The real number system can be sliced and diced in an infinite number of ways. In other words, you can't list all the different sets of numbers in the real number system. However, here are some of the commonest:
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Real numbers are closed under addition and subtraction. To get a number outside the real number system you would have to use square root.
It would be a bit stupid to call a system the real number system if real numbers were not a part of it!
The real number system is a number system using the rational and irrational numbers.
real number system (diagram) and explain it
There are many different ways to divide the real number system into sub-groups. Perhaps the most obvious division is, positive numbers and negative numbers.