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Let the latter distance be represented by b.

Then, on the principle above, AC is the sum of these distances, and this sum is represented algebraically by a b or a b.

(It will be seen that the distance BC is again measured from B in its own proper direction, and that the resultant distance AC of the point C is again the sum of the line AB and BC.)

There will evidently be three cases, viz. :—

1. When the distance BC is less in magnitude than AB, in which case the point C is on the right of A, and the distance AC is positive.

2. When the distance BC is equal in magnitude to AB, in which case the point C coincides with A, and the distance AC is zero.

3. When the distance BC is greater in magnitude than AB, the point C being then on the left of A, and the distance is negative.

Now, a b in all these cases represents the distance AC. It therefore admits of intelligible interpretation whether b be less than, equal to, or greater than a.

And, since the distance AC is obtained in the first two cases by subtracting the distance BC from that of AB, and in the second case by subtracting as far as AB will allow of subtraction, and measuring the remainder to be subtracted in an opposite direction, it follows that

The sign, which, standing before a letter, is a symbol of quality, becomes at once a symbol of subtraction in all cases when the quantity in question is placed immediately after any other given quantity with its proper sign of affection.

Hence also we may conclude that the addition of a negative quantity is equivalent to the subtraction of the corresponding positive quantity.

5. We may prove

in a similar way that

The subtraction of a negative quantity is equivalent to the addition of the corresponding positive quantity.

Let, as before, a represent the distance AB, measured from the point A to the right,

and let it be required to subtract from

A

a the distance represented by

b.

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Now, in the last article, we added a distance to a given distance AB, by measuring the second distance in its own direction from the extremity B. We shall therefore be consistent if we subtract a given distance -b by measuring this distance in a direction exactly opposite to its own direction, from the same extremity B.

Now, the direction of - b is to the left. If, therefore, we measure a distance BC to the right, equal in magnitude to the distance to be subtracted, we obtain a distance AC which is correctly represented by a correctly represented by a + b, b) = a + b.

a - (

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( − b).
(b).

But AC is also and hence it follows that

We may apply the above principle to all magnitudes which admit of continuous and indefinite extension; as, for instance, to forces which pull and push, attract and repel; to time past and time to come, to temperatures above zero and below zero, to money due and money owed, to distance up and distance down, &c., in all which cases, having represented one by a quantity affected with a + sign, we may represent the other by a quantity affected with a sign.

6. In expressing the sum of a number of quantities, the order of the terms is immaterial.

We will take, as our illustration, a body subject to various alterations of temperature, and we will suppose the temperature of the body, before the changes in question, to be zero or 0°. Let the temperature now undergo the following changes-viz., a rise of a°, a fall of b°, a fall of c°, and, lastly, a rise of d°. Let us consider a rise as positive, and therefore a fall as negative. We may then represent these changes respectively by + a, b, c, + d.

And it is further evident that the resulting temperature will be represented by the sum of these quantities, which, as previously written, will be a b - c + d.

But again, it is plain that the resulting temperature of the body will not be affected if these changes of temperature take place in the reverse order, or in any other order. Thus, suppose the temperature first falls c°, then rises a°, then rises do, and, lastly, falls b°, it is evident that the final temperature will be the same as before. And the sum of the quantities - c, + a, + d, - b, represents this final temperature.

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Now, expressing the sum of these quantities by writing them (Art. 4) side by side with their proper signs of affection, and in the order in which they stand, we have c + a + d

b for the sum.

It therefore follows, since we might have chosen any other order of these terms with a similar result, that the sum of any number of quantities, + α, -'b, - c, d, may be expressed by writing the terms side by side with their proper signs of affection in any order whatever.

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Nevertheless, for convenience, and for other reasons, we write the terms generally in alphabetical order, or we arrange them according to the power (see Art. 16) of some particular letter.

7. We may sum up the results and remarks of the last four articles as follows:

1. Positive and negative are used in exactly opposite

senses.

2. The sign before an algebraical quantity affirms the quality of the quantity as represented without the sign in question.

Thus, + (+ a) 3. The sign

=

=

+ a, and + (-b) b. before an algebraical quantity reverses the quality of the quantity as represented without the sign in question.

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a, and (-b)

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4. The algebraical subtraction of a quantity is the same as the addition of the quantity with the sign of affection reversed.

5. The algebraical addition of quantities is expressed by writing the quantities down side by side with their proper signs of affection. And they may be written down in any order, though we generally write them in alphabetical order, or arranged according to the power (Art. 16) of some letter.

Brackets.

8. Brackets(), { }, []—are used, for the most part, whenever we wish to consider an algebraical expression containing more than one term as a whole.

Thus, if we wish to express that the quantity 3 a + 76 is to be added as a whole to 4 a, we write

4 a + (3 a + 7 b),

and, while inclosed within brackets, we think and speak of

3a + 7b as one quantity.

Again, if we wish to express that b

c is to be subtracted

from a, we write

a

(b - c).

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c)

Let us consider what is the result of subtracting (b from a. We may evidently, if we please, subtract b first, then afterwards c from the quantity so obtained, without affecting this result.

Now, we know by Art. 7 (4.) that this is equivalent to band+c successively.

adding the quantities Now, the sum of a,

We have therefore a

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(bc)

= α b + c.

We observe that the sign of b within the brackets is ⇓, and that of c is -, whereas, in our final result, these signs are both reversed. And we hence arrive at the following important principle:

When a minus sign stands before a bracket, its effect on removing the brackets is to reverse the sign of affection of every term within.

And it is evident that we may, by a similar course of reasoning, arrive at a principle equally important, viz. :

When a plus sign stands before a bracket, its effect on removing the bracket is to affirm the sign of affection of every term within.

We shall show, in Art. 9, the use of brackets in expressing the product or quotient of quantities.

Though, as stated above, brackets are, for the most part, used to group together a whole a number of quantities, they are sometimes used to inclose single terms. Thus, in Art. 5, we have the expression a (-b). (-b). Now, the brackets are used here to express that the negative quantity is to be subtracted as a negative quantity. And, in the same way, the expression a +(-b) indicates that the negative quantity (-b) is to be added to the quantity a. When one pair only is required we generally use the brackets (); if,

however, a quantity already in brackets is to be inclosed in a second pair, we use { }, as in the expression—

3a - {6b+ (4cd)}.

If a third pair be required we use the brackets [ ], and finally, we sometimes find it convenient to group a number of terms by means of a vinculum, thus—

4 x - [6 x - {5y + (3≈

2y].

7 x − y)} + 2 y

It must be remembered that the vinculum has in an expression exactly the same force as brackets.

9. We shall, in this Article, show how to find the value of a few algebraical expressions, as illustrations of the foregoing principles :

Ex. 1.-If a = 1, b = 2, c = 4, find the value of

3 a +56 + 7 c.

We have only to substitute the value of the letters in the given expression, putting a sign of multiplication to avoid ambiguity.

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= 26 54.

The negative quantity is here the larger, and exceeds the positive quantity by 28, and hence (Art. 4 (3),) the result will

be negative.

We therefore have

=

4 x + 3y 9

=

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3, z = 0, find the value of―

Ex. 3.-If x

1, y

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= 3 × 1 - { 2 × 3 + (6 × 0 − 5 × 1 −

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