Hurwitz theorem (number theory)
The mathematics knows a number of sets which the name of Adolf Hurwitz are linked. The Hurwitz's theorem of number theory concerns the so-called diophantine approximations of irrational numbers , so the approximation of irrational numbers by fractions . The sentence specifies an upper limit for the quality of the approximation.
The sentence
The sentence can be formulated as follows:
For every irrational number there are infinitely many fully abbreviated fractions , which
fulfill.
In the proof of the theorem developed by Scheid, properties of the Farey sequences are used in a decisive way .
Goodness of the ceiling
The constant is sharp, so in general it cannot be replaced by a better constant. This can be proven using the irrational number (known in connection with the golden ratio ).
For a single number there may be better approximations, e.g. B. for Liouville numbers . If it is an algebraic number , the exponent of cannot be improved according to the Thue-Siegel-Roth theorem .
Related results
literature
- Adolf Hurwitz: About the approximate representation of irrational numbers through rational fractions . In: Math. Ann. , 39, 1891, pp. 279-284
- Harald Scheid : Number Theory . 3. Edition. Spectrum Academic Publishing House, Heidelberg (among others) 2003, ISBN 3-8274-1365-6 .
- Jurjen Ferdinand Koksma : Diophantine approximations . 3. Edition. Springer-Verlag, Berlin 1936.
- William Judson LeVeque : Fundamentals of Number Theory . Addison-Wesley, Reading MA 1977, ISBN 0-201-04287-8 .
Individual evidence
- ↑ Harald Scheid: Number theory . 3. Edition. Spectrum Akademischer Verlag, Heidelberg (among others) 2003, ISBN 3-8274-1365-6 , p. 64 .
- ↑ Harald Scheid: Number theory . 3. Edition. Spectrum Akademischer Verlag, Heidelberg (among others) 2003, ISBN 3-8274-1365-6 , p. 64-65 .
- ↑ Harald Scheid: Number theory . 3. Edition. Spectrum Akademischer Verlag, Heidelberg (among others) 2003, ISBN 3-8274-1365-6 , p. 65 .