Transcendental numbers powered by Cantor's infinities
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- Опубликовано: 17 окт 2024
- 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
gdz.sub.uni-goe...
Here is a link to one of the most accessible writeups of proofs that e and pi are transcendental: sixthform.info/...
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:
maths.org.au/in...
(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!
P.S.: Since somebody asked, I got the t-shirt I wear in this video from here: www.zazzle.com...
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.
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.
Man, I love you, I was studying transcendental numbers this week for research purposes and yesterday I thought: "RUclips need more videos about transcendental numbers". BOOM, you did it.
Well, I hope you got something out of this one. Quite happy with the focus ending up on using the diagonal argument in a constructive way which is very different from how it is usually used :)
Indeed very interesting, liked the introduction to measure also. If I could suggest a topic of video that RUclips lacks: Euler-Mascheroni constant.
I think Numberphile did a video on the Euler-Mascheroni constant. Did not watch it though. If you watch it can you let me know how deep it went. Would definitely be a nice topic to cover :)
Doctor, doctor, the patient seems to be responsive to our messages in the form of irregularly spaced light flashes! 100% record of the cerebral activity right now!
@@renatofernandes1086 I agree! Right now all I can say of this constant is that it sounds like a fancy pasta name 😂
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 RUclips 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 :)
Since somebody just asked, today's t-shirt I got from here: www.zazzle.com.au/polygnomial_t_shirt-235678195975837274
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.
regarding the .99999999... =1.000000 Look, just because you look like Lex Luthor doesn't mean you need to undermine the fabric of reality. Bald people can be nice too, just look at Captain Picard.
or Ghandi :)
what course do you teach?
At the moment I am teaching calculus, linear algebra and a unit called the Nature and Beauty of mathematics in which I do whatever I like (lots of fun stuff :)
Ghandi if that's how you pronounce it, but its actually Gandhi (give some load on the "dhi" part as if you were trying to speak "thee" without long ee and in low pitch)
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.
@n124lp That’s probably, why the EU has started censoring various Maths-websites: They don’t want people to get too smart, because then we could start questioning their propaganda 🤔.
The first time I came across this video, I thought it was ingeniously presented but didn't see the value in it. Lo and behold, I came across an equation I can't solve algebraically, and now I need to really understand transcendatal functions. So step one: transcendental numbers. Thank you for posting this video, the second time watching your presentation is giving me a new appreciation for this subject. Best wishes to you all!
At 16:57 I was expecting him to say "But, the margin is too small to contain my proof"
actually in an early draft that's exactly what I wanted to say :)
My immediate thoughts too, "I have put together a truly marvelous proof but this video is too small to contain it".
If Fermat had only been a man saying he has a proof he can't show, I'm not sure we would remeber him the way we do
Fermat actually developed several early theorems in Number Theory. For example, every prime of form 4k+1 is sum of 2 squares, and p divides a^p-a for all a and p. He also developed a method called infinite descent, to prove the lack of solutions for an equation.
Fermat became important not because he didn't solve a problem, but because he did advance math a lot, and because he stated a problem that is interesting and hard to solve. Just like Riemann's hypothesis, which is remembered because it's very useful and very hard, not becuase Riemann said "Eh, must be good"
+Michael Bishop Fermat wasn't bullshitting. He wrote that comment down in his own, private copy of a book. He probably found out a hole in his proof later, couldn't fix it, and said no more about it. There's no particular reason to go back and edit your own marginal notes...
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.
What do you think about the idea that the product of the tangent of 36 and the tangent of 72 equaling the √5 ?
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!
Actually, the whole part about the Aleph-null set of countable infinities' length = 0 helped me understand the Aleph sets more easily. It didn't feel counter-intuitive, but rather logical. Thanks a lot, Mathloger, keep up the good work!
Mission accomplished :)
"... of super dense mathematical pain :) " .....
I laughed out loud at that, too.
So ... can we have a proof where between each "major" step of the proof there are an infinite number of "minor" steps of the proof? (where, of course, the "minor" steps are themselves divided by infinitely many "very minor steps" and so on?) or, as we say is Frankfurt: huh??
Lmao I rewound and relistened to this part. Then I saw this comment.
You have a knack for explaining things in a way that instantly make sense (I have one of those minds that cant just accept what people tell me - I need to not only know how but also why it works - gave me alot of trouble in school). THANKYOU SO MUCH FOR EVERYTHING YOU HAVE TAUGHT ME!
My thoughts, exactly. I actually tried to pressure my high school Maths teachers to tell me, *_WHY_* a/0 is such a taboo, and not just infinity; which they failed to comply. So, after a while, I figured: ”Screw it!”, and went to figure it out, myself; coming up with basically the same explanation Mathologer gives us: Since division is really just reversed multiplication, think about multiplying some number x with 0, and getting: x*0 = a ≠ 0. Well, that’s just impossible; so, we can’t have gotten to a situation, in the first place, where we could divide some non-0-number a with 0; and therefore, a/0 is nonsense. That was so hard for my ”teachers” to explain, yet a high schooler could figure that out 😑.
x = 4 - gnome/2
:)
RandomisedRandomness I was going to comment that. You beat me to it!
What?
i thought gnome=8-2x
anyone please explain :P
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...
Finally something about transcendental numbers!
Thanks for this! I've wanted to understand this concept for years and years.... You are like a gift to humanity!
14:49 "4 pages of super-dense mathematical pain"
Reminder of Why i Love Math :)
Thank you so much for your great contribution to the world of math and science and your great favor to the mathematics students through out the world.
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.
Which youtube channe;s is he not acknowledging?
He's savage hahahaha loved the sassines.
yes but it's true... but it's ok really, most people do get outraged when something counterintuitive turns up, it's pretty natural.
Doc Daneeka I was definitely confused about .999999... being 1 at first, but it makes sense
Just like Comma 22.
Really enjoyed this video, put a lot of things in order for me. Louisville theorem was plaguing for like 8 years now on how to fit it in. Really good video
8:55 ''if your life should ever depend on it..., you know it might happen''
0.o
I recall hearing a story about a russian physicist/mathematician Igor Tamm who apparently was captured by some criminals. To prove that he actually was a mathematician he was told to calculate the error in truncation of Taylor series (Taylor's theorem): succeed and he is free to go, fail and he would be shot. He actually did the calculation and was set free.
(I couldn't find a source for this, but I like the story anyway.)
Were the criminals mathematicians too? Because if not, how did they verify his proof? :q He might get shot anyway even being right if they wouldn't understand the proof :J
that story is probably bullshit no offense
I want to play a game...
@@Fircasice Live or die. Make your choice.
Great video! Wondering about Liouville numbers for a long time, this made me understand for the first time.
Hey Mathologer! Great one, as usual..
Since you talked about Cantor's diagonalization in this video a lot, I think you should also make a video where you talk about Godel's Incompleteness Theorem and Turing's Halting Problem; as you know, both of those essentially used the diagonal argument in their proofs. I believe you laid the set theoretical foundations for these two, although you might need to explain some symbolic logic and (or) automata theory in that case.. It would be great!
Best
Definitely on my to-do-list :)
Since some numbers can represented in two different ways, such as 1 being 1 point 0’s or 0 point 9’s, are we really sure that the diagonal number we create that we think is non-algebraic because it can’t occur in the list, might actually occur in the list in an alternate form?
I LOVE YOU MATHOLOGER
me 1.999999999999999
He's married :)
@@davelowinger7056 it should be me 1.99999....
If you don't give the dots, its doesn't reach 2 😁
@@sadkritx6200 Well not as much as Joe siu
Me 1,9999999999999999999…
being a student its really inspiring for me ... and I am really thankful for the free measure theory lessons.. 😘
Great video, as always! Everything is pretty understandable to me, though I had known the most part of these things prior to the video.
What I found new to me was the countability of the set of algebraic numbers and the uncountability of the set of transcendental numbers. I think that the proof of former set being countable maybe was a bit too fast for people who are not familiar with the table method of listing rational numbers (via diagonals). But I guess even then it's still understandable if watched twice.
When you talked about the measure of rational numbers, I think it would be also useful to compare it to the measure of real or irrational numbers. On the one hand, it would further illustrate the point that there are unspeakably "fewer" rational numbers than there are, say, irrational numbers. On the other, it would provide additional insight about the notion of measure to people who are not experienced in math.
The proof that the set of transcendental numbers is uncountable could look as follows, going by the method of contradiction. I will denote the set of transcendental numbers by T.
Suppose that T is actually countable. Then it can be listed, i.e. there exists a sequence of transcendental numbers a1,a2,a3,a4,..., such that T = {a1, a2, a3, ...}. Furthermore, we already know that the set of algebraic numbers is countable, that is there exists a sequence of algebraic numbers {b1, b2, b3, b4, ...} which includes all algebraic numbers.
But then we can construct a sequence {c1, c2, c3, c4, ...}, where c_i = a_{(i+1)/2} if i is odd, and c_i = b_{i/2}, if i is even. In other words, {c1, c2, c3, c4, ...} = {a1, b1, a2, b2, a3, b3, ...}, so it lists all algebraic numbers AND all transcendental numbers. Therefore, the union of these two sets should be countable. However, the union of these sets is the set of real numbers, ℝ, which we know to be uncountable - contradiction.
Thus, T is actually uncountable.
P.S. "diagonalization would produce a real number outside the set of real numbers" - I really liked this way of formulating it :)
P.P.S. Starting to work on the subtitles in infinitesimal amount of time...
+Danil Dmitriev Glad you like the video and thank you very much for the subtitles on the Rubik's cube video that you did the other day :)
The thing about that "diagonalization would produce a real number outside the set of real numbers", though, is that not everyone introduces the real numbers as the set of all numbers of the form z.d0d1d2d3... with z an integer and every d_n a digit (0-9). If I remember correctly, there were at least two other possible definitions of the reals (Cauchy completion of rationals, "supremum completion", i.e. finding the smallest superset S such that all subsets of S that have an upper bound in S have a least upper bound in S). So that argument hinges on what exactly your definition of "the reals" is.
Yes, that's true. I think that hardcore mathematics usually introduces the set of real numbers as a field which satisfies the Completeness Axiom, which also has at least two ways of being formulated. The completion of rationales which you mention is one of them, if my memory about the abstract algebra course still works fine. In this sense, it's more like a coincidence that our usual sense of real numbers happens to satisfy this more formal definition.
I just liked the way that Mathologer said this, that's why I noted it :)
for a general audience, identifying R with decimal expansions is fairly natural...
My feeling is that this identification ought perhaps to be made explicitly within the video, perhaps using a construction along the lines of "if we could list the Real numbers, the decimal number that we construct from the diagonal would be not be a Real number. But since all decimal numbers *are* real numbers we have a contradiction which proves that we can't, after all, list the Real numbers".
This is great, I'd never seen Cantor's diagonal argument used to prove the existence of irrationals and transcendentals.
Nice isn't it :)
If you listen closely, you can hear the impending stampede of Cantor cranks and 0.999... = 1 deniers.
:)
Friend, can you replicate your Pi Measurement to using a ruler with higher precision? When you did that, did your "rational pi" value change? If so, how can you explain that a _constant_ sometimes has one value, and sometimes another? Cheers
Someone here would not miss out a minute of pi day
@Slimzie Maygen *finite polynomial with algebraic coefficients.
An infinite polynomial can be shown to converge to pi.
@Slimzie Maygen not a mathematical voice here, but if pi was wrong already at the third decimal digit, most of our modern buildings would have crumbled , and even some professional works of carpentry. Not to mention nanotechnology, molecular & cellular biology, and other microcosmic fields where the "traditional" pi is applied just fine up to many digits.
Great work. I think you have succeeded in getting over this tricky subject in an accessible way. x
A great video involving my two favourite things in maths, transcendental numbers and Cantor's infinities. Thank you Mathologer.
Burkard, you are so worth subscribing to! You have such good, in-dept videos about really nice subjects! Thanks man!!!
mr. Mathologer thanks for your interesting videos
Magnificent, as always. The most wonderful book ever written on this subject is 'An Introduction to the Theory of Numbers' by Hardy & Wright (that's the legendary English mathematician, G H Hardy). This book includes rigorous proofs of the transcendence of e, pi and Liouville's Constant (the latter being far more accessible than the other two). Indeed, the proof for pi shown very fleetingly in this video is the one in this book, so Mathologer is obviously a fan!
5:48 Anyway, I’m a primary school kid, but this transcendental stuff really blows my mind
Btw, good video. I’ve watched all your videos from the beginning and they are amazing. At the beginning, I didn’t get the meaning of this thingy. Your videos have an easy to hard level. Now I get more of this and I know a lot more of Math now. Thank you!
Glad the videos work so well for you :)
Great video as always (I know this comment comes a bit late)! What is even more mind-blowing (and would require another Mathologer video!) is that even the set of numbers that we can describe with a finite number of words (and symbols) is countable. This implies that the vast majority of all real numbers remains for us a complete mystery beyond our grasp.
To be more precise, let us decide a set of symbols, for example S=English Alphabet U Digits U parentheses U logical connectives (and, or, not, imply) U {symbol for belonging} U quantifiers. We may want to add some more special symbols (for example +, /, etc.) so in the end we may agree that S has at most - say - 200 symbols (or in general N symbols). You can list all the possible finite "sentences" made with the symbols from S starting from length 1 (each individual symbol), then length 2 etc. So, all the set of finite sequences of symbols from S is countable. Some of these sequences will not make much sense (e.g. "2+"). Some others will represent a number (for example "pi" will be pi, "square root of 2" will be the square root of 2 and so on). Well, the numbers represented by these sequences will constitute a countable set, thus a negligible part of all the real numbers.
Notice that the same applies if instead of a finite set S we use a countably infinite set S. The resulting set of sequences is the union of S, SxS, SxSxS (Cartesian Products) etc. which is a countable infinity of countably infinite sets, thus it is countable as well and as a consequence it covers 0% of all real numbers!
I bet that this will beat the 0.999... = 1 incredulity...
5:38 "... aimed at primary school kids." Oooooh, sick burn ;D
It just had to be said :)
I don't disagree :) But you seemed to secretly yet visibly enjoy saying that ;)
Oh, I definitely enjoyed saying that :)
I don't understand why there's any controversy. Of COURSE zero point all nines is going to equal one. It's nine ninths.
Ian Kjos, nine ninths of climate scientists are convinced of man-made climate change and the internet still finds a way to turn this into a huge controversy. So there.
"Algerbraic", Irrationality is continuous analog?
I think the Mathologer videos are how I became aware of the "inward" pointing, at probability one existence-potential of rationals, existing at connection-singularity, and "outward" pointing toward a vanishing point of zero irrationality of continuous infinity from at a point of origin, because it's not this clearly expressed in the library of books I've read before. The active illustrations are of great importance to visualization. Thank you.
Countably rational, Uncountable irrationality because it's the closed probability of one or open unlimited potential of infinity, which is the situation for production of stratified information density phases and "evaporating" exclusion phases dispersing, but never vanishing absolutely..., and reflecting in infinite containment.
It could be inferred that Measurement Theory conferres density and intensity of probability and functional activity on a number position(?).
From this: "Rational" is inclusively, linearly aligned, with certainty of prime probability one quantization, and "Irrational" is the reciprocal direction of alignment, diminishing probability from one, in infinite qualification.(if the language is sufficiently definitive?)
If the precise area of the circle is exactly that of a square, then the irrationality of Pi implies that the alignment of the circumference is tangential, bifurcated, potentially disconnected/discrete but reflected in alignment from unity, (=quality of "i"), while the square is contained certainty of aligned probability with connection.
(If QM-TIME is a "mechanism", then it's mathematical analysis of elements is in philosophical terms of an engineered, virtual-work, machine of probability in possibility. Science Unification project..)
..the origin of the idea that eternity-now superposition is e-Pi-i quantization resonance. Mathematics is this relative rate of time duration pulses/rates in an infinite spectrum of infinities.
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"!
Well, if you are interested in the books that I've written have a look here www.qedcat.com/books.html and if you are interested in all sorts of other things that I've been doing pre-RUclips check out this website: www.qedcat.com :)
OMG! This is gold. Thank you!
This is great! Sorry, I'm arriving late to the party...
Superb, as always. Thank you very much. Perhaps you could complete this video expanding on the statement "some algebraic numbers can not be expressed in terms of +/-*sqrt? Also, what happens with the roots of polynomials if their coefficients are not natural numbers?
"some algebraic numbers can not be expressed in terms of +/-*sqrt" This is actually something very deep and the proof that such algebraic numbers exist is the solution to another very old problem. A bit of a holy grail for somebody like myself who is into coming up with good explanations of complicated material. Pretty high on my list of things to do :)
"I have constructed a marvelous proof of the transcendence of the Louisville number, which this video is too short to contain"
Your videos are great. Much more engaging then my analysis lecturer!
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!
exactly :)
alkankondo89 or you could say that if the transidentals were coutable, then you could just enumerate by taking a algebraric number, then a trancedental, then a algebraric and so on.
or what?
Mathologer how do you prove that real numbers are only algebraic or transcendental? (the basis of this proof)
I guess the transcedantal numbers are specificly defined as the set of all real numbers that are not algebraic?
why pull out a theorem from set theory? Just count the elements using some of the tricks described in the video!
So thinking about irrationals, transcendentals Prof N.J.Wildberger talks about 'non-computable' numbers vastly out numbering all others, but what is the significance of this ? I presume we're calling irrationals and transcendentals 'computable' in a sense ?
1:54 "There is really no reason to suspect that there exists irrational numbers." Ironically, what you showed right before saying this gives a good reason to believe there could be irrational numbers. Numbers you listed contain only those rational numbers whose divisor is power of two, which clearly doesn't contain all the rational numbers (for example 1/3), and still they are infinitely dense. So you just gave an example that shows that infinitely dense /= rational numbers, so why then should rational numbers = whole number line? ;)
While talking about cardinalities almost nobody proves countability of algebraic numbers set, but Mathologer does. Perfect)
Another great video
9:48 So after the list of rational numbers we list the irrational as the Sqrt of these numbers....than we can make a list of numbers to have the all possible combinations of Sqrt numbers ( as:...√(2),√(1/2) ...) (if i understand well) ....
but then we can list also numbers like the Golden ratio? How we can do that ,cause that numbers are composed of more operation like Sqrt ,+, :, x ? Very nice video btw
Nice t-shirt, where did you buy it @Mathologer?
I got it from here www.zazzle.com.au/polygnomial_t_shirt-235678195975837274 :)
Sadly it's false advertising, there's only one gnome so it's a mognomial :'(
Clever shirt though
Adel D well x = 1 is also technically a polynomial so... but it would be even better if the gnome was above the x (x to the power of gnome). Yes i am petty
@@KosteonLink *Monognomial.
i knew algebraic numbers were countable, and i'd seen the method of constructing the list of rational numbers, and i had been wondering what one method of making the list for algebraic numbers would look like, and this video answered my question very clearly. thanks
"Don't you think this is amazing?" , I agree with excitement, pull up my phone to text my math friends that there as many natural numbers as their are natural numbers. Then I realize I have no math friends.
Hey I'll be a math friend. Math is cool, and I need a math friend
@@jetison333 S A M E
I do enjoy your videos as a Math teacher I find your style and technique very informative and clear.
That's great :)
this whole time i didnt realize i was getting GNOMED
Dear Mathologer, nice video as always, but I have a question... Around 14:10 you say "If you restrict our attention to some finite interval, and pick a random number inside this interval, in a paradoxical, but very precise sense you have a zero chance of picking any algebraic number." Well, I am interested is there a way to pick a truly random number from an uncountably inifinite set of real numbers that are in a finite interval? (by truly random I mean that all numbers in the interval have equal probability to be chosen)
4:58 isn't 2/2 equal to 1, not 2?
he did not consider 2/2, he skipped it, he took 2/1
Sooo is the irrational number obtain by cantor diagonalization of the rational spiral algebraic or transcendental?
I love a good squaring-the-circle joke :)
So do I 😅.
This video was a treat to watch. But that's in some measure because I already know my way around the basics of theory of computation. I could relax and enjoy the guided tour of the terrain, and the various charming asides.
For someone just setting out, it would be hard to parse the narrative well enough to identify the centrally important elements. And yet I have to concede that these elements make better sense in context than when served up on their own. How about this? Keep everything just as it is, but provide a synopsis at the end which repeats your original graphics, showing a fast path to (1) enumeration of the rationals, (2) enumeration of the algebraic numbers, (3) diagonalization of this emumerated set, (4) the resulting partition of the total space.
Please be sure to reuse the graphics as a way of cueing the viewer to the earlier material. I think it would help a lot of people to gain confidence in the essential material.
Take care. Keep doing this stuff!
Your shirt is incorrect, it contains only one gnome, so it isn't "poly"
It's a first degree polygnomal.
Just as we all are a first degree polyhuman (or polyAI for the bots out there, but those could actually be higher degrees too).
You didn't count the gnome who wears it ;)
Sci Twi he's not a gnome, he's an elf. stop being a fantacist
A Master's class, as usual ! GREAT !!
Does it have a proof :
any number with an infinite non-repeating string of only 0s and 1s is transcendental.
exp : 0.0110111001010110... (continues randomly) is transcendental.
It seems that I forgot to add "s'il vous plaît" xD even If I don't think it will change the rate of replies...
There are no known counterexamples, and this is conjectured to be true, though I don't think there's a proof for all such decimal expansions.
Drew Duncan,
Thank you for you answer,
Do you have any links on research papers in this topic ?
How do you know it is conjectured to be true ?
arxiv.org/abs/0908.4034
No. Some would be algebraic.
Greetings. Thank for your excellent contribution to the public knowledge. Your videos lectures are excellent and informative. I have a couple questions,
1) at 3:5, time of your video you mentioned........ ....My question "is there any algebraic number that cannot be written as such rooty expression?"
2) At 2:52...... you mentioned, ."all numbers, without exception, that can be written........... are algebraic" 1) in contrast to 2) I a bit confuses me. Thank you.
Every number which can be expressed as a solution to a polynomial equation with integer coefficients is algebraic. That's the very definition of an algebraic number. On the other hand, there exist algebraic numbers which can't be expressed as a formula with integer coefficients using addition, multiplication, subtraction, division, and n-th roots. One example is the real solution to the equation x^5 − x − 1 = 0. (You can't convert this equation to equation x=... using the previously mentioned operations.)
"... aimed at primary school kids." 🤣
Wow. this is one of your best videos. I didn't know about that measure theory thing, wow. I'm dazed.
All of the transcendental numbers described were computable. The uncomputable numbers are even more "badly behaved". As difficult as it is to prove that a number is transcendental, it's even harder to find one that is uncomputable.
Even though almost all reals are uncomputable, it's not possible, by definition, to grab an uncomputable real number and list all of its digits. All we can do is describe an uncomputable number by defining it indirectly such as via computer programs that may or may not halt.
And yet despite their ephemeral nature, every uncomputable number can be approximated to arbitrary precision by rational numbers.
And of course, even the computable numbers are countable, since we can list the programs that compute them. In fact even the *definable but noncomputable* numbers are, since we can list their definitions. So really, even though we assume uncountable sets of numbers to exist, it's literally impossible to give an example a specific number that isn't in a cleaner, countable subset.
+MrCheeze A nicely written wikipedia article on the topic: en.wikipedia.org/wiki/Definable_real_number
anon8109 You can go a step even further and consider the set of undefinable numbers. It's fairly clear that the set of possible definitions is countable. Imagine that you have them written out into a document, which is scanned into a computer. Then the resulting file will be a string of 1s and 0s, which can be trivially mapped to a unique real number between 0 and 1. This means the set of possible files is countable, which, if you make the reasonable assumption that there aren't any "magic documents" that are somehow both readable and yet cannot be represented by any form of digital photograph, means that the set of possible definitions is also countable.
Of course, narrowing down which documents are valid definitions and which are gibberish is impossible - just think about one with really bad handwriting that no one can decide for certain if that digit is a 1 or a 7 - but we don't need to. If all the documents, including the gibberish ones, are countable, then any particular subset you claim to be the valid definition ones will be countable as well.
Anyway, these undefinable numbers are interesting in that it's completely impossible to ever give an example of one - to be able to do so would imply you have some way of defining that very number. They are completely untouchable by any mathematical formulation. And yet, you can produce them at will. For instance, let's say you roll a die over and over, writing down each roll as a digit. The number you end up with as you continue rolling forever will be undefinable. The key is that there's no way some other person could independently produce the exact same number. Their dice will roll different numbers, and if the see the first 10 rolls you made, they can't determine what the 11th should be unless they see you roll it too.
Now here's a real poser: Is it possible to have a number that is definable but not computable? That is, something that can be proven to exist and be unique, but with a value so convoluted that no computation can ever approximate its value. Actually, I can think of one.
Consider the Halting Problem, which states that it's impossible to produce an algorithm that determines whether a given program will terminate. Now suppose we define an integer to be the number of programs of a given "size" that terminate. This number is unique for a given program "size", since every program will either terminate or it won't. However, figuring out such a number in general for sufficiently long programs would require solving the Halting Problem!
Now, you might argue that although the halting program is unsolvable in general, specific programs can be proven to never terminate, so how do you know that for a given "size" you have any programs that can't be reasoned about? The simple solution to this is to produce a number by somehow combining all the termination counts. Maybe set the nth digit to the first digit of the termination count of programs of "size" n. Then to calculate this number, you must solve the Halting Problem in general for all programs.
+MrCheeze What do you mean by "cleaner"?
Steve: Well, every real number is - I assume - in _some_ countably infinite set. So by "cleaner" I just mean that every number we can talk about or access is _also_ in a countably infinite set with a fairly simple definition, e.g. the set of definable numbers.
Thank you, I really love your videos, this is one of my favorites, enlightening and funny
4:56 2/2 ≠2
he skipped it cause it was already counted
A/V mismatch.
Not even for very large values of 2...
@@baruchben-david4196 😅
Master class video.
What I didn't get in this video that why transcendent numbers are uncountably infinite.
4:57 2/2=2?
Skytern... he is sliding to 2/1
Fun fact: There exists a real number from an interval between 0 and 1 which contains within its decimal representation all rational numbers from the same interval. It is in the following sense: split a real number x into countably many real numbers. x1 is the number formed from the digits of x at decimal positions 1, 3, 5, 7, ... (positions not divisible by 2). x2 is the number formed from digits of x at positions 2, 6, 10, 14, ... (divisible by 2 but not by 4). x3 is the number formed from digits at positions divisible by 4, but not by 8, and so on.
In the same sense, there exists a real number containing all algebraic numbers from 0 to 1. On the other hand, there doesn't exist a real number containing all real numbers from 0 to 1.
Леди и джентльмены, которым хотелось бы увидеть больше субтитров на русском языке к другим видео на этом канале, - пожалуйста, оставляйте ваши комментарии здесь. В данный момент я планирую поработать над видео "Ramanujan's infinite root and its crazy cousins", но если есть какие-то запросы/просьбы/предложения, с радостью их рассмотрю :)
В конечном счёте, главной моей целью являются помощь каналу и облегчение русскоязычной аудитории доступа к тому, о чём рассказывается в различных видео на нём. Поэтому если Вам хотелось бы увидеть субтитры к какому-нибудь конкретному видео поскорее, я буду рад подстроить свой план соответствующим образом :)
When going through the colum of quadratic numbers and saying 'remove those that are found in the countable rational column' is essentially an impossible task? Cantor made the square grid trick so you could "look back" on the column of inspections already completed, and thereby possible. So, to exclude a number from the 2nd column could take forever. Is there a way to do 1 rational, then 1 quadratic, and walk through the columns and rows so that a number is blocked from adding to a higher column until placed in a lower column. does this make sense?
i.e. you can only build the grid by doing a cantor diagonal walk which is countably infinit steps.
Just two remarks:
1. The solutions to Ax^2+Bx+C=0 are in the first list if and only if B^2-4AC is a square. But more importantly
2. I really only insisted on avoiding duplication to make the exposition cleaner, but the fact of the matter is that you actually don't have to worry about this. The main thing is that the list contains all the numbers you want to list (rational or algebraic in the case of this video). It does not matter at all how many times they pop up for diagonalization to spit out a number outside the list. So what this means is that for the quadratic list you simply list ALL the real roots, the same for the cubic etc. :)
"With a bit of common sense "
Mathologer it ain't that common
Yes, sadly common sense is not that common :)
why is it named common sense if it isn't common ?
Using the same mechanism used to list the fractions, it seems to be possible to list all the real numbers between 0 and 1, exhausting all digits, decimal by decimal The list would go like this:
* First decimal (9 numbers): 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9
* Second decimal (99 numbers): 0.01, 0.02, ... 0.09, 0.10, 0.11, 0.12, ..., 0.97, 0.98, 0.99
* Third decimal (999 numbers): 0.001, 0.002, ..., 0.999
* ...
One possible problem with this list is that each item in it has a finite number of decimal digits, where real numbers, like pi/10, have infinite number of decimals. But, since the list itself is infinite, couldn't we argue event pi/10 is in it?
This is a clever thought. Can someone please explain what is wrong with it?!
You correctly identify the issue with your argument. None of the real numbers with infinitely many decimal places are on the list.
And no, just because the list is infinite does not imply that they are on it.
Another way to think about countability is this way: A set is countable if you can order it in such a way that, starting from the first element, you can count up to any element you want in a finite amount of time. If you think about it in terms of lists, this is the same as saying that each individual element is in a finite position on the list.
For example, you can do this with the integers: 0, 1, −1, 2, −2, etc. Every integer n is in a finite position. 0 is the 1st number on the list. If n is a positive integer, then it is the (2n)th number on the list. If n is a negative integer, it is the (2|n|+1)st number on the list. So we see that every integer appears in a finite position.
The same thing can be done with the rational numbers. Look at the spiral given in this video. Every symbol in the grid will be reached in a finite number of steps. You can see this by taking any symbol of the form n/m where n and m are integers. Now, consider |n|+|m| = k. Notice that −k/0 is the last symbol you will ever reach where the sum of the absolute value of the numerator and denominator add up to k. Notice that −k/0 is inside a square of side length 2k+1. There are (2k+1)^2 numbers in this square, and every number n/m with |n|+|m| = k appears inside of it. Thus, for any rational number n/m, we know it must appear before the (2(|n|+|m|)+1)^2-th position on the list. Therefore, every rational number appears in a finite position on the list.
As you correctly pointed out about your own listing attempt, none of the real numbers with infinitely many digits appear in a finite position on the list. So the listing technique you gave does not work.
Proof for 12:00 is actually trivial. Let assume, that transcendental numbers are countably infinite. Now I can construct a list taking first algebraic number, then first transcendental number, then scnd algebraic, scnd transcendental and so on. Now I have a countable list of all real numbers, but such not exists.
yes, a simple back and forth counting argument is all that's needed. This is what Mathologer expected to find in the comments, but I think you're the first to see it.
the very best explain of e i have every seen
Nice job. I knew essentially all of this, but to see it presented so briskly and clearly is quite a pleasure. There is a feeling that I have, however, that there is a great deal about this, and about number theory in general, such that our situation regarding it, the understanding we are stuck with, is aesthetically repulsive, to state it in very strong terms. It is as though the last time people had this reaction was when the Pythagoreans threw Hippasus out of the boat for proving the existence of irrational numbers, as if since then people have reconciled themselves to living with this situation in which, for example, a random place on the number line is guaranteed to be transcendental, with all that that implies about its recalcitrance to our management. Another way of saying this is to imagine that we encounter intelligences that are as different from humans as humans are to dogs, for example. These might be AIs of the future, or visiting aliens; doesn't matter. My hope would be that they would tell me they have "numbers" that don't have these properties of the transcendentals, even if I am as incapable of understanding it as my dog is of understanding Cantor. Just to know that the world isn't so crazy would be a great relief.
Absolutely loving this channel.
A thought that I have always found intriguing - Between any two transcendentals you can find a rational. Between any two rationals you can find a transcendental and yet there are more transcendentals than rationals.
+Sir Lew
Just a thought ...
Every rational number can be written in decimal notation; its decimal notation always repeats (either in all 0's or in a finite set of repeating decimals).
Every irrational number (hence, also every transcendental number) can be written in decimal notation; its decimal notation never "repeats".
Don't you think that this fact leaves FAR more possibilities for irrational numbers?
On the other hand, I DO understand your intriguing thought ... ;-)
Oh yes. The instinctive feeling is to say - "there are more transcendentals between the two rationals you chose than there are rationals between the two transcendentals you chose"... but then you just apply the statement again. All those extra transcendentals I just mentioned... they ALL have rationals between them!
That is why intuition is not a good guide for non-finite maths.
Cantor was a boss. Huge fan.
Mathologer, may I suggest that you look into doing a video on idoneal numbers (as conjectured by Guass and Euler) and how the Riemann hypothesis plays into finding the last idoneal number? Loving this vid so far, oh, and thanks! :)
I'll put it on my list of things to ponder :)
Mathloger , your "anyway" and hand movement is my favourite part of your videos. But if ve leave this "joke" aside ,your way of teaching reminds me of my favourite professor at my university.
muchas gracias por el video, me acercó a la obra de Cantor y sus trabajos sobre el infinito, obra de grandes implicaciones filosóficas; ya que es a lo que me dedico.
How would you make the spiraly walk with 4d+ space? What do you call the property (or maybe the proof name) that a polynomial of degree n has at most n real solutions (any video on complex solutions soon btw?) Why am I not convinced that the diagonal number at 6:15 is necessarily irrational and different from the countably infinite set? Many other questions, but great video as always.
Hi sir!! May I know Which software do you use for such animations and video making please ??
While I learned about the concepts of countable and uncountable infinities quite a long time ago (about 15 years or so ago), I was surprised relatively recently (a year or two ago) to learn that the set of algebraic numbers is countable. I hadn't actually encountered nor realized this before, and just assumed that algebraic numbers were uncountably many.
But then I realized that there's a simple argument that can be made to demonstrate their countability (at least for me). I already knew that the set of all possible finite strings of characters is countably infinite. Thus it was just a matter of realizing that every single algebraic numbers can be represented by a finite polynomial. Thus their countability became immediately clear.
And of course this immediately meant that the principle can be generalized: Every single set of numbers, where every number can be represented with a finite representation, is countable. For example, the set of computable numbers is countable. The set of definable numbers is countable.
This may be trivially evident to somebody who already understands this, but to me it was a recent realization.
WarpRulez Good insight! I would just add two clarifications. 1. The alphabet must be at most countably infinite, which is true here since you can take the alphabet to be finite, and 2. An algebraic number is not uniquely determined by the polynomial of which it is a root, as there are multiple roots. However, this is not a big problem. Just order the roots in some well-defined way, say by dictionary ordering on their coordinates, and then append the number of the root at the end of the polynomial.
We got the proof just recently and it made me remember this video. Thanks mathologer
Liouville's constant for a given base b:
sum_(n=1)^inf b^-(n!).
If we interpret it as a binary number, it's actually more like 0.765625... in decimal.
Which shows that 0.765625… has to also be transcendental. 🙂
Could you provide a link for that proof of π being transcendental? I'd like to give it a read even though I might not understand most of it :p also awesome video! really easy to follow and understood everything, keep it up!
I'm going to have to watch this video again
I've always had an affinity for what I call the Indescribable Numbers. In order for an actual person to precisely define a specific real number it requires them to use a finite sequence of characters to produce a definition of the number. Any number that can be specifically described in a given language would be called Describable, so for example all algebraic numbers are describable (since they can be defined as solutions to given equations) as can numbers like pi, e, Liouville's number and any other number that can be specified uniquely using English and mathematical symbology. Since such definitions are finite sequences of a finite set of characters there are only a countable number of such possible definitions and therefore the number of Indescribable numbers is uncountable.
Thus in a sense the number of specific numbers we can ever theoretically specifically calculate or talk about or even specifically mention in a proof or anywhere else is merely an infinitesimal countable subset of the total set of all real numbers. Not only can "most" numbers not ever be defined in a given language, it's impossible to prove a specific number is Indescribable because in order to do so you would need to be able to describe the number in the hypothesis of your proof which would indicate that it is instead describable. Similarly no indescribable number is the limit of a describable sequence of describable numbers.
So not only are the bulk of the real numbers not rational or algebraic, they're literally not even numbers you can specifically discuss or write an algorithm to calculate. A given random real number has zero chance of ever being something "special" or even specifically used in a proof. In a very real sense most numbers exist beyond our ability to individually grasp them even in principle.
I've always been a little wary of describable/indescribable numbers. Something about the definition seems too lax. It makes me worry that the definitions may allow for some nasty self-reference. Do you know if this is not a problem?
Edit: Sorry, that was a sloppy comment. Allow me to be clearer. Here's why I'm concerned about describable numbers:
Consider the positive integers. Then there are positive integers which can be described in twenty or fewer words in English. We will make an argument that actually _all_ positive integers can be described in twenty or fewer words in English. Suppose the set of positive integers which cannot be described in twenty or fewer words in English is nonempty. Then since the set of positive integers is well-ordered, there is a smallest element in this set. Therefore, there is a unique "smallest positive integer which cannot be described in twenty or fewer words in English", which is a description in twenty or fewer words in English. This is a contradiction. Hence, every positive integer can be described in twenty or fewer words in English. But if there are only finitely many words in English, then there are only finitely many twenty-or-fewer-word descriptions in English. Therefore, it is impossible for all infinitely many positive integers to be describable in twenty or fewer words in English. What gives?
The thing that gives is self-reference. Describability/indescribability is itself describable. The definition of describable is actually too loose in that it reasonably admits self-reference.
But this example isn't exactly the same as what you're talking about in terms of describable numbers. So I wonder if perhaps the describable numbers avoid this sort of issue of self-reference.
MuffinsAPlenty There are subtleties to it, when you look at it formally you have to remember that the concept of something being definable needs to be formalized within the language of the model of set theory you are talking about. If you try to talk about something like “the set of definable sets” then you can run into problems where the concept of definable isn’t itself definable within the language of set theory, it’s a second order concept above it. In other words if you ask the question “is the set of sets that aren’t definable sets included in itself?” you encounter a paradox.
There are formal treatments of this though. As I recall there is a decent Wikipedia page on Definable Sets that outlines the different ways to talk about definability of numbers. In fact there are some counterintuitive results that if I recall right prove that if you are talking about definability in terms of the ability to define a set theory model then you can show that there must exist models of set theory for any given real that can define that real.
Thank you very much for the information! I'll definitely look into that.
@@Bodyknock All this really rings a loud ”Russell’s Paradox” -bell, in my brain. 😅
Nice! Looking forward to the video with the proof you prepared
Maybe you are going to go into this in another video but there is something I find even more mind-blowing than the fact that the algebraic numbers are a subset of the complex numbers of measure zero. This is the fact that π and e, while being transcendental can still be defined. We can give an algorithm that could, given infinite memory and infinite time, compute these numbers. And even with finite memory and time, we can compute these numbers as precisely as we want. However, since the number of possible algorithms is countable (beautifully shown with Gödelization or intuitively by realizing that each computer program is just a binary integer in that computer's memory), the number of computable numbers is countable and therefore also has measure 0.
And to make this a little less abstract, this means that the number of _thinkable_ numbers is also countable. If you picked a rational number at random, you'd be almost certain to have a number that no human could ever even come up with a way to describe.
*"And to make this a little less abstract, this means that the number of thinkable numbers is also countable."*
That would be hard to say without knowing exactly what a thinkable number is, but I doubt it.
If the thinkable numbers were countable, we could list them all. Georg that list and we get a new number. We thought of it, so it's thinkable. But it's not in the list of thinkable numbers. Contradiction. The thinkable numbers are not countable.
The concept of thinkable numbers is not less abstract than that of computable numbers. That can be seen from your comparison of the cardinality of computable numbers and programs stored in a computer memory.
Well, I skipped over assuming that the Church-Turing thesis is correct because I think it's so obvious but I doesn't even have to be, because the number of thinkable numbers is not only countable but finite. While an abstract machine can be infinite in size, the human brain takes up a finite volume so there's a finite number of configurations of elementary particles in that volume and therefore a finite number of states.
And even if we allow for infinitely sized brains, there's still just a countable number of states that it can be in. So if you don't argue that brains are only there to give neuroscientists something to do while the real thinking happens in some soul-spirit-ghost-mind, the Church-Turing thesis automatically follows.
However, the confusing part here might be that it's possible to formally define non-computable functions. So in that sense, it's possible to define (and therefore think of) a non-computable number. That's of course not what I meant when I used the informal term "thinkable number." Because, duh.
Squaring the circle at the start.. that was funny!!
My favourite video on transcendental numbers is Vihart's "Transcendental Darts", mostly because it mentions a feature of transcendental numbers that I haven't come across before: No matter what notation you come up with, you will only ever be able to use it to represent a countable (i.e. measure 0) subset of all real numbers.
So, you can only ever write down, or generate, exactly 0% of all real numbers.
Well, not least because the truly uncountable set is made out of undefinable numbers... but mostly because even writing down or generating all the rationals between 0 and 1 would take an infinite time - countable or not.
Yup!
The link to the course on measure theory is broken. Is it still possible to access it?
Clear and convincing ! Congratulations !!
So excellent. So wonderful. You contribute to world digital knowledge. great.
Ausgezeichnet! Recht vielen Dank fuer diese Erklaerung! 😀🤗
I wish YOU would give a free course on measure theory...on RUclips. Though it might be a little high-level for us math amateurs. Personally, I'd take the chance on my head catching fire and watch it. I know it's a bit of an "ask."
So would I 🤩.
I have to say, these concepts seem (even to an engaged observer normally taken by this stuff) to be dealing with arbitrary semantics. One "infinity" is "larger" than another "infinity"... well that's silly, they're both infinite, what good is it to declare one is "larger"? What good is it to say that some numbers are "transcendental"? Are we just taking the endless continuum of numbers are arbitrarily grouping them according some some method we thought of ("natural", "algebraic", etc...)? Does this relate somehow to the physical world?