endpoint will lie at the point labeled by the length of the segment. To encompass negative numbers, you must give your segments more structure. You must provide them with an orientation—a beginning and an end, a head and a tail. These oriented segments may be represented as arrows. The positive numbers are then represented by arrows that begin at the origin and end at the positive number that gives their length. Negative numbers are represented by arrows that begin at the origin and end at the negative number. That way, 4 and –4, for example, have the same length but opposite orientation. (Note: For clarity, arrows are shown above rather than on the number line.)

Suppose I want to compute 4+3 on the number line. It is difficult to add the arrows when they both begin at the origin:

But the arrows may be moved left or right, as needed, as long as they maintain the same length and orientation. To add the arrows, I move the second arrow so that it begins at the end of the first arrow.


The result of the addition is an arrow that extends from the beginning of the first arrow to the end of the second arrow.

This geometric approach is quite general: It works for negative integers and rational numbers, although in the latter case it is hard to interpret the answer in simple form without dividing the intervals according to a common denominator.

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