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10 The corresponding algebraic statements would be

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 other transactions, or perhaps wholly counterbalance a sum gained.

(ii) Suppose a man starting from a given point 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

In each of these cases the man's absolute distance from the starting-point is the same; but by taking the positive and negative signs into account, we see that $$-30$$ is a distance from the starting-point equal in magnitude but opposite in direction to the distance represented by $$+30$$. Thus the negative sign may here 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 a subtractive quantity is always opposite in character to an additive quantity of equal absolute value.

21. . When any number of quantities are connected by the signs $$+$$ and $$-$$, the resulting expression is called their algebraic sum. Thus $$11a - 27a + 13b - 5b$$ is an algebraic sum. This expression, however, is not, as will be shown, in its simplest form.