The main subgroup is the rational numbers. The set of irrational numbers is not closed with regard to addition basic arithmetical operations and so does not form a group.
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The set of real numbers can be divided into rational numbers and irrational numbers.
The real numbers are divided into rational numbers and irrational numbers.
Real Numbers cannot be the square root of a negative number. Real Numbers are not divided by zero. Basically, Real Numbers cannot be anything that is undefined.
Yes
Yes. Real numbers can be added, subtracted , multiplied and divided by nonzero numbers in an ordered way. x, for example, is not a real number.