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
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.

[@terryendicott2939]

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

[@joshuacoppersmith]

I laughed out loud at that, too.

[@beaker_guy]

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??

[@EWischan]

Lmao I rewound and relistened to this part. Then I saw this comment.

[@AM-ip2ey]

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!

[@Mathologer]

Mission accomplished :)

[@RavenLuni]

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!

[@PC_Simo]

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 😑.

[@AlecBenzer]

This is great, I'd never seen Cantor's diagonal argument used to prove the existence of irrationals and transcendentals.

[@Mathologer]

Nice isn't it :)

[@joesiu4972]

I LOVE YOU MATHOLOGER

[@davelowinger7056]

me 1.999999999999999

[@scitwi9164]

He's married :)

[@sadkritx6200]

@@davelowinger7056 it should be me 1.99999.... If you don't give the dots, its doesn't reach 2 😁

[@davelowinger7056]

@@sadkritx6200 Well not as much as Joe siu

[@PC_Simo]

Me 1,9999999999999999999…

[@rodovre]

Great video! Wondering about Liouville numbers for a long time, this made me understand for the first time.

[@MikeRosoftJH]

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.

[@michaelleventeris644]

A great video involving my two favourite things in maths, transcendental numbers and Cantor's infinities. Thank you Mathologer.

[@davidwilkie9551]

"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.

[@skylermagnificent5422]

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!

[@Mathologer]

Glad the videos work so well for you :)

[@riccardosarti3234]

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...

[@starfishsystems]

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!

[@Math_oma]

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

[@Mathologer]

:)

[@derekdonner3115]

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

[@koenth2359]

Someone here would not miss out a minute of pi day

[@user-me7hx8zf9y]

@Slimzie Maygen *finite polynomial with algebraic coefficients. An infinite polynomial can be shown to converge to pi.

[@artemiostriantafyllou7986]

@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.

[@sadusee]

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!

[@BarryBranton]

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

[@siener]

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.

[@dlevi67]

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.

[@ffggddss]

Yup!

[@DjVortex-w]

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.

[@voteforno.6155]

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.

[@RadicalCaveman]

I wish YOU would give a free course on measure theory...on KZbin. 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."

[@PC_Simo]

So would I 🤩.

[@enricolucarelli816]

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?

[@Mathologer]

"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 :)

[@myreneario7216]

Proof that the union of two countable sets is countable: Use the löwenheim skolem theorem to get a countable model of set theory. Also make it transitive by using a mostowski collapse or something like that. Then make up some dubious philosophical argument, that this countable model is the intended model of set theory. For example you could say, that because of Occam's razor, we should use the smallest possible model of ZFC as the indended model of set theory. Since our model of ZFC is countable, any set in that model will also be countable, because it´s a subset of our countable model. Since we say that this model is the intended model of set theory, we then get the wonderfully confused claim, that all sets are countable (or at least countable according to our background theory). The sets might look uncountable relative to our model, but they´re countable "in reality". But if all sets are countable, then of course the union of two countable sets is countable. QED So you see, to prove that the union of two countable sets is countable you don´t need to do this clever trick, where you make a list where you alternate between the elements of the first and the second set. You can just need to combine the löwenheim-skolem theorem with some severe philosophical confusion, and then you instantly get the result. Did I do this correctly?

[@richardschreier3866]

Alles klar, herr kommissar, (By which I mean "So far, so good.") I especially liked the demonstration that the set of algebraic numbers is countable. Can't wait to see what a proof of transcendence looks like! (Although having seen that a proof that pi is transcendental is four pages of mathematical pain, I am a little intimidated!)

[@Mathologer]

That proof will be a LOT more accessible than the one for pi :)

[@glennjohnson4919]

I can’t imagine studying mathematics now. Mathologer not only makes it fascinating, he makes it clear. It’s almost cheating!

[@jimcarroll779]

the very best explain of e i have every seen

[@dhoyt902]

"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.

[@jetison333]

Hey I'll be a math friend. Math is cool, and I need a math friend

[@averagemilffan]

@@jetison333 S A M E

[@themathedumacator2611]

There is an interesting conclusion arising from a countably infinite list having measure zero. That result itself makes sense, because each rational number is a point on the number line, and a point has no dimension; that is, no length. So with a countably infinite set like the rational numbers or the algebraic numbers, you could list the lengths of all of those countably infinite numbers (each being 0), and then add them together to get the measure of the entire list. (and adding together any sized list of zeros will result in 0) But the interesting part is that, because a number line (or a portion of it) clearly does have length and is not made up of anything other than points (that is, there are no spaces), the uncountably infinite numbers that remain are of such great "magnitude" that all of its members together have length, even though each member does not. Basically, it's like adding up so many zeros together that you somehow end up with a quantity that is non-zero.

[@voteforno.6155]

The Math Edumacator One of the important properties of a "measure" is what is called "countable additivity", i.e. if you have a countable collection of pairwise disjoint measurable sets (i.e. no overlap between any two sets), then you can find the measure of the union by adding up the measures of the sets. This does not apply to uncountable collections, however. Which is why you can't union all the singleton points on the line together and say the line has measure zero.

[@nullvoid12]

Squaring the circle at the start.. that was funny!!

[@PC_Simo]

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

[@DDranks]

Real numbers as an uncountable set really overwhelm me in a sense (because not only they include familiar numbers, but also numbers that are totally inexpressible, beings that we can define using language/notation not better than "real numbers that we can't define using language/notation better than this sentence".), so lately I've been thinking of numbers that include all the numbers we care about when doing numeric things (not abstract set theory thought experiments). I've been thinking that defining numbers as Turing machines that output digits (or generally, Cauchy series) produces us a listable set of numbers that include any number that is expressible and that we care about. The problem seems to be that because of the halting problem, we can't be always sure whether a given Turing machine will continue producing digits or not. But it is still an interesting approach: you can also classify numbers according to the computational complexity of producing the decimal expansion, and have a hierarchy of numbers that are gradually harder and harder to "reach" for. Then we reach the veil of chaotic incalculability, beyond which there's only a Lovecraftian sea of inexpressible madness of exotic numbers.

[@Mathologer]

ah fellow Lovecraft fan :)

[@phpngpl]

I think only set theorists truly care about Chaitin constants and such things you're talking about, and very few understand what set theorists are producing today. No disrespect to Cantor and his contemporaries, yes, these ideas are crazy cool but concrete math is super hard already, but I also can build stuff with it. I can program with it. I can control hardware to work for me. Sadly, most practitioners of the newer stuff, like category theorists, tend to take similar risks when they don't learn from the mistakes of the mathematicians of the past.

[@austinconner2479]

The definition you propose makes sense, despite the halting problem. It could be that what you propose is equivalent to real numbers in a constructive axiomitization of set theory, although I'm not an expert in logic...

[@drakezhard]

I would throw in the complex numbers as that's where the inclusions really lie, the complex numbers are the algebraic closure of the reals, this follows from the fundamental theorem of algebra. That said, I really like the explanation.

[@anon8109]

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.

[@MrCheeze]

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.

[@anon8109]

+MrCheeze A nicely written wikipedia article on the topic: en.wikipedia.org/wiki/Definable_real_number

[@Keldor314]

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.

[@stevethecatcouch6532]

+MrCheeze What do you mean by "cleaner"?

[@MrCheeze]

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.

[@Lightn0x]

Sooo is the irrational number obtain by cantor diagonalization of the rational spiral algebraic or transcendental?

[@tjfrye11]

4:58 isn't 2/2 equal to 1, not 2?

[@imadhamaidi]

he did not consider 2/2, he skipped it, he took 2/1

[@micmoo40]

Your section on Liouville's constant should have probably noted that the number shown with the "ocean of zeros" is in base 2.

[@Mathologer]

No, the original Liouville constant is in decimals. However, interpreting this number as being written using other bases, e.g. 2 also yields transcendental numbers :)

[@skytern1838]

4:57 2/2=2?

[@AaronHollander314]

Skytern... he is sliding to 2/1

[@mheermance]

This is one of those watch twice Mathologer videos. So this is my second time through.

[@iamrepairmanman]

Your shirt is incorrect, it contains only one gnome, so it isn't "poly"

[@frechjo]

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).

[@scitwi9164]

You didn't count the gnome who wears it ;)

[@iamrepairmanman]

Sci Twi he's not a gnome, he's an elf. stop being a fantacist

[@itsiwhatitsi]

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

[@oparkagaming6678]

Thank you sir

[@PC_Simo]

2:40 Mathologer: **Randomly generates a polynomial equation, whose solution approximates the golden ratio 🏵️.**

[@nejisamakage]

Nice t-shirt, where did you buy it @Mathologer?

[@Mathologer]

I got it from here www.zazzle.com.au/polygnomial_t_shirt-235678195975837274 :)

[@KosteonLink]

Sadly it's false advertising, there's only one gnome so it's a mognomial :'( Clever shirt though

[@2001ivar2001]

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

[@PC_Simo]

@@KosteonLink *Monognomial.

[@breathless792]

one way I think you could prove that transcendental numbers are uncountably infinite, is as follows: start by assuming that transcendental numbers are countably infinite premise 1) transcendental numbers are countably infinite, (the assumption) premise 2) algebraic numbers are countably infinite (already proven) premise 3) real numbers are uncountably infinite (already proven) premise 4) transcendental numbers are the real numbers with the algebraic numbers removed (the definition of transcendental numbers) premise 5) if you combine two countably infinite sets together, the result will also be countably infinite (proof as follows: take each set and put them side by side now, take the 1st number from each list, (as the first two numbers in new list) then the 2nd number in both lists (as the next two) etc... now you can rearrange this into a list making it countably infinite) now you can combine the transcendental numbers and the algebraic numbers together into a countably infinite set (just combine premises 1,2 and 5 by putting the two lists side by side) now this super set is the real numbers (premise 4) according to these 4 premises (1,2,4 and 5) we have a) real numbers are countably infinite but according to premise 3 we have b) real numbers are uncountably infinite which is a contadiction now the only unproven premise is premise 1 (that transcendental numbers are countably infinite,) therefore: that must be the false premise, therefore: transcendental numbers are uncountably infinite,

[@Mathologer]

Yep :)

[@yacines3180]

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.

[@fCauneau]

Clear and convincing ! Congratulations !!

[@scottmuck]

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?

[@joshuacoppersmith]

I am curious about why Mathologer chose to list the rationals using a spiral on the whole lattice rather than just picking a quadrant and using the same switch-back diagonals he uses for the algebraic listing--I think that's how I saw it in an E.T. Bell book back in the day. It seems like sign issues are easily handled, and the spiral's 0 denominators are kind of messy. If someone asks me for a list of insects and I include 7-legged insects and say, "just ignore the insects that don't exist" I think they would doubt me.

[@MuffinsAPlenty]

My guess is that he wanted to give a nice method of listing without having to worry about creating two separate lists and then stringing those two separate lists together. Also, while you have some awkward things like "fractions" with denominator 0, it's not very different from taking the method you mention and skipping fractions which are equivalent to those that you've already listed.

[@rohansharma1250]

"With a bit of common sense " Mathologer it ain't that common

[@Mathologer]

Yes, sadly common sense is not that common :)

[@michaelempeigne3519]

why is it named common sense if it isn't common ?

[@raizo-ftw]

"it is impossible to square a circle" so, ideally, if i take a perfect sphere of extremely thin surface (almost no thickness at all, impossible) and of volume approx between 3-4L filled with superfine particles smaller than the planck length (obviously impossible as well) and fill it inside a square container of 4L volume, does that mean that I just squared a circle in 3 dimensions and got a square container that contains exactly pi amount of volume? (obviously I do but, you know, some of you might highlight or add some extra things to this thought experiment)

[@FF-pv7ht]

the problem is no matter how hard you try, you can never reach exactly pi due to the accuracy constraints this universe gives you Or as someone else put it (and I dont remember the exactl calculation or number, sorry) calculating pi after a certain amount of digits is most likely a very useless task "for the real world", since you can only use the whole universe and the smallest particles and arrange them so you can get a circle with x digits exact to pi itself After that x'th digit (assuming you dont find infinitely much space or infinitely small particles) its impossible to improve that "accuracy"

[@fawzibriedj4441]

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.

[@fawzibriedj4441]

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...

[@drewduncan5774]

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.

[@fawzibriedj4441]

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 ?

[@drewduncan5774]

arxiv.org/abs/0908.4034

[@chumsky8754]

No. Some would be algebraic.

[@shaantubes]

Please make a video regarding pi = 22/7. I know its not equal but then would love to see a video by you

[@MikeRosoftJH]

Pi is not 22/7, pi is irrational (and transcendental). 22/7 is an approximation of pi correct to 2 digits after the decimal point.

[@kl45gp]

Отличное объяснение !

[@michaelhanford8139]

Uncountably infinite sets...Sesame Street's The Count's personal hell. 😭

[@Gold161803]

One issue I had: the process of spiraling out from the origin of a lattice grid is not as straightforward when done in dimensions higher than two. Can you explicitly provide a generalized procedure for listing all ordered n-tuples?

[@Mathologer]

Sure, it's just a lot more tedious than the 2d spiral. E.g. in 3d systematically inspect all integer triples (points with integer coordinates) via expanding cubes of integer diameter. Within every cube order the finitely many integer triples contained in it in lexicographic order, etc.

[@dariuszspiewak5624]

This is a fantastic job you're doing with this videos. Kudos to you and the whole team behind this. I think maths should be popularized more and more in our modern societies which are becoming more and more stupid on average. Maths is the only way to the Ultimate Wisdom (whatever that means). That's my opinion and nobody has to agree with it :)

[@michaelhanford8139]

Ultimate wisdom or knowledge? Try breath meditation. 😉

[@TristanBomber]

I'm taking up your challenge at 12:04. If the transcendental numbers were countably infinite, you could do the same trick as with the algebraic numbers by going across diagonals to make a one-dimensional list of transcendental numbers. Then, you could "shuffle" the algebraic and transcendental numbers together by alternating between algebraic and transcendental numbers. This would create another one-dimensional list, and so it would be countably infinite. But this would be all the real numbers, and we know the real numbers are uncountable! Since this creates a contradiction, it is therefore proved that the transcendental numbers are uncountably infinite.

[@Mathologer]

Exactly :)

[@osenseijedi]

Noob question : I was wondering (not sure if it makes sense tho), when we say a number is transcendental, does it apply to all numbers in all bases ? I mean, maybe its stupid, but wouldn't a base PI take care of the transcendentalness (is that a word?) of PI itself ? Because in base PI, 10 = 3.1415... in base 10. Can we even have fractional bases, let alone transcendental ones?

[@Mathologer]

There are numbers and then there are various representations of numbers. Transcendence is a property of the numbers and does not have anything to do with the different representations of numbers for example in terms of different bases. Considering bases other than natural number ones is possible but I don't think it gives any more insight into any of the things that we are talking about in this video :)

[@snelo67]

The weird thing I find is that nearly every number is transcendental. But almost every number we think of (know) isn't transcendental. Besides e and Pi, who can name other (non related) transcendental numb.

[@vcaro12]

The link to the course on measure theory is broken. Is it still possible to access it?

[@SocratesAlexander]

7:52 Algebraic #s are countably infinite 10:48 Real #s are uncountably infinite 11:44 Transcendental #s are uncountably infinite.

[@MrBorceivanovski]

Great explanation #

[@thomasluo2821]

Great video!

[@eXtremeDR]

Very mathy - my brain clearly prefers geometry over numbers. How can I define a particular range within π? For example I want to define an interval with the rhythm 2,2,3 digit of π starting at digit x of π and limited to digit y of π? Where x and y can be rational or irrational numbers. For example: start 5th digit, end 11th digit of π with rhythm 2,2,3 -> resulting set 2,5,8 0123456789012345 3141592653589793 0101001 And is π infinite in every possible number system?

[@pythagorasaurusrex9853]

Refering to your little excercise to proove the uncountability of the transcendental numbers (TN). I have something in my mind, but I am not sure of my logic is right. Here is my solution: Proof by contradiction: Let's assume the TN are countable. Then there exists (for every TN) a polynomial p with rational coefficient so that this chosen TN is a solution of p(x)=0. But this contradicts the main property of the TN. Conclusion: The set of TN is uncountable. I hope am right.

[@MCNarret]

Are all irrational normal numbers multi-dimensional? As in it is better to represent them as a shape of numbers are than a sequence? If I draw a finite line representing infinite length and the end of this pine repeats the beginning of the line but if you pick any point on the line it could represent another line at a 90 degree angle of any finite or infinite length. pi in this would turn into 3.141...3141... repeating... it would be a repeating number of infinite separation.

[@soostdijk]

Mathematicians always mess up completely when they start to include physical arguments in their sandbox. Counting is an operation and a primary operational fact here is that no matter how many irrationals you have to count, you will always have an integer available to count it with. Always. Therefore, the claim that there are more irrationals or reals than integers or rationals is nonsense and the whole Cantor argument fails.

[@MarkusBohunovsky]

Naive question of a non-mathematician: It seems that we tend to "construct" real numbers by simply generating infinite series of digits (such as the number with ones at the factorials). BUT: Just because we can think of pretty much any kind of infinite series of digits, can we be sure that such a series of digits really represents a unique real number--or in other words: Can we be sure that the imagined series of digits really represents a UNIQUE point on the number line that cannot be expressed by any other series of digits (potentially even an infinite number of infinite digit combinations?) Clearly, the fact that 0.9999999... = 1.00000000... (or 1) already proves that a unique combination of infinite digits does not mean a unique point on the number line. But do we know that there aren't (many) other (much less obvious) ways where different combinations of digits really represent the same number? In fact: How DO you translate an infinite number of digits into a specific coordinate on the number line? It seems to me that the only thing we can say of any specific point on the number line is, that it is NOT the one represented by the irrational number we are thinking of (by whatever small amount) Does my question make sense?

[@DutchMathematician]

+Markus Bohunovsky "It seems that we tend to "construct" real numbers by simply generating infinite series of digits" This is (in my opinion) the easiest way to "construct" a/the notion of real numbers, even for (non-math) university students. (a more robust way is using Dedekind's cuts, or equivalence classes of Cauchy sequences, but that certainly is way more complicated) "Just because we can think of pretty much any kind of infinite series of digits, can we be sure that such a series of digits really represents a unique real number" Actually, such an infinite series of digits DOES represent a unique real number: in R that is. It might not be a rational number, though (e.g. the infinite set of decimals of √2, those of e or π, do not "add up" to a rational number). The fact that such a sequence of digits represents a (unique) real number is quite profound and is related to the notion of "limit", as well as the fact that R is (what mathematicians call) complete. (see e.g. en.wikipedia.org/wiki/Completeness_of_the_real_numbers; beware, this is mathematically profound!) "Can we be sure that the imagined series of digits ... cannot be expressed by any other series of digits" Actually, we can (apart from your example given: 0.99999... = 1.0000... as well as 0.240000... = 0.23999..., etc.). Suppose that a real number x ϵ (0,1) (just take the interval (0,1) for simplicity) can be represented by an infinite series of digits in two different ways: x = x1 = 0.d1 d2 d3 d4... e.g. x = 0.1239... x = x2 = 0.c1 c2 c3 c4... e.g. x = 0.1239... (Note: I don't know how to use super-/subscripts here (I used Word, but super-/subscripts do not seem to translate well here, unfortunately); therefore interpret e.g. cn+3 as c(n+3), where "n+3" should be read as a subscript; powers - usually written as a superscript - have been expressed by the "^" symbol: 10^n should be read as "10 to the power of n") Now consider the first digit where the two representations x1 and x2 differ, say at the nth place. This means that d1=c1, d2=c2, ... dn-1=cn-1, but dn and cn differ. Suppose that dn>cn (otherwise, reverse x1 and x2). The least these two digits can differ is by 1 (since they are integers). Now look at the difference between x1 and x2 (which should be 0, since they are equal representations of the same number x): 0 = x1 - x2 = (dn-cn)*10^-n + (dn+1-cn+1)*10^(-n-1) + (dn+2-cn+2)*10^(-n-2)+(dn+3-cn+3)*10^(-n-3) +... Hence, 0 = 0*10^n = (x1 - x2)*10^n = (dn-cn) + (dn+1-cn+1)*10^-1 + (dn+2-cn+2)*10^-2 + (dn+3-cn+3)*10^-3 + … = (dn-cn) + Δ where Δ = (dn+1-cn+1)*10^-1 + (dn+2-cn+2)*10^-2 + (dn+3-cn+3)*10^-3 + … We already know that dn-cn is at least 1. How large can Δ be? Well, |Δ| = | (dn+1-cn+1)*10^-1+ (dn+2-cn+2)*10^-2+(dn+3-cn+3)*10^-3 + … | n (so that Δ = -1, exactly “compensating” the difference dn-cn = 1). Hope that helps …

[@gddeen1]

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.

[@Mathologer]

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. :)

[@christopherwalsh3101]

Ahhhhh, I just got it! it's a gnome, polyGNOMial! Hahahaha! I can't wait for the perfect timing to drop that in math class!!!! BOOM!!!!!! *That's me dropping the mic*

[@PC_Simo]

6:30 I wonder, what happens, if I write the list in binary 🤔.

[@ismetpilev869]

Nice one!

[@frechjo]

Hi Mathologer! Any chance on covering (some of) Gödel's Incompleteness theorems? I know it wouldn't be and easy task... but this is the best place I can think of to even ask!

[@Mathologer]

I've had a look at it. Doable but a major pain. Really need to build up some courage for that one first :)

[@frechjo]

Nice to know! :D I knew this was the right place to ask, thanks for the answer! Edit: Now I'm exited!

[@thomasa5619]

Could you do a video on how trig works? Like, how i type "sin(π/4)" and a calculator resolves 0.707...?

[@pierreabbat6157]

The diagonal argument, as you've presented it, fails if the first number is 1/10 and all nth numbers after that have 9 in the nth position. To fix this, to the nth digit of the nth number on the list, add 5 mod 10. This guarantees that the diagonal elephant will differ from the nth number by at least 4e-n. This does not work in base 2 or 3, but does work in all higher bases (in base 5, you can add 2 or 3, it doesn't matter). I recently saw on Wikipedia two transcendental numbers which aren't anywhere near normal (their digits are all 0 or 3) whose product is 1. They are 3.30033000000000033003300... and 0.3030000030300000....

[@ffggddss]

"... add 5 mod 10." Or add 1 mod 10. [+1 mod b works in any base, b ≥ 2.] Or my favorite trick: replace digit d with (9-d). [Of course, this trick (b-1-d) works only in even bases.] But your scenario *doesn't* lead to a failure of the diagonal argument, because the new number will be all 0's; but that number isn't in the list you've described. So a list of all the rationals can't be like you've postulated.

[@Mathologer]

There is no problem with the diagonal argument as I present it in the video. But the point you make is still interesting because it shows that the diagonal argument can also be used to show all sorts of interesting facts about countable sets of numbers. For example, that it is impossible to list the rational numbers in decimal form as I described it in such a way that you get a number on the diagonal that ends in a tail of 9s, etc. :)

[@NezumiM]

I love this channel!

[@Aufenthalt]

Another great video of Mathologer

[@runrickyrun157]

Because transcendental numbers are also representations of physical properties, like say pi, would it be incorrect to say we can't ever truly, no matter how small the scale of our instruments ever becomes, determine that a circle (or many other objects with transcendental properties--like an imaginary circle drawn and measured in space) has at least one of its (probably all) its components as non-transcendental? (My logic here is pi = C/d, so at least one, and a hunch says both, of those numbers has to be transcental, otherwise it would be algaberic/algabreic which by definition is not trancendental) Basically, knowing the measure of things exactly is an intrinsically unknowable thing, like mass and direction. Measurement itself is likely transcendental, always. Am I reading too much into it?

[@AgaresOaks]

You don't even need to list the algebraic numbers. Once you have that the reals are uncountable, just show the number of ways you can express a number is countable. Given a finite alphabet, you can enumerate expressions by going from strings of length 1, then 2, etc. etc. and it should be obvious each step contains at most finitely many expressions. But things like "the smallest solution of this polynomial" are a perfectly valid way to express a number.

[@Mathologer]

Of course, you can show all this in about a zillion different ways. In fact in textbooks the proof that the algebraic numbers are countable is a one-liner. However we are on youtube and are trying to explain things so that as wide an audience as possible can understand this. Also, one very important point I wanted to make in this video is that Cantor's diagonalization argument translates into a way to explicitly construct numbers with certain properties. Most people seem to be under the impression that it's only good for proving the existence of these numbers or to create contradictions as in the standard proof that the real and uncountable :)

[@KattarJ]

Professor Charles Xavier is teaching us math!

[@Mathologer]

:)

[@Constantin-kz7pb]

I do not understand why at 6:42, we know that this number is not at fraction. I dont see why their shodnt be a number that has 0 at the same decimal place. But that was necessary to conclude our number is not a fraction. I get that you could just change every number highlightet in green but not that method.

[@sohamsamaddar4759]

I found a proof for Transcendental numbers are uncountably infinite (Not sure if this is correct) We know that transcendental numbers cannot be the roots of a polynomial with integer coefficients, but we do know it can be the roots of a polynomial with real coefficients. So, we can find the roots of Ax + B for all real A and B, and then list the transcendental numbers from the roots, but we know that the set of Real numbers are uncountably infinite. Hence we cannot list all of A,B and hence transcendental numbers are uncountably infinite.

[@thomaskember4628]

Are all transcendental numbers either pi or e, or some expression containing pi or e, or maybe both, for instance pi + e, or Louisville numbers, or expressions contains Louisville numbers? Are there no other ways of expressing and approximating transcendental numbers? If this is the case, it must be that between any two numbers, whether they are transcendental or not, there is an infinite number of transcendental numbers that cannot be indicated in an expression, be it an algebraic formula or an infinite expansion. Therefore these numbers will be forever unknown to us.

[@MuffinsAPlenty]

It's a hard question to answer because the concept of "containing pi or e" is vague. For instance, you could be silly and say x = x + π − π. Then that's an expression "containing π". So in some sense, the answer is yes. But I think your question is something like: Can you get all transcendental numbers using only the rational numbers, powers of e, powers of π, and multiples all of the things previously mentioned? If that's the case, then the answer is no. I'm not going to prove it for you here, but in abstract algebra, you can talk about a notion of "transcendence degree" of a field extension. It's basically how many distinct numbers you need to add to your base field to be able to get everything in the larger field using the base field, powers of elements you add, and multiples of all of the things previously mentioned. If π and e were enough, then the transcendence degree of the real numbers over the rational numbers would be at most 2. However, it has been proven that the transcendence degree of the real numbers over the rational numbers is infinite.

[@paulbaker916]

Brilliant.

[@bobengelhardt856]

12:08 "... the set of ... rational numbers ... is vanishingly small compared to the set of ... irrational numbers ..."! If I take the set comprised of rational and irrational numbers and randomly choose one, the probability that it is rational is infinitesimal?

[@Rawdetzky]

I love ur channel.

[@mechuniversal]

Is there an updated link to the measure theory course?

[@Myrslokstok]

Excelent!

[@jth4242]

You could have mentioned using the completeness axiom here.

[@shakkathechannel6909]

That polignomal right there really cut through

[@anandchitrao4032]

If transcendental numbers are countable, then consider any list of them. Also consider a list of algebraic numbers. We make a new list by placing all algebraic numbers in the odd position of the list and place all the transcendental numbers in the even position. Thus we get a list of all real numbers which is not possible

[@Mathologer]

Exactly :)

[@scottcampbell2707]

If you use base e or base pi, do all the base 10 algebraic numbers become transcendental?

[@MuffinsAPlenty]

Most of mathematics does not at all depend on base. The concepts of algebraic and transcendental have nothing to do with base.

[@TylerHNothing]

can you make a video about fields and group theory?

[@TylerHNothing]

and how the real numbers tie in

[@Cubinator73]

_Prove for the uncountability of the transcendental numbers:_ Suppose the transcendental numbers were countable, meaning we can list them. Let A be a list of all algebraic numbers and let T be a list of all transcendental numbers: Now we construct a new list R containing all entries from A and from T: We add the first element from A to R, then we add the first element from T to R, then the second from A, then the second from T, and so on. Now our list R contains all algebraic numbers and all transcendental numbers and thus all real numbers. But this contradicts the uncountability of the real numbers, because they cannot be listed, so the transcendental numbers must be countable.

[@Mathologer]

Exactly :)

[@Metallleute]

@ 10:29 The important question is: Are there more ways to list all algebraic numbers than there are algebraic numbers?

[@voteforno.6155]

TheMatths I believe for any infinite set X, the cardinality of the set of permutations on X is equal to 2^|X|, so the answer would be yes.

[@fibbooo1123]

I dont think so. The proof I have is full of letters, but bare with me. We are trying to see if the number of (bijective) functions from the natural numbers (positive integers) back to the natural numbers is infinite. if we have some function f(u)=v, this can be thought of as us moving the algebraic number at spot u to spot v, and the bijective means there is exactly one u such that f(u)=v. Say by way of contradiction that we have a complete list of functions, and we can also enumerate them. So the first one is g1, the second is g2, etc. So the function gk(k) gives us whatever the k-th function on our list is, evaluated at k. We are going to construct a function h as a bijection from the natural numbers to the natural numbers that is not on our enumerated list. We are going to go in order, and define what h(a) is as a goes from 1 to infinity. In particular, we are going to define h(a) to be equal to the following thing: 1) h(a) is not equal to h(b), where b is an integer from 1 to (a-1) 2 )h(a) is not equal to ga(a)- that is, the a-th function on our list evaluated at a 3) h(a)Is equal to c, where c is the smallest positive integer that doesn't conflict with either of the other 2 requirements. So, we have to check some things: Is this function one to one? That is, if h(s)=h(t), does s=t? Basically, do two different numbers map to the same number? by 1), it is not Is the function onto? That is for any given integers q, does there exist a natural number p such that h(p)=q? I couldnt prove this one as nicely as I would like. For instance, if gk(k)=0 for all k, there would be no number such that h(p)=0. However, there could be at most 1 of these "holes", because if there were 2 "holes", the next function on our infinite list would be guaranteed to fill one of them, by 3). This proof is not complete because 1 number might not be on our list, but I'm not worried about that- though maybe I should be. Thus, our function should be on our list, because it is a bijection from the natural numbers to the natural numbers. But, is this function on our list? No, because by 2) this function has at least one input different from every function on our list. Thus, there are more than a countably infinite number of functions from the natural numbers to the natural numbers, and thus more than a countably infinite number of ways to list the algebraic numbers-i.e. more ways of listing the algebraic numbers than there are algebraic numbers.

[@fibbooo1123]

Here's how you patch the onto-hole: Suppose by way of contradiction that gk(k) = p for all the functions in the list for all k>=r Then swap the functions g_r and g_r+1 in your list. Because these functions are bijections, g_p cannot map p + 1 to p, since it already maps p to p and g_p+1 cannot map p to p, since it already maps p + 1 to p. Furthermore, since all functions on our new list have for k>r+1 gk(k)=p, we will never get another hole (though we do have to define h again over all of these)

[@voteforno.6155]

Let's assume AC. Let X be any infinite set with cardinality k. Let k! be the cardinality of the set of all permutations on X, i.e. the set of all bijections from X to itself. Since every bijection on X is a map from X to itself, clearly we have k!

[@sajjadakbar6649]

i love your videos

[@rossholst5315]

So my problem with Cantor’s method of diagonals, is there is no way to prove you are starting at an end. There should be an infinite amount of points that are arbitrarily similar to any other point. There is no way to eliminate starting spots that are arbitrarily similar yet distinctly different. Imagine reading a number. We start just to the left of the decimal, it’s 0, we read out for the next 10 trillion digits, all zeros. So now we look to the right of our decimal point. First digit is a zero and we read the next 10 trillion digits to the right all zeros as well. So 10 trillion digits to the right and left of the decimal point all zeros. So clearly the only location that I can reasonably be at, is the point 0? Clearly that is not necessarily true. After all 10 trillion digits is a full 0% of the total possible digits. And yet I could be at the point 0, statistically unlikely, but distinctly possible. But, there will be no way to eliminate the infinite amount of points that resemble any other point. There are an infinite number of points that could be unique and yet remain impossible to differentiate. My other idea is simply to imagine listing all of the real numbers between 0 and 1. Now all of these numbers can be listed as some infinite sequence of digits listed in decimal notation. Now if we multiply all of these decimals by infinity, they will all become some integer because everything is a factor of infinity. 0 will become 1 and 1 will become infinity. After all 0 is actually 0.000000…1.

[@michaelcolfin8464]

Every once in a while I hit upon a 'fact' that gets my mind reeling. There is a theory that due to the infinite number of universes out there that somewhere this is an identical person who is doing exactly the same thing as I am at the same time. There is in fact an infinite amount of them just because of the nature of infinity. But I have a problem with it so I tried to simplify this down to numbers. If the above is true, I reasoned than for a transcendental number such as pi to be truly transcendental (non-repeating) then any arbitrary string of numbers must be contained in it. I found the first million digits of pi and tried random 5 digit numbers and they were all there, so shouldn't that be for any arbitrary long finite string numbers? If not that means that just because something is infinite doesn't mean something has to happen at all, or could happen only once. Is my logic faulty that for pi (or e or the square root of 2 etc.) to be truly transcendental means that all finite strings of numbers must be found it it somewhere?

[@FF-pv7ht]

yea it should afaik if you make up some arbitrary way of translating pi into letters you will find every piece of human literature ever written and every piece that will still be written in that number, it just takes really long. Im not sure if you can consider that "true" information about the universe though, I mean nobody can really prove to you multiverses are thing, im quite sure nobody can even prove to you this universe alone is infinite

[@dlevi67]

Not necessarily. What you are describing is a so-called "normal" number. It is not known whether pi, e or sqrt(2) are normal. However, there are plenty of transcendental numbers that are not normal: Liouville's constant is one, and the one that Burkard derived through diagonalisation isn't either (they both contain only 1 and 0). Incidentally, much the same may well apply to universes... depending on what are the "differences" between one universe and another one. If all universes have physical laws like ours, then it's possible that "repeats" happen, but I don't think anyone knows whether that's the case.

[@29jonesie]

No, an infinite set is not necessarily exhaustive.

[@jellocraft34]

The one part of this video that isn't obvious to me was proving the algebraic numbers are countable. It is evident that you can enumerate the lattice points in 1D and 2D space, but it doesn't appear trivial to assume that the lattice points in N dimensions are countable (even though that's what my gut and this video are both telling me). Can anyone help with this?

[@Mathologer]

Well, fix a number r and consider the set of lattice points in n-space at distance r from the the origin. Can you see that there are only finitely many of these lattice points? If you can, just enumerate things by first parsing the lattice points at distance 1, then those at distance 2, then those at distance 3, etc. :)

[@jellocraft34]

That makes sense, thanks! Fantastic video by the way

[@romajimamulo]

Imagine the following. You throw a dart at a dart board. Imagine, still, that you could measure where that dart landed, infinitely precisely. What is the chance it lands on exactly a rational number? Well, if it did, no matter how closely you measured, you'd see the same digits repeating forever, or zeros repeating forever. If your measurement started with a pattern, every additional digit is a chance that pattern could break. Since you never run out of chances a pattern is broken, eventually you'll run into one of them and hence, it can't be rational

[@frankharr9466]

12:08 So countably infinite times uncountably infinite equals uncountably infinite? I kind of wish you'd talked more about the idea of being uncountably infinite. I believe you, I'd just like to hear more because I'm not sure I fully (or even basically) understand it. Oh! on a second viewing, now I get it. I'd still like to know more.

[@danildmitriev5884]

Intuitively - yes, but some people here will warn you that it is not the rigorous way to claim such things, because infinity is not a number but just a concept.

[@frankharr9466]

Oh, totes. I was a German major and aspiring book designer. I don't claim to have a rigorous proof of anything. In particular, I don't know how to define "times" in this instance. It just sounded like something that could be proved, should one have the book-larning and the inclination for it.

[@danildmitriev5884]

I understand, I just warned you so that you're ready :) If you're interested, I can tell you how you could say it formally. At 12:08 Mathologer implies that the *union* of two sets (i.e. combining their elements into one set) - the set of transcendental numbers and the set of algebraic numbers - is uncountably infinite. In general form, if we have two sets - A and B, where A is countably infinite and B is uncountably infinite - if we consider the set C = A ∪ B (∪ denotes the union of sets), then set C will be uncountably infinite, just as set B. Furthermore, if we consider a set D = B \ A (D consists of elements from B which *are not* contained in A, so we basically "exclude" A from B), then D will also be uncountably infinite. So, adding or taking away a countably infinite set to or from the uncountably infinite set doesn't affect the uncountability of that set.

[@frankharr9466]

"I understand, I just warned you so that you're ready :)" Fair enough. Thank you. It's not like anyone will notice, though. And if they do? Hey, I know I don't know what I'm talking about. " . . . So, adding or taking away a countably infinite set to or from the uncountably infinite set doesn't affect the uncountability of that set." That makes sense. I'm down with that. It did make me wonder if it was closer to addition than multiplication. However, if you look at probability theory something and something else requires multiplication. Something or something is addition. So that works for me. Thank you.

[@myreneario7216]

@Danil Dmitriev I never quite understood, what do people mean when they say "Infinity is not a number but a concept". Can you explain me what it´s supposed to mean? Firstly, what does "Infinity is not a number" mean? Infinity is not a real number. But the same holds true for most complex numbers and most p-adic numbers. Nonetheless these are called numbers. Also there are number systems containing infinite numbers, like for example the cardinal numbers, the ordinal numbers, the surreal numbers. There´s also an infinity in projective geometry and on the riemann sphere. So what do you mean when you say "Infinity is not a number"? Secondly, what does "Infinity is a concept" mean? What is a concept? Is there some definition of a concept anywhere?

[@DustinRodriguez1_0]

How inefficient is Cantors process of listing the rational numbers (considering every time you have to throw out a number you already listed or can't include like divide-by-zero) and is there a better method? It makes me uncomfortable to just say "oh, ignore the divide by zeroes and duplicates, its still fine".

[@Mathologer]

Actually, you can leave all the duplicates in, it does not make any difference. But if you are after a really neat way of enumerating the rational numbers I don't think anything beats the enumeration using the Stern-Brocot tree: en.wikipedia.org/wiki/Stern-Brocot_tree :)

[@DustinRodriguez1_0]

Oh very neat! I like that one much better! Probably just my Computer Science education makes me more comfortable with binary trees and such. Thanks for pointing that out, I'd never even heard of it before.

[@MrZyroid]

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

[@Mathologer]

actually in an early draft that's exactly what I wanted to say :)

[@__gavin__]

My immediate thoughts too, "I have put together a truly marvelous proof but this video is too small to contain it".

[@keivanroboubi4409]

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

[@andreychen6523]

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"

[@RadicalCaveman]

+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...

[@randomisedrandomness]

x = 4 - gnome/2

[@Mathologer]

:)

[@theomeletteguy9353]

RandomisedRandomness I was going to comment that. You beat me to it!

[@-_Nuke_-]

What?

[@lucamuscarella4085]

i thought gnome=8-2x

[@-_Nuke_-]

anyone please explain :P