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Quadratic irrational numbers are perhaps the most basic examples of irrational numbers that arise as basic physical ratios: think of $\sqrt{2}$ as the distance between the corners of a square to the length of a side for example.

Similarly, the number $\pi$ is the ratio of the circumference of a circle to its diameter. Even $e$ can be realized in this sense as the distance $a$ such that $\int_{1}^{a}\frac{1}{x}dx=1$. Are there similar understandings for other irrationals?

Can Liouville's number be expressed as the ratio of two measurements of a geometrical object?

I will add my inspiration for asking this question in the hopes of encouraging more discussion that can help me focus these ideas. While pondering the argument for the existence of two irrational numbers $a$ and $b$ such that $a^b$ is rational using $\sqrt{2}^\sqrt{2}$ and the "law of the excluded middle", I wondered at the linked wikipedia article's note that an "intuitionist" would not accept this argument as proof. I feel like the only way to reject this argument as proof would be to question the assumption that $\sqrt{2}^\sqrt{2}$ exists as a real number, and in fact must be either rational, or irrational. (Any intuitionists want to way in?)

This led me to ask myself, how do I know that $\sqrt{2}$ exists? I of course immediately answered that it was the ratio of the diagonal of a square to the length of a side! This led me to consider the "existence" of other irrational numbers as "some sort" of "physical ratio."

In considering this I would (for now) accept the description of $e$ above as being a "physical ratio."

Can anyone help me define what I mean by these "physical ratios?" I suppose I am looking for some argument for the "existence" of otherwise incomprehensible numbers, and whether or not every irrational could be explained in such a way.

I of course must note that this is incredibly vague and that the only one who can truly decide what I mean is myself, but any advice would be great!

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There's probably an interesting question here, but perhaps you should be more specific about just what exactly you are allowing to be done physically. I realize that with a ruler and compass you can draw a curve whose length is $\pi$. What precisely is one allowed to do here in creating geometric objects? Are you admitting any curve length (or ratio of curve lengths) that can be constructed with a ruler and compass? –  Benjamin Dickman Dec 19 '12 at 6:30

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The answer is probably no.

In his paper Transcendence of Periods: the State of the Art (Pure and Applied Mathematics Quarterly, Volume 2, Number 2, p. 435-463, 2006), Michel Waldschmidt conjectures that no period is a Liouville number (see questions 2 & 3 in the introduction). This expectation is supported by the main advances of transcendental number theory in 20th century, notably Baker's theory of linear forms in logarithms and its extensions to commutative algebraic groups.

Here, the word period has to be understood in the sense of Kontsevich and Zagier: a complex number whose real and imaginary parts are values of absolutely convergent integral of rational functions with rational coefficients, over domains in $\mathbf R$ given by polynomial inequalities with rational coefficients. (The preceding definition is quoted from the paper Periods and elementary real numbers by Masahiko Yoshinaga, who was apparently the first to prove that periods belong to the field of elementary complex numbers, those whose real and imaginary parts can be effectively approximated by Cauchy sequences of rationals.)

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This is great input and I will be looking into this in detail! (By the way I am currently trying to narrow down my interests and this is exactly the kind of feed back that I was looking for). However, a quick look into this tells me that it is expected (conjectured by who?) that $e$ and $\frac{1}{\pi}$ are not in fact periods. I have edited my original question to include more background, and while I feel like periods would be examples of the numbers I am considering, I may actually be thinking of a larger collection. –  intuilog Dec 20 '12 at 5:44

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