Real, complex, quaternion, etc. But more specifically it's a transcendental real number, not being the root of an integer polynomial (and certainly not the ratio of two integers).
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Pi is an Irrational number, which is one of the two subcategories of real numbers.
Pi is both an irrational number and a transcendental number.
The mathematically correct answer is: any set that contains it. For example, it belongs to the set of all numbers between -3 and +2, the set {0, -3, 8/13, sqrt(97), pi}, the set {0}, the set of the roots of x3 - x2 + x = 0, the set of all integers, the set of all rational numbers, the set of all real numbers, the set of all complex numbers.
It belongs to the set of prime numbers
Rational and Real numbers