans. 4 rods, and the whole length will be 12 rods.

Suppose the length 8 rods, and the breadth 5; how much must be added to the length, that the field may contain 30

square rods?

Here appears an absurdity in supposing something to be added to 40 to make 30. The result shows that we must add 2 rods, that is, subtract 2 rods, which is in fact the case;

for

40-5 X 2 = 30.

Let the question be proposed as follows. There is a field 8 rods long and 5 wide; how much must be subtracted from the length, that the field may contain 30 square rods?

405 x 30

x = 2.

The value of x is now positive, which shows that the question is correctly expressed.

There is a field 8 rods long and 5 rods wide, how much must be subtracted from the length, that the field may contain 50 square rods?

40 5 x 50

X=- 2.

Here again the value of x is negative, which shows some inconsistency in the question.

The inconsistency consists in supposing that something must be subtracted from 40 to make 50. In order to correct it, suppose something added. That is, put into the equation + 5 x instead of

5 x.

Hitherto we have treated of negative quantities only in connexion with positive. They arise from the necessity of expressing subtraction by a sign, because it cannot actually be performed on dissimilar quantities. They are only positive quantities subtracted, and in their nature they differ in nothing from positive quantities. In that connexion we discovered rules for operating upon the quantities affected with the sign

It may sometimes happen as we have just seen, that by some wrong supposition in the conditions of the question, the quantities to be subtracted may become greater than those from

which they are to be subtracted, in which case the whole expression taken together, or which is the same thing, the result after subtraction, will be negative. This is what is properly called a negative quantity.

A negative quantity cannot in reality be a quantity less than nothing, but it implies some contradiction. It answers to a figure of speech frequently used. If it is asked, how much a man is worth who owes five thousand dollars more than he can pay, we sometimes say he is worth five thousand dollars less than nothing, instead of changing the form of expression and saying, he owes five thousand dollars more than he can pay.

If any thing is added to a number, properly speaking it must increase the number; if we add nothing, it is not altered. It is impossible to add less than nothing; but by a figure of speech we may use the expression, add a quantity less than no'thing, to signify subtraction."

As these negative quantities may frequently occur, it is necessary to find rules for using them.

In the first place, let us observe, that all negative quantities are derived from endeavouring to subtract a larger quantity from a smaller one. The largest number that can actually be subtracted from any number, is the number itself. Thus the largest number that can be subtracted from 5 is 5; the largest number that can be subtracted from a is a itself. If it be required to subtract 8 from 5, it becomes 5 5-3

-- 3; the 5 only can be subtracted, the 3 remains with the sign, which shows that it could not be subtracted. If 5 be subtracted from 8, the remainder is 3, the same as in the other case except the sign.

In the same manner, if it be required to subtract b from a, b being the larger the remainder will have the sign—, that is, a-b will be a negative quantity.

Suppose b

-am; then a b -- m. That is, whether a be subtracted from bor b from a, the numerical value of the remainder is the same, differing only with respect to the sign. It is required to add the quantity ab to c.

The answer is evidently c+a-b.

Now if a is greater than b, the quantity c + a-b, is greater than c, by the difference between a and b; but if b is greater than a, the quantity is smaller than c, by the difference between a and b. That is, if

Hence, adding a negative quantity, is equivalent to subtract ing an equal positive quantity.

In the above example of the field, in which the length was 8 rods and breadth 5, it was asked, how much must be added to the length, that it might contain 30 square rods. The answer

was

2 rods.

-2; which was equivalent to saying, you must subtract

It is required to subtract ab from c.

The answer is evidently c a + b.

Now if a is greater than b, the quantity cab is less than c by the difference between a and b, but if b is greater than a, the quantity is larger than c, by the same quantity.

Let α- b = — m which gives

then ca+b=c+m.

—a+b=m

Hence, subtracting a negative quantity, is equivalent to adding an equal positive quantity.

In the example of the field, in which the length was 8 rods and the breadth 5, it was asked, how much must be subtracted from the length, that the field might contain 50 square rods. The answer was 2 rods, which was equivalent to saying that 2 rods must be added to the length.

A is worth a number a of dollars, B is not worth so much as A by a number b of dollars, and C is worth c times as much as

Now if a is greater than b, the quantity a c-be will be positive; but ifb is greater than a, then a b is negative, and also a c-be is negative.

That is, if B is in debt, C is e times as much in debt. Hence if a negative quantity be multiplied by a positive, the product is negative.

A gentleman owned a number a of farms, and each farm was worth a number c of dollars, which was his whole property. He hired money and fitted out a number b of vessels, and each vessel was worth as much as one of his farms. All the vessels were lost at sea. How much was he then worth. b times c dollars. That is, a c C

He was worth a lars.

bc dol

Now if the number of farms exceeded the number of vessels, he still had some property, but if the number of vessels exceeded the number of farms, (that is, if b is larger than a,) the quantity a c - b c is negative, and he owed more than he could

pay.

Hence if a positive quantity be multiplied by a negative the product will be negative.

value as b―a, they differ only in the sign.

Hence (a - b) c + (b − a) d = — cm+dm = m (d-- c).

Now if d is greater than c, (which is the case when c— negative,) the quantity m (dc) is positive.

d is

Hence if a negative quantity be multiplied by a negative, the product will be positive.

Another demonstration. Suppose both ab and c d to be negative, as before; then b -a and d c will both be po

sitive, and their product will be positive.