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:

$\alpha =a+{\frac {1}{q_{1}}}+{\frac {1}{q_{1}q_{2}}}+\cdots +{\frac {1}{q_{1}q_{2}\cdots q_{n}}}+\cdots$.

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 $q_{v+1}=2q_{v}^{2}-1$. The ansatz

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

leads to:

$\alpha _{1}={\frac {q^{2}}{q^{2}-1}}={\frac {2q^{2}-1+1}{2q^{2}-1-1}}$.

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.

Prof. Epstein (Straßburg), privy counsil Hensel (Marburg) and Prof. Dr. Edler (Halle) participated in the discussion.

The remark of Prof. Epstein (Straßburg) should be mentioned. He notes how, by a minor modification of the method, Cantor's product expansion as well as the expansion by unit fractions stops for the case of a rational number.

The speaker replied that this also follows from his expansions, but that he emphasizes to get an infinite expansion in any case.