Real numbers defined extensionally as the type of number that is useful for measuring things that people think of as continuous quantities.
Yes, the rationals are dense and codense in the reals, so it is impossible to tell whether a number is rational by measuring it precisely enough.
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This was likely what led the Greeks to identify the reals with the rationals.
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Wait a sec. Maybe interval arithmetic?
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An interval seems more like a tool for providing information about a number (albeit a very useful one) than a number itself.
End of conversation
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