When is a number line not a number line?

The number line is a seemingly simple object: a straight line with two points marked 0 and 1. Those two points are the seeds of great complexity, however. Whole numbers are located at positions marked off by iterating the interval. Fractions are located at equal subdivisions of the spaces between whole numbers. Flip all those numbers to the other side of 0 and you have negative rational numbers. Then, although the line is completely dense with rational numbers, you find you can sneak between them with infinite decimal expansions to define a whole universe of irrational numbers. Given all of these layers of complexity, when exactly is the right moment to introduce this marvelous object to students?

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