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Good Math by Mark C. Chu-Carroll

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The Reals, Constructively

Finally, the constructive definition—a constructive definition is a procedure for creating the set of real numbers. We can think of the real numbers as the union of several different sets.

First, we’ll take the integers. All of the integers are real numbers, with exactly the properties they had as integers.

Then we’ll add fractions, which are formally called rational numbers. A rational number is defined by a pair of non-zero integer numbers called a ratio. A ratio n/d represents a real number that when multiplied by d gives the value n. The set of numbers constructed this way ends up with lots of equivalent values, such as 1/2, 2/4, 3/6, and so on. As we did with the integers, we’ll define the rationals as a set ...

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