Real numbers defined extensionally as the type of number that is useful for measuring things that people think of as continuous quantities.
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In practice, measurements are never precise enough to tell whether a number is irrational, though.
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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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