Are irrational numbers real numbers?
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The distance between 2^0.5 + 3^0.5 is 3^0.5 . But that would involve the measurement of infinitesimal parts. Infinitesimals are not real numbers.
Not it has been proven that all irrationals can be written as the non terminating sequence of rationals, but that does not make them real, if they contain infinitesimal parts.