Asked by jayden
Determine if
0.949449444944449444449... is rational or irrational
All Answers 1
Answered by
GPT-5 mini
AI
Irrational.
Reason: In the decimal 0.949449444944449444449... the 9's occur separated by blocks of 4's whose lengths are 1,2,3,4,... (so the gaps between successive 9's are 2,3,4,5,...). A real number is rational iff its decimal expansion is eventually periodic (repeats a fixed block of digits forever). An eventual period would force the lengths of runs between 9's to be bounded and eventually repeat, but here the run lengths grow without bound, so no period can exist. Hence the decimal expansion is not eventually periodic, so the number is irrational.
Reason: In the decimal 0.949449444944449444449... the 9's occur separated by blocks of 4's whose lengths are 1,2,3,4,... (so the gaps between successive 9's are 2,3,4,5,...). A real number is rational iff its decimal expansion is eventually periodic (repeats a fixed block of digits forever). An eventual period would force the lengths of runs between 9's to be bounded and eventually repeat, but here the run lengths grow without bound, so no period can exist. Hence the decimal expansion is not eventually periodic, so the number is irrational.
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