The unit for real numbers is the number 1.
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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.
No, it is imaginary. Irrational numbers are a subset of real numbers Real numbers and imaginary numbers are sets without any overlap.
The real number system is a number system using the rational and irrational numbers.
All real numbers can be represented in the decimal system. Complex numbers can be represented by a pair of numbers in the decimal system.
It would be a bit stupid to call a system the real number system if real numbers were not a part of it!