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".

Fermat will likely go down as the biggest, best bullshitter in math. "I have a wonderful proof of this assertion, but its too big to write in a youtube comment" Now do I get to have my name remembered by millions for hundreds of years?

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"

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 KZfaq 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 😂

:)

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

What?

i thought gnome=8-2x

anyone please explain :P

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.

@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 🤔.

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.

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

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.

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 😑.

Mission accomplished :)

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)

Glad the videos work so well for you :)

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.

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!

+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".

Definitely on my to-do-list :)

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-KZfaq check out this website: www.qedcat.com :)

OMG! This is gold. Thank you!

This is great! Sorry, I'm arriving late to the party...

:)

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.

Michael Bishop The cardinality of the continuum is c. The assertion that c is equal to aleph-1 is precisely the continuum hypothesis. And there are uncountably many aleph numbers, in fact, aleph (alpha) is defined for every ordinal number alpha.

OH? Then please tell me of aleph-2.718281828.....

Michael Bishop 2.71828... is not an ordinal number.

2.718281828... is not an ordinal.

" And there are uncountably many aleph numbers, in fact, aleph (alpha) is defined for every ordinal number alpha." It would then seem to me that there are a countably many aleph numbers.

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

@@jetison333 S A M E

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

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.

That's great :)

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

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

he skipped it cause it was already counted

A/V mismatch.

Not even for very large values of 2...

Skytern... he is sliding to 2/1

Yes, sadly common sense is not that common :)

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

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.

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.

Nice isn't it :)

Good luck with reading that proof :)

hahah i just read theorem 204 and 203 and undestood probably 40% the hardest parts that werent down my level were the formula manipulations and some theory about integral polynomials but well at least i satisfied my curiosity!!!!

I'll put it on my list of things to ponder :)

I already made it into a t-shirt :)

"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 :)

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.

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

Ultimate wisdom or knowledge? Try breath meditation. 😉

+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.

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!