rules in Boxes 3–1 to 3–5 all hold. In both systems, all arithmetic is determined by these rules.

Finally, the two procedures actually produce the same system. The end result is essentially the same, whether one first annexes the negatives and then the fractions, or the other way around. The hard part is making sure that you can actually do it—that there really is a system in which you can add, subtract, multiply, and divide, and where all the rules work in harmony to tell you how to do it. Mathematicians call this system the rational numbers.

Arithmetic into Geometry—The Number Line

The rational numbers are harder to visualize than the whole numbers or even the integers, but there is a picture that lets you think about rational numbers geometrically. It lets you interpret whole numbers, negative numbers, and fractions all as part of one overall system. Furthermore, it provides a uniform way to extend the rational number system to include numbers such as p and that are not rational;9 it provides a link between arithmetic and geometry; and it paves the way for analytic geometry, which connects algebra and geometry. This conceptual tool is called the number line. It can be seen in a rudimentary way in many classrooms, but its potential for organizing thinking about number and making connections with geometry seems not to have been fully exploited. Finding out how to realize this potential might be a profitable line of research in mathematics education.

The potential for organizing thinking about number and making connections with geometry seems not to have been fully exploited.

The number line is simply a line, but its points are labeled by numbers. One point on the line is chosen as the origin. It is labeled 0. Then a positive direction (usually to the right) is chosen for the line. This choice amounts to specifying which side of the origin will be the positive half of the line; the other side is then the negative half. Finally, a unit of length is chosen. Any point on the line is labeled by its (directed) distance from the origin measured according to this unit length. The point is labeled positive if it is on the positive half of the line and as negative if it is on the negative half. The integers, then, are the points that are a whole number of units to the left or the right of the origin. Part of the number line is illustrated below, with some points labeled.10



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