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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?

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    $\begingroup$ 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. $\endgroup$ – Arthur Dec 2 '17 at 10:06
  • $\begingroup$ 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,..." $\endgroup$ – Carey G. Butler Dec 2 '17 at 10:08
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    $\begingroup$ "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. $\endgroup$ – littleO Dec 2 '17 at 11:05
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    $\begingroup$ $\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 $\pi$. $\endgroup$ – user491874 Dec 2 '17 at 11:24
  • $\begingroup$ I changed the "area" and "arc", to which you refer, to "area component" and "arc component" to more clearly reflect my meaning. $\endgroup$ – Carey G. Butler Dec 4 '17 at 15:09
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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.

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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.

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