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CHAPTER II.

NEGATIVE QUANTITIES. ADDITION OF LIKE TERMS.

17. In his arithmetical work the student has been accustomed to deal with numerical quantities connected by the signs + and - ; and in finding the value of an expression such as 14+73 - 33.+ 6 – 4} he understands that the quantities to which the sign + is prefixed are additive, and those to which the sign

is prefixed are subtractive, while the first quantity, 14, to which no sign is prefixed, is counted among the additive terms. The same notions prevail in Algebra; thus in using the expression 7a+36 – 4c – 2d we understand the symbols 7a and 36 to be additive, while 4c and 2d are subtractive.

18. But in Arithmetic the sum of the additive terms is always greater than the sum of the subtractive terms; and if the reverse were the case the result would have no arithmetical meaning. In Algebra, however, not only may the sum of the subtractive terms exceed that of the additive, but a subtractive term may stand alone, and yet have a meaning quite intelligible.

And so it is usual to divide all algebraical quantities into positive quantities and negative quantities, according as they are expressed with the sign + or the sign – ; and this is quite irrespective of any actual process of addition and subtraction.

This idea may be made clearer by one or two simple illustrations.

I. Suppose a man were to gain $100 and then lose $70, his total gain would be $30. But if he first gains $70 and then loses $100 the result of his trading is a loss of $30. The corresponding algebraical statements would be

$100-$70 = +$30,
$70 – $100= -$30,

and the negative quantity in the second case is interpreted as a debt, that is, a sum of money opposite in character to the positive quantity, or gain, in the first case; in fact it may be said to possess a subtractive quality which would produce its effect on a future transaction, or perhaps wholly counteract a future gain.

II. Suppose a man were to walk along a straight road 100 yards forwards and then 70 yards backwards, his distance from the starting-point would be 30 yards. But if he first walks 70 yards forwards and then 100 yards backwards his distance from the starting-point would be 30 yards, but on the opposite side of it. As before we have

100 yards - 70 yards= +30 yards,

70 yards – 100 yards= -30 yards, and here we see that the negative sign may be taken as indicating a reversal of direction.

Many other illustrations might be chosen; but it will be sufficient here to remind the student that some consistent interpretation can always be given, when necessary, to the negative quantities he may meet with. In applying the rules we are about to give for the addition and subtraction of algebraical quantities, it is not necessary to recur constantly to the nature and meaning of the quantities employed, whether positive or negative, though it will be found by experience that these rules will always be justified when any of the symbolical results of Algebra are interpreted into the language of common life.

19. DEFINITION. When terms do not differ, or when they differ only in their numerical coefficients, they are called like, otherwise they are called unlike. Thus 3a, 7a; 5a2b, 2a2b; 3a’b, - 4a’b2 are pairs of like terms; and 4a, 36; 7a2, 9a2b are pairs of unlike terms.

RULES FOR ADDITION OF LIKE TERMS.
RULE I. The sum of a number of like terms is a like term.
RULE II. If all the terms are positive, add the coefficients.
Example. 8a +9a=17a.

The truth of this will be recognised by the beginner when he remembers that 8 lbs. added to 9 lbs. gives 17 lbs., and so

8a +9a=17a. Similarly

8a+ 9a + a + 2a + 7a=27a.

NEGATIVE QUANTITIES.

ADDITION OF LIKE TERMS.

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RULE III. If all the terms are negative, add the coefficients numerically and prefix the minus sign to the sum.

Example. The sum of - 3x, – 5x, 7x, - 2 is – 16x.

For a sum of money diminished successively by $3, $5, $7 and $1 is diminished altogether by $16.

RULE IV. If the terms are not all of the same sign, add together separately the coefficients of all the positive terms and the coefficients of all the negative terms; the difference of these two results, preceded by the sign of the greater, will give the coefficient of the sum required.

Example 1. The sum of 17x and - 80 is 9x, for the difference of 17 and 8 is 9, and the greater is positive.

Example 2. To find the sum of 8a, – 9a, – a, 3a, 4a, – 11a, a. The sum of the coefficients of the positive terms is 16,

negative ..... 21. The difference of these is 5, and the sign of the greater is negative; hence the required sum is - 5a.

We need not however adhere strictly to this rule, for since terms may

be added or subtracted in any order, we may choose the order we find most convenient.

Note. The sum of two quantities numerically equal but with opposite signs is zero. Thus the sum of 5a and 5a is 0.

Example 3. Find the sum of fa, 3a, – ja, – 2a.
The sum

=3a - 21a
=1}a
=ža.

EXAMPLES II.

Find the sum of

1. 5a, 7a, lla, a, 23a.

2. 4x, x, 3x, 7x, 9x. 3. 7b, 101, 11b, 96, 26.

4. 6c, 8c, 2c, 150, 19c, 100c, C. 5. -3x, - 5x, - 11x, -7x. 6. – 56, - 66, - 116, - 186. 7. - 31, -77, -y, - 2y, - 4y. 8. -C, - 2c, - 50c, – 13c. 9. - 116, -56. - 36, -6.

10. 5.x, - 3x, 2x, 11. 26y, -lly, - 15y, y, - 3y, 2y.

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12.5f, - 9f, - 3f, 218, – 30f. 13. 2s, - 3s, s, -S, - 5s, 5s. 14. 7, lly, 16y, 3y, - 2y. 15. 5x, 7X, 2x, 7x, 2x, – 5x. 16. 7ab, 3ab, 5ab, 2ab, ab.

Find the value of 17. - 9.82 +11x2 + 3x2 - 4x2. 18. 3a-x – 18a-x + a2x. 19. 3a3 - 7a - 84? + 2q3 – lla'. 20. 4.23 – 5.43 – 823 – 7.x3. 21. 4u2b2 a2b2 - 7a2b2 + 5a262 a2b2. 22. – 9.44 – 4.x4 – 12.04 + 13x4 – 7.x4. 23. 7abcd llabcd 41abcd +2ubcd. 24. 5x }x+x+3x.

25. a+ga-ta. 26. 5b+{6 - 36+21 16+1b. 27. - 5x2 – 2x2 – 3x2+2° +3+2+11.2.4. 28. - ab- fab- }ab - jab-fab+ab+bab. 29. sx- *x+3x-2x+ 11x - 3x+x. 30. - 22-32-422-12-2?.

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CHAPTER III.

SIMPLE BRACKETS. ADDITION.

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20. WHEN a number of arithmetical quantities are connected together by the signs + and -, the value of the result is the same in whatever order the terms are taken. This also holds in the case of algebraical quantities.

Thus a-b+ç is equivalent to a+c-b, for in the first of the two expressions b is taken from a, and c added to the result; in the second c is added to a, and b taken from the result. Similar reasoning applies to all algebraical expressions. Hence we may write the terms of an expression in any order we please.

Thus it appears that the expression a b may be written in the equivalent form 6+a.

To illustrate this we may suppose, as in Art. 18, that a represents a gain of a dollars, and -b a loss of b dollars : it is clearly immaterial whether the gain precedes the loss, or the loss precedes the gain.

21. A bracket () indicates that the terms enclosed within it are to be considered as one quantity. The full use of brackets will be considered in Chap. vii.; here we shall deal only with the simpler cases.

8+(13+5) means that 13 and 5 are to be added and their sum added to 8. It is clear that 13 and 5 may be added separately or together without altering the result. Thus

8+(13+5)=8+13+5=26. Similarly a +(6+c) means that the sum of b and c is to be added to a. Thus

a+b+c)=a+b+c.

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