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19. 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 $$1\tfrac{3}{4} + 7\tfrac{2}{3} - 3\tfrac{1}{8} + 6 - 4\tfrac{1}{5}$$ 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, $$1\tfrac{3}{4}$$, to which no sign is prefixed, is counted among the additive terms. The same notions prevail in Algebra ; thus in using the expression $$7a + 3b - 4c - 2d$$ we understand the symbols $$7a$$ and $$3b$$ to be additive, while $$4c$$ and $$2d$$ are subtractive.

20. 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.

Hence all algebraic quantities may be divided 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.