Yes.
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All prime numbers are rational.
No prime numbers are irrational: All prime numbers are integers, and all integers are rational, since they can be expressed as themselves divided by 1.
All prime numbers are rational number because they can be expressed as 'top heavy' fractions as for example the prime number 3 as a fraction is 3/1
There is no specific name. Such numbers include all irrational numbers, all non-integral rational numbers, 0 and 1.
All rational and irrational numbers are real numbers.