All the Numbers - Numberphile

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  • @brogcooper25
    @brogcooper25 5 років тому +4572

    It's reassuring to hear a mathematician say they read a math paper and couldn't comprehend it.

    • @sorenlily2280
      @sorenlily2280 5 років тому +317

      It's absolutely true. It's like a native english speaker listening to a really heavy accent, like a heavy irish, or austrailian accent. If you've never really been exposed to that kind of accent before (that area of mathematics), you won't have a damn clue what they're saying, even though you're a native english speaker (mathematician). If you listen to some lighter accents, you can train your ear to eventually understand the heavy accent, but it's not easy. And unfortunately, even when you understand one heavy accent, it really doesn't help you with most other accents.

    • @heimdall1973
      @heimdall1973 5 років тому +85

      @@sorenlily2280 That sounds more like the language that lawyers speak and that you encounter in terms and conditions... Give me maths any day!

    • @redbeam_
      @redbeam_ 5 років тому +29

      I find it kind of scary...

    • @angelmendez-rivera351
      @angelmendez-rivera351 5 років тому +27

      Barrack Obama Vlogs Eh, no. Scientific papers are rigorously written. People are simply not properly educated to understand them.

    • @General12th
      @General12th 5 років тому +67

      @@angelmendez-rivera351 In this case "people" includes professional mathematicians as well. Math is a subject with so much breadth _and_ depth that folks in one field can be newcomers in other fields.

  • @filipw9973
    @filipw9973 4 роки тому +3644

    "You like math? Name every number then."

    • @hkayakh
      @hkayakh 4 роки тому +81

      -infinity to +infinity

    • @CaptHayfever
      @CaptHayfever 4 роки тому +164

      @@hkayakh: That's only the reals.

    • @hkayakh
      @hkayakh 4 роки тому +36

      @@CaptHayfever i is in there, if it weren't then it isn't

    • @maxonmendel5757
      @maxonmendel5757 4 роки тому +24

      I wish I could upvote this twice.

    • @Ryanisthere
      @Ryanisthere 4 роки тому +80

      @@hkayakh imagine a square with the two points
      (∞, ∞i) and (-∞, -∞i)
      thats all the numbers
      well until we get into quaternions

  • @tonyhakston536
    @tonyhakston536 5 років тому +2668

    0:15 There are only three whole numbers: 11, 17, and 3435.

  • @robertofontiglia4148
    @robertofontiglia4148 5 років тому +3123

    "An infinite series that gives you pie." -- Isn't that the Great British Bake-off ?

    • @serglian8558
      @serglian8558 5 років тому +26

      yeah, I guess

    • @oldcowbb
      @oldcowbb 5 років тому +8

      it ended after they changed the host

    • @SimplySara55
      @SimplySara55 5 років тому +7

      No. -_- The name of the food and the name of the number are homophones. "Homophones" are words that have identical pronounciations but vary in spelling. "Pi" = the number and "pie" = the food.

    • @fghsgh
      @fghsgh 5 років тому +51

      @@SimplySara55 r/whooosh
      also, try not to annoy people by responding to their comments 6 months after they've written them
      EDIT 2 years later: sorry

    • @SimplySara55
      @SimplySara55 5 років тому +6

      @@fghsgh I am sorry I do not see every single UA-cam comment the exact moment they are posted. :)

  • @MikuJess
    @MikuJess 5 років тому +4325

    So the majority of numbers are normal and noncomputable, but we don't know a single one? It's like... the mathematical version of dark matter.
    Dark mather.

    • @henrymick9648
      @henrymick9648 5 років тому +57

      Lol, you commented on the TwoSet Video aswell.

    • @superposition2644
      @superposition2644 5 років тому +45

      It's kind of like that, except with no dark energy or mass or photons or space-time or transfinite ordinals.

    • @eventhorizon853
      @eventhorizon853 5 років тому +72

      Pretty much, and just like the whole dark matter fiasco it looks more like a coping mechanism for our lack of understanding rather than a reasonable explanation.

    • @brcoutme
      @brcoutme 5 років тому +7

      What they didn't show is what/if we know numbers are not normal numbers for the non-trival cases. That is to say, we don't know weather or not all transcendental numbers or computable numbers (that are outside of our transcendental numbers) are normal numbers. Rational (and thus, whole) numbers, are trivial to see that they are not normal numbers. (Thus, why Matt did not draw any intersection into them for his Normal numbers circle).

    • @brcoutme
      @brcoutme 5 років тому +7

      @Ron Maimon I'm not going to lie most of that was over my head, but I did follow the bit about how to guarantee an uncomputable number is also a normal number by simply placing the digits of a known normal number into the digits of an uncomputable number (even though we can not actually compute it obviously). Not familiar with the Liouville numbers, but I'll take you word that it is a transcendental number that definitely isn't a normal number. I agree that the video would have been better giving these examples at least.

  • @davidlittlewood2860
    @davidlittlewood2860 5 років тому +2306

    -We're going to do all the numbers
    -We're not going to do Complex numbers
    Oh

    • @coopergates9680
      @coopergates9680 5 років тому +192

      Quaternions.... octonions.... infinite cardinals and ordinals...
      Or versions such as p-adic and quote notation

    • @bogdandamaschin9381
      @bogdandamaschin9381 5 років тому +44

      Complex numbers do not exist technicaly speaking

    • @General12th
      @General12th 5 років тому +198

      @@bogdandamaschin9381 All numbers are made up.

    • @noelkahn4212
      @noelkahn4212 5 років тому +6

      @Cooper Gates technically the infinite cardinals, and ordinals aren't numbers that would be computable or normal I think

    • @EebstertheGreat
      @EebstertheGreat 5 років тому +22

      @@noelkahn4212 There is not the same notion of computability for cardinal numbers that there is for real numbers, but there is a similar notion for ordinals. Finite ordinals (natural numbers) are all "computable" in any sense, since you can compute them by just supplying all the digits. Uncountable ordinals are not computable. But it turns out that not all countable ordinals can be "computed" either, given the appropriate meaning of the word. Using a generalizaiton of Turing computability called hyperarithmetic, you can construct notations and prove theorems for all recursive ordinals. But you cannot do that for non-recursive ordinals, the first of which is called the Church-Kleene ordinal. Countable ordinals larger than this can be considered non-computable.

  • @Aceronian
    @Aceronian 5 років тому +1908

    I see Matt is trying to one up the other numberphile presenters by talking about *ALL THE NUMBERS*

    • @kapitantaryfa
      @kapitantaryfa 5 років тому +79

      he should put them in a magic square

    • @General12th
      @General12th 5 років тому +63

      Then Tony will come back with a video about *ALL THE OTHER NUMBERS*

    • @standupmaths
      @standupmaths 5 років тому +72

      Aceronian “ne up them? I’m trying to up them by an uncomputable amount.

    • @sevret313
      @sevret313 5 років тому +12

      All they need to do to up him again is to solve his mistakes in the Parker Square.

    • @rq4740
      @rq4740 5 років тому +11

      @standupmaths
      I’m afraid your letters have gone off eating each other again, Matt 😂

  • @Ken.-
    @Ken.- 5 років тому +611

    3:42 _Rap Lyrics_
    Which? We don't know
    Pi to the e
    We don't know
    e to the e
    We don't know
    Pi to the Pi
    We don't know
    Right, these are all in the cusp!

    • @Razorcarl
      @Razorcarl 4 роки тому +20

      Wow

    • @isaacmiles-watt2758
      @isaacmiles-watt2758 4 роки тому +11

      We know!

    • @GaryFerrao
      @GaryFerrao 4 роки тому +27

      4:19 there's st… there's a list; here's the only ones we know, and THAT'S IT.
      4:25 Graham's number, in here. Googolplex, in here.

    • @jan-wito
      @jan-wito 4 роки тому +4

      Looking for a math rap? Watch 3blue1brown's poem on e to the pi i

    • @spongebobbatteries
      @spongebobbatteries 4 роки тому +2

      _bars_

  • @mkaali
    @mkaali 3 роки тому +702

    I love how mathematicians discovered the rarest group of numbers and decided to call them 'normal numbers'.

    • @jasondeng7677
      @jasondeng7677 3 роки тому +77

      12:34 not really the rarest but yeah... still a strange name to choose for this kind of like obscure category

    • @Tom-u8q
      @Tom-u8q 2 роки тому +79

      They're not rare almost all numbers are normal. If you were to randomly pick a value from a distribution it would be normal with probability 1.

    • @d3xCl34n
      @d3xCl34n 2 роки тому +7

      they describe the normal - 1 tree 2 monkeys 6 bananas (thats the logic).

    • @mihailmilev9909
      @mihailmilev9909 2 роки тому +16

      @@d3xCl34n *w h a t*

    • @mihailmilev9909
      @mihailmilev9909 2 роки тому

      @@d3xCl34n banana monkey brain neuron activation

  • @martinwalls64
    @martinwalls64 5 років тому +971

    I love how the code on the laptop animation actually does compute pi when you run it! Attention to detail!

    • @forstnamelorstname4169
      @forstnamelorstname4169 5 років тому +59

      So does the recipe.

    • @rocketlawnchair9352
      @rocketlawnchair9352 4 роки тому +66

      And written in Python, making the whole thing a play-on-words. I love numberphile.

    • @fanq_
      @fanq_ 4 роки тому +27

      @@rocketlawnchair9352 python most likely because Matt knows and uses python to play around and research videos

    • @AssemblyWizard
      @AssemblyWizard 4 роки тому +20

      5:05

    • @toniokettner4821
      @toniokettner4821 3 роки тому +1

      by just looking to the right you'll be surprised that the 3.14... gives exactly that away

  • @Mmmmmmkai
    @Mmmmmmkai 5 років тому +300

    "this is where numbers are, and we have none" is so funny to me

    • @ModernandVintageWatches
      @ModernandVintageWatches 2 роки тому +1

      I am amazed with all the science channels, the breakthrough in physics, medicine etc are almost daily basis now

  • @kennyearthling7965
    @kennyearthling7965 Рік тому +78

    I would love them to make a sequel to this, including the imaginary, hypercomplex numbers and hyperreals and asurreals etc.

  • @basapon7074
    @basapon7074 5 років тому +1763

    So "all the numbers", but not quite. So it's like a Parker Diagram then.

    • @TheMrvidfreak
      @TheMrvidfreak 5 років тому +52

      For sure. Even the Parker Square, drawn on a non-cube for the occasion, can be seen present at the birth celebrations of another of its kind at 12:07.

    • @Filip6754
      @Filip6754 5 років тому +23

      He hasn't begun with the naturals either.

    • @Ploppism
      @Ploppism 5 років тому +13

      A Parker Circle?

    • @SassInYourClass
      @SassInYourClass 5 років тому +18

      Chvocht - Also no direct mention of integers. He just kind of halfway acknowledges them exist without labeling them.

    • @davecrupel2817
      @davecrupel2817 5 років тому +16

      I take it we're never letting Matt live this down...

  • @EebstertheGreat
    @EebstertheGreat 5 років тому +507

    e was the first number that arose "naturally" in math to be proven transcendental, but the actual first numbers were the Liouville numbers in 1844, deliberately constructed for the purpose of being transcendental.

    • @guillaumelagueyte1019
      @guillaumelagueyte1019 5 років тому +49

      Artificial numbers heh

    • @vivekmathur3514
      @vivekmathur3514 5 років тому +1

      Ceski.

    • @alansmithee419
      @alansmithee419 5 років тому +2

      @@guillaumelagueyte1019 so...
      Numbers?
      Literally all numbers.

    • @Cythil
      @Cythil 5 років тому +33

      It is quite funny that we see numbers as "Artificial" or "Natural" when we just mean by that they where ether constructed specifically for the purpose of creating number that fits a category, or was number that we had constructed for a different purpose that was later found out to belong to one of the categories. Maybe not the best terminology but it sort of feels right anyway. ^_^

    • @EebstertheGreat
      @EebstertheGreat 5 років тому +22

      @@Cythil Pretty much. e is a useful constant in many ways, and its transcendence is the type of problem mathematicians were really interested in. Liouville defined his numbers just to demonstrate that transcendental numbers exist; they have no other known practical use. It's sort of like pointing out that 0.123456789101112131415... is normal. This is true, and it's trivial to show, but it isn't exactly a useful result in the study of normal numbers.

  • @cube2fox
    @cube2fox 5 років тому +76

    "As mathematicians we're thinking we are getting somewhere, but up until now we have found none of the numbers."

  • @barefootalien
    @barefootalien 5 років тому +766

    |*facepalms*| Mind blown in the first thirty seconds. Decades of math and science, a full understanding of what rational numbers are, and only when he says, "The rational numbers-those that are *ratios* ..." do I finally make the connection between those two words...
    Thanks, Matt!

    • @t.c.bramblett617
      @t.c.bramblett617 5 років тому +29

      I remember when I made that connection too, it was one of the big epiphanies. lol
      as a non-math student or professional, I also got my mind blow quite late in life by Euler's formula, and I think the biggest mind blow moment I can remember regarding math was learning about Cantor's infinities

    • @ryanoutram7059
      @ryanoutram7059 5 років тому +5

      "Ratio" came first too! :)

    • @tcoren1
      @tcoren1 5 років тому +7

      Barefoot the way I heard it, the ancient greek (or whoever), weren’t big fans of irrational numbers, and felt they didn’t make sense-they were “irrational”, and that’s were the term comes from

    • @t.c.bramblett617
      @t.c.bramblett617 5 років тому +7

      @@tcoren1 Yeah the Greek term is "alogos" for irrational or unknowable. "ir/ratio" is Latin and was the translation used by later Renaissance mathematicians

    • @brooksolomon7663
      @brooksolomon7663 5 років тому +2

      Is the "golden ratio" rational or irrational? That was the first question that came to my right after he said that

  • @Matias_Zimmermann
    @Matias_Zimmermann 5 років тому +649

    In the article "Borel normality and algorithmic randomness" Calude proved that every Chaitin's constant is normal.
    So, exist a non computable number, which is normal.

    • @superposition2644
      @superposition2644 5 років тому +18

      :O

    • @thefamousarthur
      @thefamousarthur 5 років тому +14

      Random Decimals: 2.817316571046953926392639363856293619263625287483748846362515375828402010164936492638262748392

    • @thefamousarthur
      @thefamousarthur 5 років тому +7

      And so on.

    • @randomdude9135
      @randomdude9135 5 років тому +20

      I didn't fully understand what computable and an noncomputable numbers are. Can some one clearly explain? :/

    • @WaffleAbuser
      @WaffleAbuser 5 років тому +110

      @@randomdude9135 If there exists an algorithm to compute a number's digits, then it is a computable number. If no algorithm can exist, it's uncomputable.

  • @hyungilkoo9340
    @hyungilkoo9340 4 роки тому +565

    5000 years ago: we need something to help count stuff! Let’s call it numbers!
    Now, in 2020: we don’t know most of the numbers!

    • @aldobernaltvbernal8745
      @aldobernaltvbernal8745 4 роки тому +4

      but we don't lol

    • @KombatGod
      @KombatGod 4 роки тому +38

      I just discovered a new number!
      1278603764680367894927767590382684995837376374858483735241790693752137800965358000000000000010000100100100006594762729191661916151881161681948583826261515618010100101000101110000001001111111106648493025858493028475749374748387384847641324422048487646483929201.003
      Yes, it's a new number. It's nothing special but it was never said nor written down in the history of mankind.

    • @hyungilkoo9340
      @hyungilkoo9340 4 роки тому +9

      KrossoverGod why is there a r in it

    • @KombatGod
      @KombatGod 4 роки тому +6

      @@hyungilkoo9340 There's no r in it.

    • @hyungilkoo9340
      @hyungilkoo9340 4 роки тому +7

      KrossoverGod yes there is there’s also an e in it

  • @DonGeritch
    @DonGeritch 5 років тому +527

    this video should be called 'None of the Numbers'

    • @jimmythewig3354
      @jimmythewig3354 5 років тому +23

      Or Parker All of the Numbers...

    • @Tjalve70
      @Tjalve70 5 років тому +40

      Infinitely few of the numbers?

    • @ZeHoSmusician
      @ZeHoSmusician 5 років тому +6

      Adding a quantum dimension to this topic:
      The video is of course titled "all the numbers"...but that's *if you don't watch it*. As soon you do, then the title changes to "none of the numbers"... :D

    • @neilgerace355
      @neilgerace355 4 роки тому

      "Almost all" numbers are transcendental

    • @luantuan1653
      @luantuan1653 4 роки тому

      'Some of the Numbers'

  • @mpupster
    @mpupster 5 років тому +728

    You should do a video about the 100 page proof in Principia Mathematika of how 1 + 1 = 2

  • @FMFF_
    @FMFF_ 4 роки тому +77

    I just love everytime a different subject illustrates this saying:
    "The more you know, the more you know you dont know"

  • @jon2431
    @jon2431 5 років тому +376

    How can one not love Matt Parker?

    • @TheOneMaddin
      @TheOneMaddin 5 років тому +26

      At times, he is a little bit too unprecise. But thats the price for being popular anong non-mathematicians.

    • @cordlefhrichter1520
      @cordlefhrichter1520 5 років тому +24

      @@TheOneMaddin Imprecise*

    • @Triumvirate888
      @Triumvirate888 5 років тому +3

      Matt Parker loves himself so much that the rest of us don't need to.

    • @jon2431
      @jon2431 5 років тому

      @@Triumvirate888 GOT EM 😂

    • @jon2431
      @jon2431 5 років тому +7

      Also, you okay buddy? Sounds like you think loving yourself is a bad thing.

  • @p11111
    @p11111 5 років тому +397

    We need a video on non-computable numbers! (please)

    • @gibrana9214
      @gibrana9214 5 років тому +47

      By uploading, through a computer, it would become... Computable?

    • @gold4963
      @gold4963 5 років тому +5

      Gibran A ...Mind-blown.

    • @Patrickhh69
      @Patrickhh69 5 років тому +10

      For example: busy beaver numbers and Rayo number

    • @KafshakTashtak
      @KafshakTashtak 5 років тому +1

      Does not fempute, does not fempute.

    • @aaaa-hj9vv
      @aaaa-hj9vv 5 років тому +22

      @@Patrickhh69 The busy beaver function is uncomputable, but the numbers themselves are computable because all integers are computable. That is, we can't compute what the numbers actually are, but we know that no matter what they are, they are computable numbers.

  • @shill2920
    @shill2920 3 роки тому +76

    I just feel awe at the fact that we created math as a concept and now its something people are working their lives to unveil because we created something, a huge set of rules and interactions that have lied out a entire infinitely sized concept that has grown larger than what the creators understand of it. The concept of math growing larger than the people who created it, now that's something.

    • @mondkalb9813
      @mondkalb9813 3 роки тому

      Math + computers = even more awe. :D
      When I got my Amiga back in the late 80s, I started exploring fractals (mainly the Mandelbrot set) and continued so later on with better and better PCs. What then took hours or days to compute, you can do now nearly in real time on modern home computers. There are videos on Youbtube showing zooms into the set to unbelievable depths.
      What struck me with amazement: Even on small home computers, when you zoom in deep enough, the whole Mandelbrot set relatively grows bigger than the entire known universe pretty fast. With 100% certainty you are looking at details, that nobody else has ever seen (though, due to the nature of the set, they all look similar).

    • @ModernandVintageWatches
      @ModernandVintageWatches 2 роки тому +1

      I am amazed with all the science channels, the breakthrough in physics, medicine etc are almost daily basis now

    • @auscaliber1
      @auscaliber1 Рік тому +7

      There is a long standing philosophical debate about whether maths is invented/created or discovered. I don't think we created maths, we just created our own sets of language and symbols to interpret it.

    • @mevideym
      @mevideym Рік тому +1

      @@auscaliber1 But we assume axioms which we deem useful and then derive true statements using logic from them

    • @cara-seyun
      @cara-seyun Рік тому +4

      We don’t create math anymore than I create a landslide by tossing a rock onto an unstable pile. I trigger things with an input, but the architecture was there the whole time.

  • @twodollars4u
    @twodollars4u 5 років тому +671

    I found hundreds of uncomputable numbers in my calculus homework

    • @saetainlatin
      @saetainlatin 5 років тому +54

      just wait when you get to differential equations, no numbers whatsoever, just uncomputable letters and variables

    • @dlevi67
      @dlevi67 5 років тому +21

      @@saetainlatin Abstract algebra I find much worse. Differential equations I can somehow "understand" geometrically (not always, and not always easily), but a variety? Or a vector space?

    • @LuigiElettrico
      @LuigiElettrico 5 років тому +4

      Uncomputable teacher xD

    • @jakedones2099
      @jakedones2099 5 років тому +1

      @@dlevi67 I agree with you

    • @lyrimetacurl0
      @lyrimetacurl0 5 років тому +1

      @@saetainlatin Then wait till you get to partial differential equations

  • @fishandchips8813
    @fishandchips8813 5 років тому +27

    Thank you SO MUCH for stretching my brain like this!!
    I am not a mathematician, nor will I ever be one, but I swear my quality of life is noticeably improved every time you guys blow my mind like this!
    I’m gonna have to go lay down for a bit and sort of digest this stuff.
    Thanks again!!

    • @WoogiBoogie
      @WoogiBoogie 9 місяців тому

      LOL @ lay down for a bit and digest this stuff.

  • @Bignic2008
    @Bignic2008 4 роки тому +64

    My takeaway is that the real numbers are far more complicated than one might think. I certainly felt a level of comfort with them when I took my first real analysis course years ago - “they’re just non-terminating decimal expansions with no repetitions” - but even that alone is an extremely deep and complicated statement. People are fooled by the simple name “real numbers” that we sort of understand them, but we just don’t. As Matt said, most reals are “dark”, and also bizarrely, there are subsets of the reals that can’t be assigned a meaningful notion of “volume”. This leads to weirdness like Banach-Tarski.

    • @isavenewspapers8890
      @isavenewspapers8890 11 місяців тому +1

      "non-terminating decimal expansions with no repetitions"
      That sounds like a description of the irrational numbers.

    • @eguineldo
      @eguineldo 8 місяців тому

      ​@@isavenewspapers8890 Irrationals definitely are like that but there are rationalsk like 1/3 which have an infinite decimal expansion.

    • @isavenewspapers8890
      @isavenewspapers8890 8 місяців тому

      @@eguineldo "with no repetitions"

    • @eguineldo
      @eguineldo 8 місяців тому

      @@isavenewspapers8890 Apologies, I guess I didn't read your comment very thoroughly. Then I would agree

    • @isavenewspapers8890
      @isavenewspapers8890 8 місяців тому

      @@eguineldo Nice.
      Technically, any terminating decimal expansion can also be made non-terminating; you just put infinitely many 0's at the end. You can even do some weird stuff like represent 1 as 0.999..., but let's not get too crazy here.

  • @eta0carinae
    @eta0carinae 5 років тому +152

    it was proven that chaitin's constants are normal in 1994

    • @cj719521
      @cj719521 5 років тому +46

      Citation needed

    • @BattousaiHBr
      @BattousaiHBr 5 років тому +7

      @@cj719521 wikipedia 4Head

    • @gabrielfrey3004
      @gabrielfrey3004 5 років тому +9

      Yes. Chaitin’s constant is normal
      Even if it was not normal, it would probably be possible to create a non computable normal number based on the Chaitin’s constant and the Champernowne constant, for example by alternating set of bits from these two numbers

    • @gabrielfrey3004
      @gabrielfrey3004 5 років тому +3

      Yevhenii Diomidov Yes, I was thinking of using the Champerowne constant construction and just adding some digits from a non computable number (or some of the non computable rules used to define a non computable number)

    • @steffahn
      @steffahn 5 років тому +2

      To add on to the "this is the only properly empty section" claim at 11:56, for which of course your comment already says it's false, we additionally have - at least according to Wikipedia (article on "normal number"s) - that "there [...] exists no algebraic number that has been proven to be normal in any base". So if Wikipedia is correct there, that's a different "properly empty section" in the sense of the video.

  • @samsulh314
    @samsulh314 5 років тому +40

    Numberphile: "ALL The Numbers!"
    Me: *heavy breathing* (Gets un-countably infinitely excited)

  • @DmitryPetrov
    @DmitryPetrov 4 роки тому +89

    "Chaitin's constant" is non-computable, and is proven to be algorithmically random (see: Downey, Rodney G., Hirschfeldt, Denis R., Algorithmic Randomness and Complexity), thus it is normal.
    So, strictly speaking, we know quite a few non-computable normal numbers - that is, Chaitin's constants Omega(F) for prefix-free universal computable functions F.

  • @bonecanoe86
    @bonecanoe86 5 років тому +216

    When I was 5 years old I started writing numbers on a paper. (1 2 3 4 etc). When I got done with one paper I'd tape another piece of paper to the bottom and continue. Eventually I had a 20 foot long roll of paper that all the way up to 1200. I then made a few other, shorter rolls. They somehow morphed into a character called "The Numbers" and his friends, and I used to write stories about them including a time where they had to escape vicious evil pianos. Fun times.

    • @tonio103683
      @tonio103683 5 років тому +3

      Reminds me of Philemon. Cool story.

    • @bb2fiddler
      @bb2fiddler 5 років тому +7

      I want to read some, link plz

    • @PhilBagels
      @PhilBagels 5 років тому +2

      When I was about 10 or 11, I wrote out a Pascal's Triangle, and taped additional pieces of paper to the bottom of it so I could keep adding more rows. It never got to 20 feet long, but it was probably over 4 feet long.

    • @jamesWilliams-py5zy
      @jamesWilliams-py5zy 5 років тому +2

      R/thathappened

    • @iqbaltrojan
      @iqbaltrojan 5 років тому +4

      awesome!

  • @_rlb
    @_rlb 5 років тому +274

    I like that you put 22/7 which is of course Parker Pi :)

    • @massimookissed1023
      @massimookissed1023 5 років тому +12

      355/113

    • @martinepstein9826
      @martinepstein9826 5 років тому +5

      333/106 is Parker 355/113

    • @dlevi67
      @dlevi67 5 років тому +3

      @@martinepstein9826 Spoken like a true numberphile.

    • @Tfin
      @Tfin 5 років тому +1

      22/7 has been pi longer than he's been alive.
      It was what we used in school before they taught us decimals.

    • @dlevi67
      @dlevi67 5 років тому +1

      @@Tfin Unusual curriculum where they teach pupils fractions and long division before decimals...

  • @flummoxedpanda
    @flummoxedpanda 4 роки тому +143

    "countable infinity land"
    I prefer the observable universe of numbers 😂

    • @RandomAmbles
      @RandomAmbles 3 роки тому +4

      Weeeeeeell... quantum mechanics currently suggests that there are continuous properties in the actual universe, which is sick, just absolutely sick.
      Like rotational, translational and Lorentz symmetry are all supposed to be continuousish.
      I'm skeptical of this, frankly, but I need to be open to the possibility that the universe is not fundamentally discrete.
      Apparently Buckminster Fuller was considering how to construct systems of physics with discrete properties, but he's pretty much unreadable.
      It's an open question.

    • @bardofhighrenown
      @bardofhighrenown 3 роки тому +1

      Hard disagree "Countable infinity-land" is the superior term.

    • @ModernandVintageWatches
      @ModernandVintageWatches 2 роки тому

      I am amazed with all the science channels, the breakthrough in physics, medicine etc are almost daily basis now

  • @leesweets4110
    @leesweets4110 3 роки тому +48

    There is actually a larger circle around the computable numbers called the set of definable numbers. Definable numbers contain all computables and is also countably infinite. The Chaitin constant is a definable non-computable.

    • @Liggliluff
      @Liggliluff 2 роки тому +7

      Can you give an example of a non-definable number? ;)

    • @sabouedcleek611
      @sabouedcleek611 2 роки тому +8

      @@Liggliluff Wait a minute...

    • @SG2048-meta
      @SG2048-meta 2 роки тому +7

      @@Liggliluff uh, the chance that the number of- oh I just defined that number, uh, the number of ways you can ea- ah just defined that as well aaaah

    • @nzqarc
      @nzqarc 2 роки тому +2

      @@Liggliluff literally point at anywhere on a ruler, the chances of the specific point being undefinable are almost 100% (unless you point at an integer)

    • @tobiaswilhelmi4819
      @tobiaswilhelmi4819 2 роки тому +2

      ​@@nzqarc Are you actually talking about the number defined as "I'm pointing at it right now"?

  • @mikey5396
    @mikey5396 Рік тому +4

    I love the little details here. Like how the drawn circles are slightly larger in the upper left area and more compressed in the lower right and the animation matches it. Also can we talk about how the camera man has continuously gotten smarter as these videos go on. His questions keep getting more and more clever.

  • @carpyet9507
    @carpyet9507 Рік тому +10

    This is just one of those videos you have to watch every year.

    • @spinecho609
      @spinecho609 2 місяці тому

      Yep, back again. I find it Lovecraftian

  • @folksyoxytocin
    @folksyoxytocin 5 років тому +50

    God, Matt Parker is truly the best.

    • @GrandMoffTarkinsTeaDispenser
      @GrandMoffTarkinsTeaDispenser 5 років тому

      He is isn't he? Man is full of joy and brightens my day to see this video, thank you Matt.

    • @joryjones6808
      @joryjones6808 5 років тому +1

      Aidan Worthington nice Feynman pic but mine’s better.

    • @folksyoxytocin
      @folksyoxytocin 5 років тому

      @@joryjones6808 Thanks bby. But mine is the best

    • @henryordish
      @henryordish 5 років тому

      u missed a comma after "Parker"

  • @jacob.gamble
    @jacob.gamble 4 роки тому +81

    Ah yes the normal numbers. Their only weakness is against fighting type numbers.

  • @EAS__
    @EAS__ 10 місяців тому +3

    This is my favorite numberphile video. Keep coming back to this.

    • @Yora21
      @Yora21 16 днів тому

      This video specifically was the direct inspiration for the mysticism in a Space Opera I am working on.
      This idea does not just apply to numbers, but to all possible concepts. Since the human brain, as a physical object, has limited computing capacity, there are many concept and complex phenomenons that we could never comprehend or even imagine. And that's most likely the vast majority of all concepts.

  • @tobiaskristianto8051
    @tobiaskristianto8051 5 років тому +29

    I love that he snickered during the -1/12 :^)

    • @kaviramyead7987
      @kaviramyead7987 5 років тому +3

      If you don't get it google zeta function regularization.

  • @fzndn-xvii
    @fzndn-xvii 5 років тому +214

    Can we get Algebraic Parker Number?

    • @FawwazSyarif
      @FawwazSyarif 5 років тому +1

      I can't believe I met you here!

    • @cristianstancu6700
      @cristianstancu6700 5 років тому +7

      Almost but not quite

    • @darealpoopster
      @darealpoopster 5 років тому +1

      Fauzan D. Rywannis Probabilistically it’s 0

    • @tryAGAIN87
      @tryAGAIN87 5 років тому

      I thought the Parker square was already algebraic, although not consistent with magic squares lol. Does that then mean the Parker square is a non-computable magic square?

  • @stupid_sleazoid2
    @stupid_sleazoid2 6 місяців тому

    I think it's the best numberphile video ever. Detailed, yet clear, easily understandable, and absolutely mind-blowing

  • @kavish8034
    @kavish8034 5 років тому +122

    "We gonna talk about ALL the numbers!!!!"
    (except the negatives)
    In other words, all the parker numbers

    • @helloofthebeach
      @helloofthebeach 4 роки тому +14

      Negative numbers have committed the unforgivable crime of being boring.

    • @adamrezabek9469
      @adamrezabek9469 3 роки тому +1

      @@helloofthebeach but without them, we have no fun with complex numbers

  • @rogerszmodis
    @rogerszmodis 4 роки тому +70

    It's amazing how we will only ever know 0% of all numbers no matter how hard we try.

    • @nathantempest9175
      @nathantempest9175 4 роки тому +5

      not exactly ) but an infinetely close number to it

    • @DemoniteBL
      @DemoniteBL 3 роки тому +18

      infinitesimal% of the numbers

    • @Cowtymsmiesznego
      @Cowtymsmiesznego 3 роки тому +10

      @@nathantempest9175 The only real number "infinitely close" to 0 is 0.

    • @Elrog3
      @Elrog3 2 роки тому +2

      @@Cowtymsmiesznego Maybe he uses hyperreals.

    • @charliedegiulio9951
      @charliedegiulio9951 Рік тому

      We have discovered infitecimal% of them

  • @jean-paulsartre602
    @jean-paulsartre602 5 місяців тому

    Thanks!

  • @MateusSFigueiredo
    @MateusSFigueiredo 5 років тому +105

    12:13 "this is completely empty" as in "we don't know any numbers that go in here", not as in "we know that zero numbers go in here".

    • @heimdall1973
      @heimdall1973 5 років тому +6

      The animation was wrong though. As it zoomed out and the "normal" circle gets relatively larger, the line should straighten and curve the other way, making it so the normal numbers are outside the circle and the circles would then indicate bubbles that are virtually nothing but we don't know anything from outside those bubbles.

    • @factsverse9957
      @factsverse9957 4 роки тому +2

      @@heimdall1973 but it gets the point across, it's not an intended pun because it's technically wrong.

    • @Cowtymsmiesznego
      @Cowtymsmiesznego 3 роки тому

      In fact, as he explained later - almost all numbers DO go in there

  • @ArlenBrackovic
    @ArlenBrackovic 5 років тому +13

    When Parker said this is beyond me... wow :D

  • @johnnye87
    @johnnye87 3 місяці тому +2

    Before watching this I would totally have assumed 1873 was an integer. Amazing that you can go through 1,872 integers, then add 1 and get a transcendental number out of nowhere. Shows you should never assume a pattern will hold forever.

  • @JanKentaur
    @JanKentaur 5 років тому +153

    Get it, 1873, 1882 and 1934 are transcendental.

    • @dlevi67
      @dlevi67 5 років тому +1

      Also 139, 1826, 1837, 1852.

  • @wafelsen
    @wafelsen 5 років тому +14

    Perhaps I have been watching too much Great British Baking Show, but I quite liked the Pi Recipe at 6:05

  • @iceymonster4675
    @iceymonster4675 2 роки тому +2

    "up until now we have found none of the numbers" - Absolutely love that line!

  • @phscience797
    @phscience797 5 років тому +89

    On the Wikipedia entry for Chaitin’s constant it says that it is indeed normal, contradicting what Matt said. What is it then?

    • @piguyalamode164
      @piguyalamode164 5 років тому +16

      That probably means that people think its normal, but we don't know, unless it has a citation.

    • @pi314159265358978
      @pi314159265358978 5 років тому +38

      ​@@piguyalamode164 It seems that there is a proof in "Borel Normality and Algorithmic
      Randomness" by Cristian Calude, 1994.

    • @Theo0x89
      @Theo0x89 5 років тому +8

      [citation needed]

    • @pietervannes4476
      @pietervannes4476 5 років тому +5

      @@pi314159265358978 Always fun to see youtubers you know in comment sections of something completely different

    • @jeffcoleman8094
      @jeffcoleman8094 5 років тому +39

      Matt is wrong. Not only are all the Chaitin constructions normal and uncomputable, *any* algorithmically random number is normal, and there are lots of uncomputable, algorithmically random numbers we can describe via computer science / information theory. I tweeted them about it already, so maybe they'll fix the video.

  • @curtiswfranks
    @curtiswfranks 5 років тому +83

    Thank you for refuting the *assumed* normalcy of π; that ALWAYS bothers me!

  • @snoodge-cv7fj
    @snoodge-cv7fj 3 роки тому +9

    I love how the cameraman is just as clueless as everyone else, it kind of acts to give the viewers some chance to comprehend the math via him asking the questions we were all thinking.

  • @GravelLeft
    @GravelLeft 5 років тому +23

    12:35 I was curious about what the statement "Most numbers are normal" means, and initially thought it meant that normal numbers are uncountable, but non-normal numbers are countable. But according to wikipedia, both sets are uncountable; in this case, "most numbers" means something different, to do with something called Lebesgue measure.

    • @sourdoughsavant22
      @sourdoughsavant22 5 років тому +14

      Intuitively, you can think of that as if you picked a random number, the probability that it is normal is 1.
      Or, if you know about integrals, if you define a function which is 1 on the normal numbers, and 0 on the non-normal Numbers, and integrate that from 0 to 1, you get 1.

    • @henrikbrautmeier6534
      @henrikbrautmeier6534 5 років тому +1

      Tbh i thought most numberfile viewer have a mathematic background. Everyday, one can learn something new

    • @GravelLeft
      @GravelLeft 5 років тому +4

      @@sourdoughsavant22 Wow, that's weird. It's as if you start with a function which is 1 for every real number, then the integral from 0 to 1 will be one, representing the area of a 1x1 square, then when you go to the integral of the function you described, it's as if you're removing an infinitesimal sliver of area from the square for each non-normal number, which there are uncountably infinitely many of. But the area still remains 1.

    • @haniyasu8236
      @haniyasu8236 5 років тому +3

      The integral idea works, but you don't need it. Another way of thinking about it is that if you take all the real numbers from 0 to 1 and try to cover it with open intervals such that no normal number is left out, the total length of those intervals will never be less than 1.
      The key thing to note is that if you try to do this with other sets of numbers (like the rational or even algebraic numbers) , you can actually cover all of the them with open sets of any total length. For rational and algebraic numbers, this is easily provable by using the fact that they are countable. However, there are uncountable sets of numbers where you can do this as well (like the cantor set), so hence why the converse about normal numbers is significant.

    • @paoloborello2530
      @paoloborello2530 5 років тому

      @@sourdoughsavant22 I'm not sure that function can be integrated with a Riemann integral

  • @wtmftproductions
    @wtmftproductions 5 років тому +158

    If Pi turned out to be "Normal" then would you be able to find Pi within itself? Would Pi be a fractal?

    • @Kycilak
      @Kycilak 5 років тому +24

      As a layman I'd say no because π would have to be recursive.

    • @yourlordandsaviouryeesusbe2998
      @yourlordandsaviouryeesusbe2998 5 років тому +13

      @@Kycilak But how can it be recursive if the digits of π itself never repeat and are infinitely many...

    • @Kycilak
      @Kycilak 5 років тому +78

      @@yourlordandsaviouryeesusbe2998 That was my point.
      More formally I would construct proof by contradiction.
      Say whole π can be found in its fractional part after some finite number n of digits from decimal point. That means that somewhere in its fractional part it continues with the same digits with which it starts. In order to contain itself whole would mean that after another n+1 digits from decimal point it would start again this sequence and so on. That would mean that digits of π are recurring which would make π rational. We have proofs that π is not rational so we have come to contradiction. Hence our assumption must be wrong and π is not contained whole in its fractional part. QED
      I hope I have not made any mistakes. Feel free to correct me. As I said I am but a layman.

    • @heimdall1973
      @heimdall1973 5 років тому +46

      What you *can* say about pi (if it's normal) is that however big (finite) chunk of pi's digit sequence you take, it will be contained elsewhere within the sequence again and again. For example, the first billion digits will be repeated infinitely many times. So will the first quadrillion digits. Or the first Graham's number of digits... Of course, not periodically.

    • @Kycilak
      @Kycilak 5 років тому +8

      @@heimdall1973 I agree, all finite sequences would be in there somewhere.

  • @ChadTanker
    @ChadTanker 4 роки тому +1

    I like the " so its tike the least efficent way to do this"
    reaction
    -> His mimik and voice for " it is"

  • @JamesSpeiser
    @JamesSpeiser 5 років тому +159

    PLEASE DO A VIDEO ON UNCOMPUTABLE NUMBERS!!!

    • @heimdall1973
      @heimdall1973 5 років тому +17

      I might give it a go when I'm not too busy. As long as there's some interest.
      There's not loads to say about them, but there is something. Shall I give it a go?

    • @robertdarcy6210
      @robertdarcy6210 5 років тому +5

      @@heimdall1973 yes

    • @heimdall1973
      @heimdall1973 5 років тому +8

      @@robertdarcy6210 I'll have to work out how to do video editing to animate the numbers and curves as is done in this video. Mathematically I already know some things I'd like to mention and how I'd like to present it...
      So... I can record myself talking and writing. But during some of the video, I would like to keep the sound and replace the picture of me with an animation - that I don't know yet how to do. I'll check what the built-in video editing software on my laptop can do...

    • @thecakeredux
      @thecakeredux 5 років тому +10

      ThreeBlueOneBrown animates his videos using a python module he wrote and it's on github. If you're into programming, it's probably the most useful tool for that purpose.

    • @heimdall1973
      @heimdall1973 5 років тому +9

      @@thecakeredux Thanks. I'll look into it. I never tried python before but it looks simple enough.

  • @andyyyz9114
    @andyyyz9114 5 років тому +94

    For me, everything outside of the "Rational numbers" circle might as well be labelled "Here be dragons" :)

    • @Cookiefz
      @Cookiefz 5 років тому +5

      What's wrong with dedekind-completeness and algebraic closure?

    • @dlevi67
      @dlevi67 5 років тому +14

      Don't have irrational fears. It's not even complex stuff.

    • @Vietcongster
      @Vietcongster 5 років тому +4

      Beyond the computable numbers should be labeled "Here be Lovecraftian Elder Gods"

    • @dlevi67
      @dlevi67 5 років тому +3

      @@Vietcongster Appropriately surreal...

    • @trondordoesstuff
      @trondordoesstuff 5 років тому +1

      @@Vietcongster Beyond computable numbers and in normal numbers should be labeled "Here be".... I actually don't know.

  • @robpuchyr7407
    @robpuchyr7407 4 роки тому +9

    “Grease a circular tin.”
    I love it!

  • @alephnull4044
    @alephnull4044 5 років тому +71

    Actually e wasn't the first to be proved transcendental, some weird decimals were.

    • @prakashlikhitkar
      @prakashlikhitkar 5 років тому +47

      Those weird decimals are called Liouville numbers.

    • @UltraCboy
      @UltraCboy 5 років тому +43

      Like the first normal numbers, the first transcendental numbers were specifically designed to be transcendental.

    • @alephnull4044
      @alephnull4044 5 років тому +1

      @@prakashlikhitkar Yep

    • @hedger0w
      @hedger0w 5 років тому +12

      Dark numbers and weird decimals, I think I had enough internet for today. And its Monday. I might be able to watch video about infinity alone on Monday but this is too much.

    • @serraramayfield9230
      @serraramayfield9230 5 років тому +3

      The username makes this better

  • @lowercaserho
    @lowercaserho 5 років тому +47

    Wouldn't it be possible to devise a normal non-computable number by defining it in terms of a known non-computable number, something along the lines of the following? Take a chaitin constant, then put a 1 between the first and second digits, a 2 between the second and third digits, and so on? Wouldn't that have to be both normal and non-computable?

    • @GeekyNeil
      @GeekyNeil 5 років тому +14

      Yes I think so, providing you continue by inserting successive integers. So after inserting 9, you insert 10. I'm guessing that's what you mean. The digits from the Chaitin constant become increasingly rare so they don't affect the normality, but they are all there so you can compute the Chaitin constant from the number you defined. Since the Chaitin constant cannot be computed, neither can your number.

    • @LordNethesis
      @LordNethesis 5 років тому +1

      Ooh, I like that. So if that were computable you could easily adjust the program to get chaitlin. You can’t, so it isn’t. Certainly it is normal to base 10, though I don’t know if it would be normal to all bases.

    • @sykes1024
      @sykes1024 5 років тому +5

      It wouldn't necessarily be a normal number. For it to be a normal number, the average frequency of each digit must approach 1/10, the frequency of each 2 digit number must be 1/100, the frequency of each three digit number must be 1/1000 and so on. However, since we know basically nothing about any of the digits of Chaitin's Constant. It's possible it could be really lopsided and slightly skew one or more of these ratios. Note that 0.0123456789 repeating is NOT a normal number because it only has the proper frequency for each single digit but no occurrences of most 2 digit and greater numbers; no 22's no 333's, no 565's.

    • @qorilla
      @qorilla 5 років тому +4

      @@sykes1024 chaitlin digits are exceedingly rare among this number's digits, since it goes like one digit from chaitlin then the next natural number which consists of more and more digits the further you go, then a single digit from Chaitlin etc. For the purpose of computing ratios, the chaitlin digits can be ignored as they have zero effect on it in the limit of infinity.

    • @sykes1024
      @sykes1024 5 років тому +1

      @@qorilla Hmmm, I guess you're right. In the limit the proportion of Chaitin digits goes to zero.

  • @HunterJE
    @HunterJE Рік тому +2

    "Champernowne's constant is one of the few numbers we know is normal" he says, writing it outside the "normal numbers" circle (and for that matter outside the computable one, too), making this in fact a Parker diagram

  • @captaincygni2162
    @captaincygni2162 5 років тому +9

    0:40
    "Circular Thingys"
    10/10 best description

    • @sugarandbones6272
      @sugarandbones6272 4 роки тому +2

      it's so much better when you realize that particular diagram is neither a venn diagram nor euler diagram

    • @isavenewspapers8890
      @isavenewspapers8890 2 місяці тому

      @@sugarandbones6272 It looks like a valid Euler digram to me. Am I going crazy?

  • @mittfh
    @mittfh 5 років тому +29

    Let's just admire the genius of the recipe at 6:04 😁

  • @alejrandom6592
    @alejrandom6592 5 місяців тому +1

    For those wondering: it is possible for two trascendentals to add up to an algebraic. Example: (e) + (1-e)

  • @HaloInverse
    @HaloInverse 5 років тому +15

    9:28 "That's an N, it's just climbing under the A" a.k.a. _Parker spelling_

  • @Eniro20
    @Eniro20 2 роки тому +9

    Could have also added definable numbers: numbers that can be defined in a formal language (so any number you can in any way define uniquely). These numbers form a countable infinity (as all formal sentences are finite strings of a finite set of symbols), so almost all numbers are undefinable, i.e. such that you cannot even specify any one of them.

    • @gofrisuto
      @gofrisuto 11 місяців тому

      What do you mean we can't define it? Un undefined number is undefined because it doesn't have a name yet, however using set theory, all numbers can be defined.

  • @joedellinger9437
    @joedellinger9437 4 роки тому

    A more elegant construction:
    1) Pick your favorite uncomputable attribute with a binary outcome. For example, the halting problem. 0=halts, 1=runs forever.
    2) Pick your favorite programming language that is written in ASCII. Fortran, say.
    3) Write the next number group from the construction of Champernowne’s constant. Start with “1” if first time.
    4) Convert that number group to Hexadecimal. Interpret the hex number as an ASCII string. Interpret the ASCII string as a Fortran program.
    5) Is the resulting program syntactically correct? Will it compile? If no, it will crash when run, which means it halts, so 0.
    6) If it is syntactically correct, if run, does it halt? (Assume if it requires input you feed it as many zeroes as it asks for.) If yes, 0. If it runs forever, 1.
    7) Add that 0 or 1 to the output string of digits.
    8) Go back to step 3.
    This number is at least more interesting. And I know the first digits of it. 102030405060708090100110120...
    The first interleaved “1” will be the program “ 1 goto1”. Eventually there will be interleaved numbers where it is difficult to prove whether it is a 0 or 1, but they will be pretty far out there. But all possible Fortran programs and whether they halt or not for inputs of 0’s are encoded in this number. Definitely uncomputable.
    But it is also a normal number. And a little more interesting.

  • @johannesvanderhorst9778
    @johannesvanderhorst9778 4 роки тому +7

    3:13 The Liouville Constant, the sum of 10^(-n!) for n running from 1 to infinity, was already in 1851 constructed and proven to be a transcendental number.

  • @xxnotmuchxx
    @xxnotmuchxx 3 роки тому +3

    This is one of my favorite videos about math. It is so mysterious and I end up with questions. I wonder if it might be easier to check if an irrational or transcendental number is normal by changing the base of the number system. We use base 10. If we use base 2, we just have to deal with 0s and 1s.

  • @Madoc_EU
    @Madoc_EU 3 роки тому +5

    Still my most beloved Numberphile video. I've watched it so many times now, it flashes me every single time.
    Whenever I feel tempted to believe that we may have maths figured out for the most part, I watch this video. And bam, I'm back at square zero.
    Really an intellectual shower if you think about it, for getting rid of primate-brain hubris.

    • @semicolumnn
      @semicolumnn 3 роки тому +1

      I know that the fact that we have none of them is scary but they’re just arbitrary numbers in R, which means they obey theorems and rules of the real numbers, and are just limits of Cauchy sequences like 3 and -1/12

    • @ModernandVintageWatches
      @ModernandVintageWatches 2 роки тому +1

      I am amazed with all the science channels, the breakthrough in physics, medicine etc are almost daily basis now

  • @ramdamdam1402
    @ramdamdam1402 5 років тому +10

    No normal uncomputable is non empty, actually it is very easy to come up with one :
    Let n be the number defined as such : at step i write the i-th digit of the chaitin number, then write the i-th natural number you've not yet written. This number is normal (by construction) and uncomputable.

    • @ChibiRuah
      @ChibiRuah 5 років тому

      That was what I was wondering too

    • @Wargon2013
      @Wargon2013 5 років тому +3

      How is it uncomputable if you can define an algorithm to produce it?

    • @fnors2
      @fnors2 5 років тому

      "computable" informally (really informally) means that you can find an algorithm to aproximate the number to 'n' decimals then halt in finite time.
      The algorithm you gave can be used to approximate the number by stopping at 'n' decimals. So it is a computable number, since Chaitin number is computable.

    • @ramdamdam1402
      @ramdamdam1402 5 років тому +1

      @@Wargon2013 it is uncomputable because if an algorithm existed to produce this number, one could deduce an algorithm to write Chaitin constant.

    • @ramdamdam1402
      @ramdamdam1402 5 років тому +1

      @@fnors2 no because Chaitin is uncomputable. I don't give a method to write it ( because it doesn't exist), i just say write the i-th decimal of Chaitin.

  • @TheTexas1994
    @TheTexas1994 5 років тому +67

    This was a Parker Square of a video for not including the negatives

  • @hersirirminsul
    @hersirirminsul 4 роки тому +1

    One way to understand a possible uncomputable number would be to imagine any number that was close to, but not actually, a transcendental number - for example pi - where your number had all pi's digits except one (or two or any other number of digits) randomly replaced with other digits. It breaks the sequence that allows you to calculate the next digit of pi, so now you have to actually >know< all the digits of this 'almost pi' number.

    • @DStecks
      @DStecks 2 роки тому

      It's like the computable numbers are just the infinitesimal handful of dark numbers that there happens to be a shortcut to. Pi is this vast, eldritch thing, that by sheer coincidence of geometry happens to be precisely the ratio of a circle's circumference to its diameter.

  • @arkantyne7122
    @arkantyne7122 5 років тому +12

    Clarification: Chaitin's Constant is NORMAL. Yay. (And it's uncomputable, so you should really keep up to date)
    And just one more thing to say:
    29 1 47
    41 37 1
    23 41 29

  • @TheOneMaddin
    @TheOneMaddin 5 років тому +25

    "e" wasn't the first number proven to be transcendental! The first number proven to be transcendental was an "artificial one" (as Matt would call it) called "Liouville's number".

    • @angelmendez-rivera351
      @angelmendez-rivera351 5 років тому +1

      TheWinter e is the first non-artificial number to be proven to be transcendental, is what he meant, and this much is true.

  • @dinamosflams
    @dinamosflams 3 роки тому

    one thing is being a phisycist or a chemyst looking at what has everyone in your field has discovered and wonder how much is yet to be found. other thing completely diferent is to look at your field of study knowing *exactly* what everyone don't know and don't even understand.
    This is a a whole other level of a beast.

  • @nivolord
    @nivolord 5 років тому +15

    Wouldn't you be able to weave Chaitin's number with Champerowne's number? Alternating between writing out n and the n-th Chaitin digit? That would be an uncomputable normal number.
    Edit: Sorry that may not be a normal number. Maybe if you increase the occurrences of Champerowne's number at later places in the digit expansion, in order to give it infinitely more weight in the limit? So you'd wait longer and longer amounts of time until adding the next Chaitin's digit. Just an idea though.
    Edit 2: Wikipedia says that Chaitin's number is normal. Now I'm just confused.

    • @angelmendez-rivera351
      @angelmendez-rivera351 5 років тому +7

      nivolord Matt Parker is wrong. Chaitin's constant is in fact normal. This is a well-known fact in mathematical computer science.

    • @nivolord
      @nivolord 5 років тому +1

      @@angelmendez-rivera351 Ah, thank you! Seemed odd there wasn't an example of such a number.

  • @meropealcyone
    @meropealcyone Рік тому +3

    Future me: "In a few years, there will be a huge online collection of videos of every imaginable kind!"
    Me in 2000: "Wow! Tell me more!"
    Future me: "You will be able to watch them at any time, for free, on devices that fit in your pocket. And you will be addicted to one specific channel--you'll watch every video it releases!"
    Me in 2000: "I'm so excited. What will it be about?!"
    Future me: "Number theory!!!"
    Me in 2000: "I..."
    Yet here I am.

  • @1CO1519
    @1CO1519 4 роки тому

    Matt Parker managed to spoil even our understanding of numbers!
    Thank you very much.

  • @EebstertheGreat
    @EebstertheGreat 4 роки тому +4

    A simple example of an uncomputable number is the number between 0 and 1 constructed by concatenating all the numbers in a sequence produced by an uncomputable function. For example, the Busy Beaver function S could give us the sequence (S(2,n)) = 6, 21, 107, 47176870, .... So we could use this to produce the number 0.62110747176870.... Since the function S is uncomputable, this number must be uncomputable as well. We know some of the digits by effectively analyzing every small 2-symbol Turing machine by brute force and for each running it until it halts or proving it does not halt. (Actually, most of the 2-symbol 5-state machines haven't been checked, so the last few digits are uncertain.) But there is no algorithmic way to do this, because the halting problem is undecidable.

  • @kiga14
    @kiga14 5 років тому +4

    There's also the describable numbers: numbers for which there is a finite description that uniquely specifies the number. So all constructible numbers are describable. Still countable, so most numbers are not describable.

    • @thomassynths
      @thomassynths 5 років тому +2

      The set you are talking about is more commonly known as the Definable Numbers.

    • @alan2here
      @alan2here 5 років тому

      Sounds like the computable numbers.

    • @angelmendez-rivera351
      @angelmendez-rivera351 5 років тому

      Alan Tennant No, because computable numbers deal with algorithms, not definitions.

  • @charles3840
    @charles3840 Рік тому +2

    My new diss against Maths majors as a physics major will be "Statistically, you guys know none of the numbers."

    • @tookie36
      @tookie36 Рік тому

      Physicists haven’t even figured out if matter exists 😂

  • @TemplerOO7
    @TemplerOO7 5 років тому +5

    It's amazing. Basically every number is an infinite series of digits that follow no underlying rule

  • @balintnk
    @balintnk 5 років тому +6

    Not that it really matters, but his name is written as Erdős, which translates to Foresty or Woody if anyone's interested :)

  • @JMUDoc
    @JMUDoc 4 роки тому +2

    6:20 I don't know if this has already been pointed out, but the orange circle is not a countable infinity; it's only countable out to the turquoise circle.

    • @renerpho
      @renerpho 4 роки тому

      There are only countably many Turing machines, so the orange circle is a countable infinity.

  • @Deejaynerate
    @Deejaynerate Рік тому +16

    I think it would be interesting to do a video on non-computable numbers. Seems like a fascinating concept that we know examples of something so seemingly impossible

  • @DeathBringer769
    @DeathBringer769 5 років тому +57

    Matt, I love you and all your Maths knowledge, but you apparently need to go read "Borel Normality and Algorithmic Randomness" by Cristian Calude, 1994. There is a proof contained within for Chaitin's constant being normal.

  • @beamish123
    @beamish123 3 роки тому +2

    I like the choice of the rational numbers, 22/7 being an approximation for pi and -1/12 being the result of summing 1 + 2 + .. + n. Maybe 7/2 & 1/17 also have special properties, but I'm not aware of these.

  • @harmony.enforcer
    @harmony.enforcer 5 років тому +5

    So basically, there's an infinitesimally small ammount of things which make sense and we can grasp, and an uncountable f**kton of infinitely large lovecraftian horrors

  • @domramsey
    @domramsey 5 років тому +32

    ..and outside all those groups? The Parker Numbers.

    • @mathgeniuszach
      @mathgeniuszach 5 років тому

      A perfect example of how I can take a joke too literally... XD

  • @rainbowinv
    @rainbowinv 3 роки тому +2

    Just saying, watching Turing and Champernowne both mentioned in the same video is quite satisfactory

  • @lawrencedoliveiro9104
    @lawrencedoliveiro9104 3 роки тому +8

    6:20 By “most” he means “100%”. The ones inside that outermost circle make up the remaining 0%.

    • @Owen_loves_Butters
      @Owen_loves_Butters 2 роки тому +1

      But that 0% is actually not 0, but an infinitesimal.

    • @_ranko
      @_ranko 6 місяців тому

      @@Owen_loves_Butters there are no infinitesimals in the real number line

  • @DekarNL
    @DekarNL 5 років тому +35

    Chaitin's constant is a normal number according to Wikipedia as the digits are equidistributed.

    • @krejcar25
      @krejcar25 4 роки тому +1

      Confusing right? Wikipedia isn't the best and most reliable source of information but perhaps Matt might care to explain :)

  • @tomkerruish2982
    @tomkerruish2982 5 років тому +2

    I think that there should be a category intermediate between constructible and algebraic numbers, which I would term radical numbers. Explicitly, they are the numbers which can be constructed from the naturals by finitely many applications of addition, subtraction, multiplication, division, and taking nth roots (for natural n). As is well-known, the general quintic with integer coefficients cannot be solved by such numbers and so are a proper subset of the algebraic numbers. Likewise, the constructible numbers are obtained from the radical numbers by allowing only square roots (possibly iterated), yet do not contain (e.g.) the cube root of 2, and thus are a proper subset. It follows that the radical numbers are indeed intermediary between the constructible and algebraic numbers.

  • @random6434
    @random6434 4 роки тому +13

    There's also the "nameable/unnameable" reals. For some logical system (I hear the kids are all into ZFC these days), the set of all finite strings of symbols in in that system that define a unique real number is only countably infinite, thus we can only uniquely define a countable subset of the real numbers. The rest are "unnameable" numbers. This set is so weird that, by defininition, cannot ever find a specific example.

    • @NYKevin100
      @NYKevin100 2 роки тому +1

      Strictly speaking, you have to be very careful about how you reason about such things, or else you run into fun problems like Richard's paradox. Ideally, you want to characterize this in terms of model theory, but that requires a lot of rigor.

  • @Milkyway_Squid
    @Milkyway_Squid 5 років тому +15

    Can't you get a normal non computable number by inserting every number between every digit of Ω?
    So in other words:
    0. Ω1 0 Ω2 1 Ω3 2 ... etc?

    • @giladu.6551
      @giladu.6551 5 років тому

      Doesn't work. Since this Chaitin thing has infinite digits, you won't ever get to the 1 at the end of it.

    • @jacobrobbins3147
      @jacobrobbins3147 5 років тому

      @@giladu.6551 I think Milkyway Squid is saying that each Ω symbol in the number represents only 1 digit of Ω

    • @giladu.6551
      @giladu.6551 5 років тому +1

      @@jacobrobbins3147 Ahh, that makes sense. I don't know if that works :)

    • @ciherrera
      @ciherrera 5 років тому +5

      Well according to the Wikipedia entry on Chaitin's number, it is normal, so...

    • @giladu.6551
      @giladu.6551 5 років тому

      @@ciherrera That's strange. It wasn't mentioned in the video. I think I trust Matt more than Wikipedia about this, though.

  • @harktischris
    @harktischris 4 роки тому

    I'd never heard of computable/uncomputable numbers before, and the idea that not only are there uncomputable numbers but they are uncountably many is truly mind boggling.

  • @xyzpdg1313
    @xyzpdg1313 5 років тому +8

    @0:01 "We're gonna do all the numbers!"
    @1:20 "We're not gonna do ALL the numbers! That'd be crazy!"