OEIS/Engel expansion - Revision history

imported>Gfis: comma

2018-04-12T14:03:45Z

comma

← Older revision		Revision as of 14:03, 12 April 2018
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, 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.		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>

imported>Gfis: q_2

2018-04-12T13:54:27Z

q_2

← Older revision		Revision as of 13:54, 12 April 2018
Line 20:		Line 20:
	:<math>\alpha=a(1+\frac{1}{q_1})(\frac{1}{q_2})\cdots</math>		:<math>\alpha=a(1+\frac{1}{q_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_1~~})\cdots</math>		:<math>\sqrt{\frac{q+1}{q-1}} = (1+\frac{1}{q_1})(1+\frac{1}{q_2})\cdots</math>
	where <math>q_1=q</math> and <math>q_{v+1}=2 q_v^2 -1</math>. The ansatz		where <math>q_1=q</math> and <math>q_{v+1}=2 q_v^2 -1</math>. The ansatz
	:<math>\frac{q+1}{q-1}} = (1+\frac{1}{q}^2)\alpha_1</math>		:<math>\frac{q+1}{q-1} = (1+\frac{1}{q})^2\alpha_1</math>
	leads to:		leads to:
	:<math>\alpha_1 = \frac{q^2}{q^2-1}} = \frac{2 q^2 - 1 + 1}{2q^2-1-1}<math>.		:<math>\alpha_1 = \frac{q^2}{q^2-1} = \frac{2 q^2 - 1 + 1}{2q^2-1-1}</math>.
	In the product expansion of the square root of an arbitrary rational number there will, beginning at a certain <math>q_n</math>, always hold <math>q_{n+v+1} = 2{q_{n+v}}^2 - 1</math>, but the proof for that seems not to be so easy.		In the product expansion of the square root of an arbitrary rational number there will, beginning at a certain <math>q_n</math>, always hold <math>q_{n+v+1} = 2{q_{n+v}}^2 - 1</math>, but the proof for that seems not to be so easy.

imported>Gfis: fertig

2018-04-12T13:50:04Z

fertig

← Older revision		Revision as of 13:50, 12 April 2018
Line 19:		Line 19:
	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})(\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 ~~ifm~~ 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_1})\cdots</math>
	where ~~...~~ . The ansatz		where <math>q_1=q</math> and <math>q_{v+1}=2 q_v^2 -1</math>. The ansatz
			:<math>\frac{q+1}{q-1}} = (1+\frac{1}{q}^2)\alpha_1</math>
	leads to:		leads to:
			:<math>\alpha_1 = \frac{q^2}{q^2-1}} = \frac{2 q^2 - 1 + 1}{2q^2-1-1}<math>.
	In the product expansion of the square root of an arbitrary rational number there will, beginning at a certain qn , always hold ~~...~~, but the proof of that seems not to be so easy.		In the product expansion of the square root of an arbitrary rational number there will, beginning at a certain <math>q_n</math>, always hold <math>q_{n+v+1} = 2{q_{n+v}}^2 - 1</math>, 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.		Prof. Epstein (Straßburg), privy counsil Hensel (Marburg) and Prof. Dr. Edler (Halle) participated in the discussion.

imported>Gfis at 13:38, 12 April 2018

2018-04-12T13:38:19Z

← Older revision		Revision as of 13:38, 12 April 2018
Line 1:		Line 1:
	English translation of Friedrich Engel's speech: ''Entwicklung der Zahlen nach Stammbrüchen.'' Verhandlungen der 52. ~~Verammlung~~ Deutscher Philologen und Schulmänner, 1913, Marburg, pp. 190-191		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==		==Expansion of the numbers by unit fractions==
Line 9:		Line 9:
	where <math>a, q_1, q_2\ldots</math> represent integer numbers and where <math>a < \alpha \leqq a+1</math>, while the numbers <math>q_1, q_2\ldots.</math> are determined iteratively by the requirement that always		where <math>a, q_1, q_2\ldots</math> represent integer numbers and where <math>a < \alpha \leqq a+1</math>, while the numbers <math>q_1, q_2\ldots.</math> are determined iteratively by the requirement that always
	:<math>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}</math>		:<math>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}</math>
	must hold.		must hold. One finds that <math>q_{v+1} > {q_v}^2 - q_v</math> must hold and that vice versa each infinite series of the form shown above which fulfills this requirement is convergent. A number <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 - q_{n+v}</math>
	One finds that must hold and that vice versa each infinite series of the form shown above which fulfills this requirement is convergent. A number ~~''a '~~ is rational if and only if beginning at a certain qn always ~~...~~ holds.		holds.

	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>.
			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.

	Now ''a''is rational if and only if beginning at a certain qn always ... 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 ..., where v 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
			:<math>\alpha=a(1+\frac{1}{q_1})(\frac{1}{q_2})\cdots</math>
	Georg Cantor remarked already in 1869 in the ''Zeitschrift für Mathematik und Physik'' that each positive number ~~''a'' >~~ 1 allows for a uniquely defined product expansion		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 ifm 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 qn are determined iteratively in the same way as ~~previously~~ described. Here ~~...~~ must hold, and ~~''a''~~ is rational if and only if beginning at a certain qn always ~~...~~ holds. The simple generation of product expansions which Cantor found for certain numbers like 2, 3, etc. is based on the fact that for each positive number ''q~~' >~~ 1 :

	where ... . The ansatz		where ... . The ansatz

imported>Gfis at 13:08, 12 April 2018

2018-04-12T13:08:36Z

← Older revision		Revision as of 13:08, 12 April 2018
Line 4:		Line 4:
	Thereafter Prof. Dr. Engel (Gießen) rose to speak about '''Expansion of the numbers by unit fractions.''' The speaker explains:		Thereafter Prof. Dr. Engel (Gießen) rose to speak about '''Expansion of the numbers by unit fractions.''' The speaker explains:

	For each positive number ~~''a''~~ there is a uniquely defined series expansion		For each positive number <math>\alpha</math> there is a uniquely defined series expansion
	:<math>x=~~\frac{1}{a_1}~~+\frac{1}{~~a_1a_2~~}+\frac{1}{~~a_1a_2a_3~~}+\cdots.</math>		:<math>\alpha=a+\frac{1}{q_1}+\frac{1}{q_2}+\cdots</math>,
	where ''a '' represent integer numbers and where a < , while the numbers ~~q ..~~. are determined iteratively by the requirement that always
			where <math>a, q_1, q_2\ldots</math> represent integer numbers and where <math>a < \alpha \leqq a+1</math>, while the numbers <math>q_1, q_2\ldots.</math> are determined iteratively by the requirement that always
			:<math>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}</math>
			must hold.

	One finds that must hold and that vice versa each infinite series of the form shown above which fulfills this requirement is convergent. A number ''a ' is rational if and only if beginning at a certain qn always ... holds.		One finds that must hold and that vice versa each infinite series of the form shown above which fulfills this requirement is convergent. A number ''a ' is rational if and only if beginning at a certain qn always ... holds.

imported>Gfis at 09:11, 11 April 2018

2018-04-11T09:11:10Z

← Older revision		Revision as of 09:11, 11 April 2018
Line 5:		Line 5:

	For each positive number ''a'' there is a uniquely defined series expansion		For each positive number ''a'' there is a uniquely defined series expansion
			:<math>x=\frac{1}{a_1}+\frac{1}{a_1a_2}+\frac{1}{a_1a_2a_3}+\cdots.</math>
	where ''a '' represent integer numbers and where a < , while the numbers q ... are determined iteratively by the requirement that always		where ''a '' represent integer numbers and where a < , while the numbers q ... are determined iteratively by the requirement that always

imported>Gfis: v2

2018-04-10T20:21:42Z

← Older revision		Revision as of 20:21, 10 April 2018
Line 1:		Line 1:
	English translation of Friedrich Engel's speech: ''Entwicklung der Zahlen nach Stammbrüchen.'' Verhandlungen der 52. Verammlung Deutscher Philologen und Schulmänner, 1913, Marburg, pp. 190-191		English translation of Friedrich Engel's speech: ''Entwicklung der Zahlen nach Stammbrüchen.'' Verhandlungen der 52. Verammlung Deutscher Philologen und Schulmänner, 1913, Marburg, pp. 190-191

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

Line 12:		Line 12:
	In the same way can be developped:		In the same way can be developped:

	Now ''a''is rational if and only if beginning at a certain qn always ... 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 ..., where v is a positive integer number.		Now ''a''is rational if and only if beginning at a certain qn always ... 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 ..., where v is a positive integer number.

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

	in which the qn are determined iteratively in the same way as previously described.		in which the qn are determined iteratively in the same way as previously described. Here ... must hold, and ''a'' is rational if and only if beginning at a certain qn always ... holds. The simple generation of product expansions which Cantor found for certain numbers like 2, 3, etc. is based on the fact that for each positive number ''q' > 1 :

			where ... . The ansatz

			leads to:

			In the product expansion of the square root of an arbitrary rational number there will, beginning at a certain qn , always hold ..., but the proof of 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.

imported>Gfis: v1

2018-04-10T19:57:51Z

New page

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

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

For each positive number ''a'' there is a uniquely defined series expansion

where ''a '' represent integer numbers and where a < , while the numbers q ... are determined iteratively by the requirement that always

One finds that must hold and that vice versa each infinite series of the form shown above which fulfills this requirement is convergent. A number ''a ' is rational if and only if beginning at a certain qn always ... holds.

In the same way can be developped:

Now ''a''is rational if and only if beginning at a certain qn always ... 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 ..., where v is a positive integer number.

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

in which the qn are determined iteratively in the same way as previously described.