Are there examples that we know a number is a rational number but we do not know what is its numerator and denominator?

Question

Are there examples that we know a number is a rational number but we do not know what is its numerator and denominator?

In order to say clearly, this number should given by a certain formula, such as $\sum_{i=1}^\infty f(n)$ (or $\int_0^\infty f(x)dx$)where $f(n)$ is a certain function so that we can calculate $f(n)$ for any given integer number $n$ (or real number $x$). Hence we avoid the answers like "the least even integer $N$ which makes Goldbach conjecture not true" or "the age when I get married".

Thanks in advance!

Answer 1

Take any Turing machine $M$, and let $h_M$ be $\frac{1}{N}$ iff $M$ halts after exactly $N$ steps, and let $h_M := 0$ if $M$ never halts. It is clear that $h_M$ is a rational number, given $M$ we can compute $h_M$ as a real number (i.e. find better and better approximations to it), but there is no general procedure to write $h_M$ as a quotient of natural numbers (because that would mean solving the Halting problem).

Edit: I might point out that there is a topological counterpart to this argument, namely the observation that the $id : \mathbb{Q}_e \to \mathbb{Q}_d$ is discontinuous, where $\mathbb{Q}_e$ are the rationals with the subspace topology inherited from $\mathbb{R}$, while $\mathbb{Q}_d$ are the rationals with the discrete topology.

Answer 2

Let $E$ be an elliptic curve over $\mathbb Q$ of (algebraic) rank $0$. Let $L(E,s)$ be its associated $L$-function. Then the quantity $L(E,1)/\Omega_E$, where $\Omega_E$ is the period of $E$, is known to be rational, even if we don't know its numerator and denominator.