# 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?

• Not all irrational numbers fulfill some relation or criteria (assuming you mean "criteria" as something identifying a single, unique irrational number). In fact, most of them don't, but we can't give you any concrete examples, because that would be a criteria that identified one of them. – Arthur Dec 2 '17 at 10:06
• en.wikipedia.org/wiki/Irrational_number stipulates them to be "In mathematics, the irrational numbers are all the real numbers which are not rational numbers,..." – Carey G. Butler Dec 2 '17 at 10:08
• "There is no precise definition of what it means" Sure there is. On p. 302 of Calculus by Spivak, $\pi$ is defined to be $2 \int_{-1}^1 \sqrt{1 - x^2} \, dx$. Other authors might start with a different definition of $\pi$, but we can prove that the various definitions of $\pi$ are equivalent. "They are stipulated or defined to be the same" They are proved to be the same, if you read the right books. – littleO Dec 2 '17 at 11:05
• $\pi$ is a *number*. Not an area. Not an arc. It is dimensionless, the dimension comes from $r$ (in your examples). It is a known number that just doesn't happen to be a ratio of two integers. There are many formulas that let you calculate it to an arbitrary number of decimals. It does appear in a lot of places in the mathematics, but this is not a wonder; it is more that we (humans) recognised a number that often pops up and gave it a name $\pi$. – user491874 Dec 2 '17 at 11:24
• I changed the "area" and "arc", to which you refer, to "area component" and "arc component" to more clearly reflect my meaning. – Carey G. Butler Dec 4 '17 at 15:09

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.

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.