Mixing
The approximation formula $left|alpha -frac{p}{q}right| le frac{1}{sqrt{5}q^2}$
$begingroup$
I have seen a result about the approximation of irrational numbers and want to find its proof.
Suppose $alpha$ is an irrational number, then there are infinitely many integers $p,q$ with $(p,q)=1$ such that
$$left|alpha -frac{p}{q}right| le frac{1}{sqrt{5}q^2}.$$
Is there a name attached to this theorem? Where can I find the proof of it?
Thanks in advance!
number-theory reference-request approximation
edited Jun 9 '16 at 23:44
Anton Grudkin
2,231719
asked Jun 9 '16 at 23:39
No OneNo One
2,0141519
$endgroup$
add a comment |
$begingroup$
I have seen a result about the approximation of irrational numbers and want to find its proof.
Suppose $alpha$ is an irrational number, then there are infinitely many integers $p,q$ with $(p,q)=1$ such that
$$left|alpha -frac{p}{q}right| le frac{1}{sqrt{5}q^2}.$$
Is there a name attached to this theorem? Where can I find the proof of it?
Thanks in advance!
number-theory reference-request approximation
edited Jun 9 '16 at 23:44
Anton Grudkin
2,231719
asked Jun 9 '16 at 23:39
No OneNo One
2,0141519
$endgroup$
5
$begingroup$
This is Hurwitz's theorem, look at refs in the link.
$endgroup$
– achille hui
Jun 9 '16 at 23:45
$begingroup$
@achillehui : do you think it requires many steps for being convinced of it once we know this inequality math.stackexchange.com/questions/1818499/… for algebraic numbers ?
$endgroup$
– reuns
Jun 9 '16 at 23:48
$begingroup$
@user1952009 I don't think so, the direction of inequality is different. In the Liouville's theorem, it gives a lower bound to how close an algebraic irrational number can be approximated by rational numbers. In the Hurwitz's theorem, it is an upper bound for arbitrary irrationals.
$endgroup$
– achille hui
Jun 9 '16 at 23:58
add a comment |
$begingroup$
I have seen a result about the approximation of irrational numbers and want to find its proof.
Suppose $alpha$ is an irrational number, then there are infinitely many integers $p,q$ with $(p,q)=1$ such that
$$left|alpha -frac{p}{q}right| le frac{1}{sqrt{5}q^2}.$$
Is there a name attached to this theorem? Where can I find the proof of it?
Thanks in advance!
number-theory reference-request approximation
edited Jun 9 '16 at 23:44
Anton Grudkin
2,231719
asked Jun 9 '16 at 23:39
No OneNo One
2,0141519
$endgroup$
I have seen a result about the approximation of irrational numbers and want to find its proof.
Suppose $alpha$ is an irrational number, then there are infinitely many integers $p,q$ with $(p,q)=1$ such that
$$left|alpha -frac{p}{q}right| le frac{1}{sqrt{5}q^2}.$$
Is there a name attached to this theorem? Where can I find the proof of it?
Thanks in advance!
number-theory reference-request approximation
number-theory reference-request approximation
edited Jun 9 '16 at 23:44
Anton Grudkin
2,231719
asked Jun 9 '16 at 23:39
No OneNo One
2,0141519
edited Jun 9 '16 at 23:44
Anton Grudkin
2,231719
asked Jun 9 '16 at 23:39
No OneNo One
2,0141519
edited Jun 9 '16 at 23:44
Anton Grudkin
2,231719
edited Jun 9 '16 at 23:44
Anton Grudkin
2,231719
edited Jun 9 '16 at 23:44
Anton Grudkin
2,231719
2,231719
asked Jun 9 '16 at 23:39
No OneNo One
2,0141519
asked Jun 9 '16 at 23:39
No OneNo One
2,0141519
asked Jun 9 '16 at 23:39
No OneNo One
2,0141519
2,0141519
5
$begingroup$
This is Hurwitz's theorem, look at refs in the link.
$endgroup$
– achille hui
Jun 9 '16 at 23:45
$begingroup$
@achillehui : do you think it requires many steps for being convinced of it once we know this inequality math.stackexchange.com/questions/1818499/… for algebraic numbers ?
$endgroup$
– reuns
Jun 9 '16 at 23:48
$begingroup$
@user1952009 I don't think so, the direction of inequality is different. In the Liouville's theorem, it gives a lower bound to how close an algebraic irrational number can be approximated by rational numbers. In the Hurwitz's theorem, it is an upper bound for arbitrary irrationals.
$endgroup$
– achille hui
Jun 9 '16 at 23:58
add a comment |
5
$begingroup$
This is Hurwitz's theorem, look at refs in the link.
$endgroup$
– achille hui
Jun 9 '16 at 23:45
$begingroup$
@achillehui : do you think it requires many steps for being convinced of it once we know this inequality math.stackexchange.com/questions/1818499/… for algebraic numbers ?
$endgroup$
– reuns
Jun 9 '16 at 23:48
$begingroup$
@user1952009 I don't think so, the direction of inequality is different. In the Liouville's theorem, it gives a lower bound to how close an algebraic irrational number can be approximated by rational numbers. In the Hurwitz's theorem, it is an upper bound for arbitrary irrationals.
$endgroup$
– achille hui
Jun 9 '16 at 23:58
5
5
$begingroup$
This is Hurwitz's theorem, look at refs in the link.
$endgroup$
– achille hui
Jun 9 '16 at 23:45
$begingroup$
This is Hurwitz's theorem, look at refs in the link.
$endgroup$
– achille hui
Jun 9 '16 at 23:45
$begingroup$
@achillehui : do you think it requires many steps for being convinced of it once we know this inequality math.stackexchange.com/questions/1818499/… for algebraic numbers ?
$endgroup$
– reuns
Jun 9 '16 at 23:48
$begingroup$
@achillehui : do you think it requires many steps for being convinced of it once we know this inequality math.stackexchange.com/questions/1818499/… for algebraic numbers ?
$endgroup$
– reuns
Jun 9 '16 at 23:48
$begingroup$
@user1952009 I don't think so, the direction of inequality is different. In the Liouville's theorem, it gives a lower bound to how close an algebraic irrational number can be approximated by rational numbers. In the Hurwitz's theorem, it is an upper bound for arbitrary irrationals.
$endgroup$
– achille hui
Jun 9 '16 at 23:58
$begingroup$
@user1952009 I don't think so, the direction of inequality is different. In the Liouville's theorem, it gives a lower bound to how close an algebraic irrational number can be approximated by rational numbers. In the Hurwitz's theorem, it is an upper bound for arbitrary irrationals.
$endgroup$
– achille hui
Jun 9 '16 at 23:58
add a comment |
1 Answer
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$begingroup$
The theorem in your question is known as Hurwitz's Theorem. It was first stated by A. Hurwitz in 1891 in the article "Ueber die angenäherte Darstellung der Irrationalzahlen durch rationale Brüche" (Eng. On the approximation of irrational numbers by rational numbers), Math. Ann. XXXIX, pp. 279-284 (1891).
Hurwitz's article can be found here: https://zbmath.org/?format=complete&q=an:23.0222.02, although the paper is in German.
answered Jan 3 at 13:58
KlangenKlangen
1,70111334
$endgroup$
add a comment |
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1 Answer
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1 Answer
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oldest
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$begingroup$
The theorem in your question is known as Hurwitz's Theorem. It was first stated by A. Hurwitz in 1891 in the article "Ueber die angenäherte Darstellung der Irrationalzahlen durch rationale Brüche" (Eng. On the approximation of irrational numbers by rational numbers), Math. Ann. XXXIX, pp. 279-284 (1891).
Hurwitz's article can be found here: https://zbmath.org/?format=complete&q=an:23.0222.02, although the paper is in German.
answered Jan 3 at 13:58
KlangenKlangen
1,70111334
$endgroup$
add a comment |
$begingroup$
The theorem in your question is known as Hurwitz's Theorem. It was first stated by A. Hurwitz in 1891 in the article "Ueber die angenäherte Darstellung der Irrationalzahlen durch rationale Brüche" (Eng. On the approximation of irrational numbers by rational numbers), Math. Ann. XXXIX, pp. 279-284 (1891).
Hurwitz's article can be found here: https://zbmath.org/?format=complete&q=an:23.0222.02, although the paper is in German.
answered Jan 3 at 13:58
KlangenKlangen
1,70111334
$endgroup$
add a comment |
$begingroup$
The theorem in your question is known as Hurwitz's Theorem. It was first stated by A. Hurwitz in 1891 in the article "Ueber die angenäherte Darstellung der Irrationalzahlen durch rationale Brüche" (Eng. On the approximation of irrational numbers by rational numbers), Math. Ann. XXXIX, pp. 279-284 (1891).
Hurwitz's article can be found here: https://zbmath.org/?format=complete&q=an:23.0222.02, although the paper is in German.
answered Jan 3 at 13:58
KlangenKlangen
1,70111334
$endgroup$
The theorem in your question is known as Hurwitz's Theorem. It was first stated by A. Hurwitz in 1891 in the article "Ueber die angenäherte Darstellung der Irrationalzahlen durch rationale Brüche" (Eng. On the approximation of irrational numbers by rational numbers), Math. Ann. XXXIX, pp. 279-284 (1891).
Hurwitz's article can be found here: https://zbmath.org/?format=complete&q=an:23.0222.02, although the paper is in German.
answered Jan 3 at 13:58
KlangenKlangen
1,70111334
answered Jan 3 at 13:58
KlangenKlangen
1,70111334
answered Jan 3 at 13:58
KlangenKlangen
1,70111334
answered Jan 3 at 13:58
KlangenKlangen
1,70111334
1,70111334
add a comment |
add a comment |
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5
$begingroup$
This is Hurwitz's theorem, look at refs in the link.
$endgroup$
– achille hui
Jun 9 '16 at 23:45
$begingroup$
@achillehui : do you think it requires many steps for being convinced of it once we know this inequality math.stackexchange.com/questions/1818499/… for algebraic numbers ?
$endgroup$
– reuns
Jun 9 '16 at 23:48
$begingroup$
@user1952009 I don't think so, the direction of inequality is different. In the Liouville's theorem, it gives a lower bound to how close an algebraic irrational number can be approximated by rational numbers. In the Hurwitz's theorem, it is an upper bound for arbitrary irrationals.
$endgroup$
– achille hui
Jun 9 '16 at 23:58