Irrationality Measure

Author

Eric W. Weisstein
January 23, 2004

This notebook downloaded from http://mathworld.wolfram.com/notebooks/NumberTheory/IrrationalityMeasure.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/IrrationalityMeasure.html.

©2005 Wolfram Research, Inc. except for portions noted otherwise

Examples

π

Max[Divide@@@Partition[Log/@Denominator/@FromContinuedFraction/@NestList[Most,ContinuedFraction[Pi,100],98],2,1]+1//N]

(π^2)

Max[Divide@@@Partition[Log/@Denominator/@FromContinuedFraction/@NestList[Most,ContinuedFraction[Pi^2,100],97],2,1]+1//N]

log(2)

Max[Divide@@@Partition[Log/@Denominator/@FromContinuedFraction/@NestList[Most,ContinuedFraction[Log[2],100],97],2,1]+1//N]

Pi

Zeilberger and Zudlin 2019

https://arxiv.org/pdf/1912.06345.pdf

N/.Solve[108N^3−2359989N^2+138304N−2048==0,N]

{n3,n1}=Union[RootReduce[Abs/@%]]

InputForm[%]

ToRadicals[%]

N[a=Log[n1]-(5/2)Log[2]+4-(π/(2(Sqrt[3])))+Log[((3(Sqrt[3]))/4)],20]

N[b=Log[n3]-(5/2)Log[2]+4-(π/(2(Sqrt[3])))+Log[((3(Sqrt[3]))/4)],20]

N[1-b/a,20]

FullSimplify[1-b/a]

RootReduce[%]

InputForm[%]