The answer is

negative sign?

2 rods. What shall we understand by this

Let us return to the original equation.

8X5+5 = 30

or 40+5 = 30.

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

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?

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

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 +5x 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 nothing, 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 endeavoring 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, α 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 a—b to c.

The answer is evidently cab.

Now if a is greater than b, the quantity cab, 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 subtracting 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; which was equivalent to saying, you must subtract 2 rods.

It is required to subtract a

b 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 a b

then c

m which gives

· a + b = m

a+b=c+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.

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 How much is C worth?

B.

B's property

C's property

=ac

bc.

Now if a is greater than b, the quantity a c

bc will be positive; but if b is greater than a, then ab is negative, and also

ас

Let

- b c is negative.

then

and

or

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?

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 cbc 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.

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

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

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

Another demonstration. Suppose both a

be negative, as before; then b

--

a and d

itive, and their product will be positive.

b and c d to

c will both be pos

This product is precisely the same as that produced by multiplying ab by cd. Therefore if two negative quantities be multiplied together, the product will be the same as that of two positive quantities of the same numerical value, and will have the positive sign.

b

It is required to find the second power of a

b is positive, then b is negative, then ba is positive.

+ b2 — 2 a b.

b, and also of

a is negative; or if a-b

Suppose

then

That is, the second power of any quantity, whether positive or negative, is necessarily positive.

The rules for division will necessarily follow from those of multiplication.

Hence the rules which apply to terms affected with the sign in compound quantities, extend to isolated negative quantities.

We might also derive the same rules in the following manner. It has been shown that a negative quantity is derived from some contradiction in the conditions of question, by which that quantity entered into the equation with the wrong sign. Now, in order to make it right, the sign of that quantity must be changed in all places where it is used. That is, if it was before added, it must now be subtracted; and if it was subtracted before, it must now be added, and that whether multiplied by another quantity or

not.

Suppose we have the equation

α x 2 x2 2 ab x=c - - d.

that we have x —— m.