2017-06-20

[public] 136K views, 3.77K likes, 46.0 dislikes audio only

Mathologer

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!