A "rational number" is a fraction, a ratio of whole numbers. In the sense that integer arithmetic is exact, rational arithmetic is also exact. (By contrast, evaluating rational numbers in floating point form entails round-off error.) In a first real analysis course, students learn that every real number can be approximated arbitrarily closely by rational numbers. The image suggests a proof by writing the set of rational numbers as a union of scaled copies of the integers. The image also depicts the complicated way real numbers are approximated by rational numbers of bounded denominator.
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