# OEIS/Engel expansion

English translation of Friedrich Engel's speech: Entwicklung der Zahlen nach Stammbrüchen. Verhandlungen der 52. Versammlung Deutscher Philologen und Schulmänner, 1913, Marburg, pp. 190-191

## Expansion of the numbers by unit fractions

Thereafter Prof. Dr. Engel (Gießen) rose to speak about Expansion of the numbers by unit fractions. The speaker explains:

For each positive number $\alpha$ there is a uniquely defined series expansion

$\alpha=a+\frac{1}{q_1}+\frac{1}{q_2}+\cdots$,

where $a, q_1, q_2\ldots$ represent integer numbers and where $a < \alpha \leqq a+1$, while the numbers $q_1, q_2\ldots.$ are determined iteratively by the requirement that always

$a+\frac{1}{q_1}+\cdots+\frac{1}{q_n} < \alpha \leqq a+\frac{1}{q_1}+\cdots+\frac{1}{q_{n-1}} + \frac{1}{q_n - 1}$

must hold. One finds that $q_{v+1} > {q_v}^2 - q_v$ must hold and that vice versa each infinite series of the form shown above which fulfills this requirement is convergent. A number $\alpha$ is rational if and only if, beginning at a certain $q_n$, always

$q_{n+v+1} = {q_{n+v}}^2 - q_{n+v}$

holds.

In the same way can be developped:

$\alpha=a+\frac{1}{q_1}+\frac{1}{q_1q_2}+\cdots+\frac{1}{q_1q_2\cdots q_n} + \cdots$.

Now $\alpha$ is rational if and only if, beginning at a certain $q_n$, always $q_{n+v+1} = q_{n+v}$ holds. For $e$ this leads to the known series expansion, and at the same time to a simple proof of the irrationality of $e$. By the way the same holds for each power $e^{\frac{1}{v}}$, where $v$ is a positive integer number.

Georg Cantor remarked already in 1869 in the Zeitschrift für Mathematik und Physik that each positive number $\alpha > 1$ allows for a uniquely defined product expansion

$\alpha=a(1+\frac{1}{q_1})(1+\frac{1}{q_2})\cdots$

in which the $q_n$ are determined iteratively in the same way as described above. Here $q_{v+1} = {q_{v}}^2 - 1$ must hold, and $\alpha$ is rational if and only if, beginning at a certain $q_n$, always $q_{n+v+1} = {q_{n+v}}^2$ holds. The simple generation of product expansions which Cantor found for certain numbers like $\sqrt{2}, \sqrt{3}$ etc. is based on the fact that for each positive number $q > 1$:

$\sqrt{\frac{q+1}{q-1}} = (1+\frac{1}{q_1})(1+\frac{1}{q_2})\cdots$

where $q_1=q$ and $q_{v+1}=2 q_v^2 -1$. The ansatz

$\frac{q+1}{q-1} = (1+\frac{1}{q})^2\alpha_1$

$\alpha_1 = \frac{q^2}{q^2-1} = \frac{2 q^2 - 1 + 1}{2q^2-1-1}$.
In the product expansion of the square root of an arbitrary rational number there will, beginning at a certain $q_n$, always hold $q_{n+v+1} = 2{q_{n+v}}^2 - 1$, but the proof for that seems not to be so easy.