Transcendental numbers powered by Cantor's infinities

In today's video the Mathologer sets out to give an introduction to the notoriously hard topic of transcendental numbers that is both in depth and accessible to anybody with a bit of common sense. Find out how Georg Cantor's infinities can be used in a very simple and off the beaten track way to pinpoint a transcendental number and to show that it is really transcendental. Also find out why there are a lot more transcendental numbers than numbers that we usually think of as numbers, and this despite the fact that it is super tough to show the transcendence of any number of interest such as pi or e. Also featuring an animated introduction to countable and uncountable infinities, Joseph Liouville's ocean of zeros constant, and much more.

Here is a link to one of Georg Cantor's first papers on his theory of infinite sets. Interestingly it deals with the construction of transcendental numbers!
Cantor, Georg (1874), "Ueber eine Eigenschaft des Inbegriffes aller reellen algebraischen Zahlen", Journal für die Reine und Angewandte Mathematik, 77: 258–262

Here is the link to the free course on measure theory by my friend Marty Ross who I also like to thank for his help with finetuning this video:
(it's the last collection of videos at the bottom of the linked page).

Thank you also very much to Danil Dmitriev the official Mathologer translator for Russian for his subtitles.

Enjoy!

These Zazzle t-shirt are very good quality, but way too expensive (at least for my taste). If you are really keen on one of their t-shirts I recommend waiting for one of their 50% off on t-shirts promotions.

[MrZyroid]

At 16:57 I was expecting him to say "But, the margin is too small to contain my proof"

[terryendicott]

"... of super dense mathematical pain :) "

[Vinvininhk]

0:11 throwing shade at other youtube math channels right off the bat. 5:46 and then continuing to throw shade at his own audience. I love how sassy this channel is.

[renatofernandes]

Man, I love you, I was studying transcendental numbers this week for research purposes and yesterday I thought: "YouTube need more videos about transcendental numbers". BOOM, you did it.

[wens]

You're THE (consistently) best math teacher I know of. Good teachers are not measured by his/her breath and depth of knowledge, but by the uncanny ability to make abstract/complex concepts essily accessible to others.

[Mathologer]

As I promised you last time, today's video is meant to be an accessible introduction to transcendental numbers. This is yet another video I've been meaning to make for a long time. Nicely self-contained as it is but I'll have to revisit this topic sometime soon since there is are many more interesting ideas I'd really like to talk about.
I don't have much time at the moment for making these YouTube videos because I am doing all my teaching at uni for the whole year in the first semester here in Australia. Five more very busy weeks until the end of this semester. Looking forward to a lot more Mathologer action in the second half of the year (fingers crossed).
As usual, if you'd like to help with Mathologer consider contributing subtitles and titles in your native language :)

These Zazzle t-shirt are very good quality, but way too expensive (at least for my taste). If you are really keen on one of their t-shirts I recommend waiting for one of their 50% off on t-shirts promotions.

[nlp]

This video touches on so much of what I love most about math and about learning math. I think it is incredible valuable to realize how our normal intuition breaks down when we start dealing with the infinite and the infinitesimal. Then we start learning about how to work around that huge handicap and build up our toolkit for working with concepts beyond our everyday understanding of the world, and we then start to build up a new intuition.

I remember how confounded and irritated I was, when I first encountered Zeno's paradoxes, and how satisfying it was, when I came to a sufficient understanding of limits and infinite sequences and series to see that there was really no paradox there at all. :-)

The other area of math, that I would put at the top of my list of favorites, is learning how to read and do proofs.

Being able to construct logical arguments, and to carefully analyze the arguments of others are incredibly valuable skills, not only when doing math and science work, but in every aspect of life.

I would say that the aforementioned lesson regarding the limits of our intuition as we deal with experiences outside of our normal lives also has value that extends well beyond math.

[Math_oma]

If you listen closely, you can hear the impending stampede of Cantor cranks and 0.999... = 1 deniers.

[cynx]

8:55 ''if your life should ever depend on it..., you know it might happen''
0.o

[BarryBranton]

What do you think about the idea that the product of the tangent of 36 and the tangent of 72 equaling the √5 ?

[prometeus]

Hi. The measure theory course link doesn´t work anymore. Could you update it, please? Yeah, I know I am asking for it four years later... but...

[SSJProgramming]

Finally something about transcendental numbers!

[kgeorg]

That T-shirt there is definitely monognomial, although I agree that the general set of equations thus described is polygnomial.

Thanks for the great videos and the clear math!

[alkankondo]

PROVING THE TRANSCENDENTALS ARE UNCOUNTABLY INFINITE:
Recall that a real number is either transcendental or algebriac, i.e. the sum of the sets of algebraics plus the transcendentals equals the set of reals. Also, a theorem in set theory states that a countable collection of countable sets is countable.

Therefore, we simply observe that, if the set of transcendentals were countably infinite (and we know the set of algebriacs is contably infinite) then the reals (their union) would also be countably infinite, which we know is not true. Contradiction!

Therefore, the transcendentals must be countably infinite. QED!

[unvergebeneid]

5:38 "... aimed at primary school kids." Oooooh, sick burn ;D

[drakelundberg]

14:49 "4 pages of super-dense mathematical pain"

Reminder of Why i Love Math :)

[ricardofabilareyes]

Dudeee, you need to write a book with the contents of your videos, like Matt Parker did with his "Thing to do and make in the 4th dimension"!

[PC_Simo]

Basically, uncountable infinity is to countable infinity, what infinity is to finite numbers 🤔. 12:30

[isEverywhere]

"I have constructed a marvelous proof of the transcendence of the Louisville number, which this video is too short to contain"

[WombatSlug]

I remember struggling with this a lot in graduate school, which is part of the reason I went into applied mathematics. It made pretty good sense to me, but I had trouble regurgitating it for exams.