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To the many people who attempted to correct me about what portion of the reals we can use, the computable numbers are exactly that portion (or, as I said, the subset the of it we are large enough to represent). Look up "computable number". Pi, and all rational multiples of pi, are in that set, for instance.

It has many interesting properties. Among those properties is that it contains every number you could every conclusively identify, and that it is only countably infinite, meaning it is much smaller than the reals. It is useful for many things (and a good thing it is, since it is the most sophisticated set of numbers we actually get to compute with), but it is no substitute for the reals.

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