If you had a case of the circumference and the diameter both being integers, then pi would be a rational number. There is no polynomial f(x) with integer coeff

If you had a case of the circumference and the diameter both being integers, then pi would be a rational number. There is no polynomial f(x) with integer coefficients such that f(pi) = 0. In the language of the post, if you have pi, and you're allowed to add, subtract, multiply, and divide by integers, and you're allowed to multiply by pi, you can never and up with 0. So pi can be expressed as the ratio of the diameter to the circumference, but when you do so, at least one of the diameter and circumference will themselves be transcendental.