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?


Two hints:
  
  
*
  
*Irrational numbers can be built from the rational numbers and then identified with specific Dedekind cuts or with specific equivalence classes of Cauchy-sequences.
This way we see an intimate relationship of the irrational numbers with the rational numbers based upon two construction principles.
  
*The book Mathematical constants by Steven R. Finch provides a thorough guided tour about named mathematical constants, many of them being irrational numbers.
Going through this book and looking for connections and different relationships
  might be fruitful and interesting.

