No they are not. The numbers Pi and e are irrational and are not radicals. There are many others.
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All rational and irrational numbers are real numbers.
All irrational numbers are not rational.
All irrational numbers are real, but not all real numbers are irrational.
No. All irrational numbers are real, not all real numbers are irrational.
There are infiitelt many subsets of irrational numbers. One possible subset is the set of all positive irrational numbers.