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: $area=πr^2$
the same as π in this formula: $C = 2\pi r$
They are stipulated or defined to be the same, but are they not acting as two completely different constants of proportionality?