π is a/an

Concept: A rational number can be written exactly as p/q for integers p and q with q ≠ 0. Its decimal expansion terminates or eventually repeats.

An irrational number is a real number that cannot be written as such an integer ratio. Its decimal expansion is non-terminating and non-repeating; integers and prime numbers are narrower integer-based categories.

Application

The number π = 3.14159265358979… is proven irrational: it has no exact representation p/q with integer p and nonzero integer q. Therefore, decimal approximations such as 3.14 and 22/7 do not change its classification.

Contrast

• A rational number requires an exact integer ratio; 22/7 is rational but is only an approximation to π.

• A prime number must first be an integer greater than 1 and then have exactly two positive divisors; π is not an integer.

• An integer has no fractional part, whereas π lies strictly between 3 and 4.

Cross-check

Among the offered categories, only irrational number agrees with the proven absence of an exact integer-ratio representation for π.

Result: π is an irrational number.