Are irrational numbers like π relationships which are established by some rule or criteria?

Question

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?

Answer 1

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.

Answer 2

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.