Liouville's number, the easiest transcendental and its clones (corrected reupload)
This is a corrected re-upload of a video from a couple of weeks ago. The original version contained one too many shortcut that I really should not have taken. Although only two viewers stumbled across this mess-up it really bothered me, and so here is the corrected version of the video, hopefully free of any more reupload-worthy mistakes. For those of you who already watched the previous version of this video, see whether you can figure out what required fixing :)
This video is all about convincing you that Liouville's number is really a transcendental number. I am presenting a proof for this fact that you won't find in any textbook and I am keeping my fingers crossed that people will agree that this is the most accessible proof of the transcendence of any specific number. Also part of this video is a nice way to create a clone of the real numbers using the Liouville's number as a template. This clone is a seriously paradoxical subset of the reals: it consists entirely of transcendental numbers (with one exception), just like the reals it is uncountably infinite AND of it is of measure 0, that is, it is hidden so well within the reals that in a sense it not even there.
The measure 0 extra video is at https://youtu.be/4ga58IP1iJU on Mathologer 2.
Liouville's original paper is here:
Liouville, J. "Sur des classes très-étendues de quantités dont la valeur n'est ni algébrique, ni même réductible à des irrationelles algébriques." J. Math. pures appl. 16, 133-142, 1851.
http://sites.mathdoc.fr/JMPA/PDF/JMPA_1851_1_16_A5_0.pdf
And if you are interested in having a look at this proof as it also appears in all the textbooks here is one possible reference:
http://people.math.sc.edu/filaseta/gradcourses/Math785/Math785Notes5.pdf
The proof that I am showing you in this video was inspired by Conway and Guy's take on the subject in their "Book of numbers". In particular, if you are familiar with this book you'll also recognise the 6th degree polynomial that I am using as one of the examples.
This week's t-shirt is from here: https://shirt.woot.com/offers/liars-paradox
You can download the comments of the original video as a pdf file here: http://www.qedcat.com/misc/comments.pdf.
Thank you very much for my friends Marty Ross for his feedback on a draft of this video and Danil Dmitriev for his Russian subtitles.
Enjoy!
Видео Liouville's number, the easiest transcendental and its clones (corrected reupload) канала Mathologer
This video is all about convincing you that Liouville's number is really a transcendental number. I am presenting a proof for this fact that you won't find in any textbook and I am keeping my fingers crossed that people will agree that this is the most accessible proof of the transcendence of any specific number. Also part of this video is a nice way to create a clone of the real numbers using the Liouville's number as a template. This clone is a seriously paradoxical subset of the reals: it consists entirely of transcendental numbers (with one exception), just like the reals it is uncountably infinite AND of it is of measure 0, that is, it is hidden so well within the reals that in a sense it not even there.
The measure 0 extra video is at https://youtu.be/4ga58IP1iJU on Mathologer 2.
Liouville's original paper is here:
Liouville, J. "Sur des classes très-étendues de quantités dont la valeur n'est ni algébrique, ni même réductible à des irrationelles algébriques." J. Math. pures appl. 16, 133-142, 1851.
http://sites.mathdoc.fr/JMPA/PDF/JMPA_1851_1_16_A5_0.pdf
And if you are interested in having a look at this proof as it also appears in all the textbooks here is one possible reference:
http://people.math.sc.edu/filaseta/gradcourses/Math785/Math785Notes5.pdf
The proof that I am showing you in this video was inspired by Conway and Guy's take on the subject in their "Book of numbers". In particular, if you are familiar with this book you'll also recognise the 6th degree polynomial that I am using as one of the examples.
This week's t-shirt is from here: https://shirt.woot.com/offers/liars-paradox
You can download the comments of the original video as a pdf file here: http://www.qedcat.com/misc/comments.pdf.
Thank you very much for my friends Marty Ross for his feedback on a draft of this video and Danil Dmitriev for his Russian subtitles.
Enjoy!
Видео Liouville's number, the easiest transcendental and its clones (corrected reupload) канала Mathologer
Показать
Комментарии отсутствуют
Информация о видео
Другие видео канала
Transcendental numbers powered by Cantor's infinitiesTimes Tables, Mandelbrot and the Heart of MathematicsFermat’s HUGE little theorem, pseudoprimes and FuturamaAll the Numbers - NumberphileEuler's real identity NOT e to the i pi = -1Secret of row 10: a new visual key to ancient Pascalian puzzlesInfinity shapeshifter vs. Banach-Tarski paradoxVisualising Pythagoras: ultimate proofs and crazy contortionsComplex number fundamentals | Lockdown math ep. 3Epicycles, complex Fourier series and Homer Simpson's orbitThe PROOF: e and pi are transcendentalNYT: Sperner's lemma defeats the rental harmony problemIrrational Roots700 years of secrets of the Sum of Sums (paradoxical harmonic series)Power sum MASTER CLASS: How to sum quadrillions of powers ... by hand! (Euler-Maclaurin formula)Transcendental Numbers - NumberphileIndeterminate: the hidden power of 0 divided by 0Death by infinity puzzles and the Axiom of Choice9.999... really is equal to 10