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(45.) It must be observed that the term addition is used in a more extended sense in algebra than in arithmetic. In arithmetic, where all quantities are regarded as positive, addition implies augmentation. The sum of two quantities will therefore be numerically greater than either quantity. Thus the sum of 7 and 5 is 12, which is numerically greater than either 5 or 7.

But in algebra we consider negative as well as positive quantities; and by the sum of two quantities we mean their aggregate, regard being paid to their signs. Thus the sum of +7 and −5 is +2, which is numerically less than either 7 or 5. So, also, the sum of

QUEST.-What is the difference between arithmetical and algebraic

addition?

+a and b is a-b. In this case the algebraic sum is numerically the difference of the two quantities.

This is one instance among many in which the same terms are used in a much more general sense in the higher mathematics than they are in arithmetic.

SECTION III.

SUBTRACTION.

(46.) SUBTRACTION is the taking of one quantity from another; or it is finding the difference between two quantities or sets of quantities.

Thus, if it is required to subtract 17 from 25, we may write it 25-17,

which equals 8.

So, also, if it is required to subtract 5a from 8a, we nay write it 8a-5a,

which equals 3a.

If it is required to subtract 5b from 8a, we write it

8a-5b,

and these, being unlike terms, can not be united in one single term. Hence, if the quantities are positive and similar, subtract the coefficient of the subtrahend from the coefficient of the minuend, and to their difference annex the literal part. If the quantities are not similar, the subtraction can only be indicated by the usual

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QUEST.-What is subtraction? If the quantities are positive and sim. ilar, how do we proceed? When the quantities are not similar?

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(47.) Let us now consider the case in which the quantities are not all positive; and let it be required to subtract 8-3 from 15.

We know that 8-3 is equal to 5, and 5 subtracted from 15 leaves 10.

But, to perform the were given, we first

The result, then, must be 10. operation on the numbers as they subtract 8 from 15 and obtain 7. This result is too small by 3, because the number 8 is larger by 3 thar the number which was required to be subtracted. Therefore, in order to correct this result, the 3 must be added, and we have

15-8+3=10, as before.

Again, let it be required to subtract c-d from a-b. It is plain that if the part c were alone to be subtracted, the remainder would be

a-b-c.

But as the quantity actually proposed to be subtracted is less than c by the units in d, too much has been taken away by d, and therefore the true remainder will be greater than a-b-c by the units in d, and will hence be expressed by

QUEST.-When the quantities are not all positive?

a-b-c+d,

where the signs of the last two terms are both contrary to what they were given in the subtrahend. (48.) Hence we deduce the following general

RULE.

Conceive the signs of all the terms of the subtrahend to be changed from + to — or from — to +, and then collect the terms together as in the several cases of addition.

It is better, in practice, to leave the signs of the subtrahend unchanged, and simply conceive them to be changed; that is, treat the quantities as if the signs were changed; for otherwise, when we come to revise the work, to detect any error in the operation, we might often be in doubt as to what were the signs of the quantities as originally proposed.

Examples.

Ex. 1. From 7x+4y take 3x2-2y.

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QUEST.-Give the general rule for subtraction. Is it best actually to change the signs of the subtrahend?

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