Are irrational numbers like π relationships which are established by some rule or criteria? Rational and Irrational of Reals
Irrational numbers appear to fill in the ‘gaps’ between Rational numbers on a Real number line. However they seem to be stipulations or definitions of relationships which are established by some rule or criteria.
Take π, for an example: There is no precise definition of what it means. This fact becomes evident over the entire known history of this ‘number’ (relationship). It’s not the computation that is the problem; rather, the definition of its meaning.
Is π in this formula:
$area=πr^2$
π as an area component
the same as π in this formula:
$C = 2\pi r$
π as an arc component
They are stipulated or defined to be the same, but are they not acting as two completely different constants of proportionality?

There are uncountably many numbers on the real line. Most of them are just there and serve as a sort of "glue" in order to make the system ${\mathbb R}$ complete. The real numbers that actually do occur in mathematics as individuals of interest  are all defined by criteria, formulas, or algorithmic procedures, etc. 
The irrational number $\sqrt{2}$ is the unique positive real number whose square is $=2$. The number $e$ is defined, e.g., as limit
$$\lim_{n\to\infty}\left(1+{1\over n}\right)^n\ .$$
For highschool purposes $\pi$ is defined as ratio circumference/diameter of arbitrary circles; but of course at a higher level we could define $\pi$ by
$$\pi=4\int_0^1\sqrt{1-x^2}\>dx$$
without reference to folklore facts of elementary geometry.
