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Good Math by Mark C. Chu-Carroll

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The Reals, Informally

An informal and intuitive way to depict the real numbers is the number line that we learned in elementary school. Imagine a line that goes on forever in both directions. You can pick a spot on it and label it 0. To the right of 0, you mark off a second spot and label it 1. The distance between 0 and 1 is the distance between any two adjacent integers. So go the same distance to the right, make another mark, and label it 2. Keep doing that as far as you want. Then start going to the left from 0. The first mark is –1, the second –2, etc. That’s a basic number line. I’ve drawn a version of that in the following figure. Anyplace on that line that you look is a real real number. Halfway between 0 and 1 is the real number 1/2. ...

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