A radian is simply a measurement unit. The relation between a radian and real numbers is similar to the relation between a degree and real numbers or a metre and real numbers.
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Richard Dedekind and Georg Cantor.
Complex numbers are a proper superset of real numbers. That is to say, real numbers are a proper subset of complex numbers.
no
Integer numbers are a subset of real numbers. Real numbers may contain fractions.
A set can contain anything, or in some cases, nothing. Real numbers are an important and useful mathematical concept, and they are among the things that could be placed in various sets for various purposes.