OEIS/Engel expansion: Difference between revisions
imported>Gfis q_2 |
imported>Gfis comma |
||
| Line 15: | Line 15: | ||
In the same way can be developped: | In the same way can be developped: | ||
:<math>\alpha=a+\frac{1}{q_1}+\frac{1}{q_1q_2}+\cdots+\frac{1}{q_1q_2\cdots q_n} + \cdots</math>. | :<math>\alpha=a+\frac{1}{q_1}+\frac{1}{q_1q_2}+\cdots+\frac{1}{q_1q_2\cdots q_n} + \cdots</math>. | ||
Now <math>\alpha</math> is rational if and only, | Now <math>\alpha</math> is rational if and only if, beginning at a certain <math>q_n</math>, always <math>q_{n+v+1} = q_{n+v}</math> holds. For <math>e</math> this leads to the known series expansion, and at the same time to a simple proof of the irrationality of <math>e</math>. By the way the same holds for each power <math>e^{\frac{1}{v}}</math>, where <math>v</math> is a positive integer number. | ||
Georg Cantor remarked already in 1869 in the ''[http://opacplus.bsb-muenchen.de/title/3084944/ft/bsb10594443?page=160 Zeitschrift für Mathematik und Physik]'' that each positive number <math>\alpha > 1</math> allows for a uniquely defined product expansion | Georg Cantor remarked already in 1869 in the ''[http://opacplus.bsb-muenchen.de/title/3084944/ft/bsb10594443?page=160 Zeitschrift für Mathematik und Physik]'' that each positive number <math>\alpha > 1</math> allows for a uniquely defined product expansion | ||
:<math>\alpha=a(1+\frac{1}{q_1})(\frac{1}{q_2})\cdots</math> | :<math>\alpha=a(1+\frac{1}{q_1})(1+\frac{1}{q_2})\cdots</math> | ||
in which the <math>q_n</math> are determined iteratively in the same way as described above. Here <math>q_{v+1} = {q_{v}}^2 - 1</math> must hold, and <math>\alpha</math> is rational if and only if, beginning at a certain <math>q_n</math>, always <math>q_{n+v+1} = {q_{n+v}}^2 </math> holds. The simple generation of product expansions which Cantor found for certain numbers like <math>\sqrt{2}, \sqrt{3}</math> etc. is based on the fact that for each positive number <math>q > 1</math>: | in which the <math>q_n</math> are determined iteratively in the same way as described above. Here <math>q_{v+1} = {q_{v}}^2 - 1</math> must hold, and <math>\alpha</math> is rational if and only if, beginning at a certain <math>q_n</math>, always <math>q_{n+v+1} = {q_{n+v}}^2 </math> holds. The simple generation of product expansions which Cantor found for certain numbers like <math>\sqrt{2}, \sqrt{3}</math> etc. is based on the fact that for each positive number <math>q > 1</math>: | ||
:<math>\sqrt{\frac{q+1}{q-1}} = (1+\frac{1}{q_1})(1+\frac{1}{q_2})\cdots</math> | :<math>\sqrt{\frac{q+1}{q-1}} = (1+\frac{1}{q_1})(1+\frac{1}{q_2})\cdots</math> | ||
Latest revision as of 14:03, 12 April 2018
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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha} there is a uniquely defined series expansion
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha=a+\frac{1}{q_1}+\frac{1}{q_2}+\cdots} ,
where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a, q_1, q_2\ldots} represent integer numbers and where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a < \alpha \leqq a+1} , while the numbers Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q_1, q_2\ldots.} are determined iteratively by the requirement that always
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha} is rational if and only if, beginning at a certain Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q_n} , always
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q_{n+v+1} = {q_{n+v}}^2 - q_{n+v}}
holds.
In the same way can be developped:
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha=a+\frac{1}{q_1}+\frac{1}{q_1q_2}+\cdots+\frac{1}{q_1q_2\cdots q_n} + \cdots} .
Now Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha} is rational if and only if, beginning at a certain Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q_n} , always Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q_{n+v+1} = q_{n+v}} holds. For Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle e} this leads to the known series expansion, and at the same time to a simple proof of the irrationality of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle e} . By the way the same holds for each power Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle e^{\frac{1}{v}}} , where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v} is a positive integer number.
Georg Cantor remarked already in 1869 in the Zeitschrift für Mathematik und Physik that each positive number Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha > 1} allows for a uniquely defined product expansion
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha=a(1+\frac{1}{q_1})(1+\frac{1}{q_2})\cdots}
in which the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q_n} are determined iteratively in the same way as described above. Here Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q_{v+1} = {q_{v}}^2 - 1} must hold, and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha} is rational if and only if, beginning at a certain Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q_n} , always Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q_{n+v+1} = {q_{n+v}}^2 } holds. The simple generation of product expansions which Cantor found for certain numbers like Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sqrt{2}, \sqrt{3}} etc. is based on the fact that for each positive number Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q > 1} :
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sqrt{\frac{q+1}{q-1}} = (1+\frac{1}{q_1})(1+\frac{1}{q_2})\cdots}
where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q_1=q} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q_{v+1}=2 q_v^2 -1} . The ansatz
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{q+1}{q-1} = (1+\frac{1}{q})^2\alpha_1}
leads to:
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q_n} , always hold Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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.