All the Numbers - Numberphile
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- čas přidán 1. 06. 2024
- Matt Parker talks about numbers - as he often does. His book "Humble Pi" is at: bit.ly/Humble_Pi
More links & stuff in full description below ↓↓↓
The book on Amazon: amzn.to/2NKposg
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A bit of extra footage from this interview: • All the Numbers (extra...
More videos with Matt: bit.ly/Matt_Videos
Transcendental Numbers with Simpon Pampena: • Transcendental Numbers...
The Mile of Pi: • Mile of Pi - Numberphile
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Yes we went for the sometimes English spelling constructable... Constructible is more common and probably better! - Věda a technologie
"You like math? Name every number then."
-infinity to +infinity
@@hkayakh: That's only the reals.
@@CaptHayfever i is in there, if it weren't then it isn't
I wish I could upvote this twice.
@@hkayakh imagine a square with the two points
(∞, ∞i) and (-∞, -∞i)
thats all the numbers
well until we get into quaternions
It's reassuring to hear a mathematician say they read a math paper and couldn't comprehend it.
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.
@@sorenlily2280 That sounds more like the language that lawyers speak and that you encounter in terms and conditions... Give me maths any day!
I find it kind of scary...
Barrack Obama Vlogs Eh, no. Scientific papers are rigorously written. People are simply not properly educated to understand them.
@@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.
0:15 There are only three whole numbers: 11, 17, and 3435.
That's why it's an Euler diagram
Gives me chicken nuggets flashbacks
Can someone please explain
Eꜰꜰi the numbers in the diagramm are examples of and not „all“ numbers of this category = Eulers diagramm
3
-We're going to do all the numbers
-We're not going to do Complex numbers
Oh
Quaternions.... octonions.... infinite cardinals and ordinals...
Or versions such as p-adic and quote notation
Complex numbers do not exist technicaly speaking
@@bogdandamaschin9381 All numbers are made up.
@Cooper Gates technically the infinite cardinals, and ordinals aren't numbers that would be computable or normal I think
@@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.
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.
Lol, you commented on the TwoSet Video aswell.
It's kind of like that, except with no dark energy or mass or photons or space-time or transfinite ordinals.
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.
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).
@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.
"An infinite series that gives you pie." -- Isn't that the Great British Bake-off ?
yeah, I guess
it ended after they changed the host
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.
@@_Sara 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
@@fghsgh I am sorry I do not see every single CZcams comment the exact moment they are posted. :)
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!
Wow
We know!
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.
Looking for a math rap? Watch 3blue1brown's poem on e to the pi i
_bars_
"As mathematicians we're thinking we are getting somewhere, but up until now we have found none of the numbers."
I see Matt is trying to one up the other numberphile presenters by talking about *ALL THE NUMBERS*
he should put them in a magic square
Then Tony will come back with a video about *ALL THE OTHER NUMBERS*
Aceronian “ne up them? I’m trying to up them by an uncomputable amount.
All they need to do to up him again is to solve his mistakes in the Parker Square.
@standupmaths
I’m afraid your letters have gone off eating each other again, Matt 😂
"this is where numbers are, and we have none" is so funny to me
I am amazed with all the science channels, the breakthrough in physics, medicine etc are almost daily basis now
I would love them to make a sequel to this, including the imaginary, hypercomplex numbers and hyperreals and asurreals etc.
I just love everytime a different subject illustrates this saying:
"The more you know, the more you know you dont know"
you might say that is "clarity" : knowing what you don't know....
@@dasguptaarup8684 noice!
This video is the humblest way of saying "I know that I know nothing"
I know.
Dang. I just now typed in this comment you made 4 years ago. Sorry. I didn't know.
So "all the numbers", but not quite. So it's like a Parker Diagram then.
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.
He hasn't begun with the naturals either.
A Parker Circle?
Chvocht - Also no direct mention of integers. He just kind of halfway acknowledges them exist without labeling them.
I take it we're never letting Matt live this down...
You should do a video about the 100 page proof in Principia Mathematika of how 1 + 1 = 2
damn thats a hard one
i only have proof for how 1=1
@@Blox117 But not in Lie algebra groups.
@@DominiqEffect *confused*
@@Blox117 Let's see it
"countable infinity land"
I prefer the observable universe of numbers 😂
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.
Hard disagree "Countable infinity-land" is the superior term.
I am amazed with all the science channels, the breakthrough in physics, medicine etc are almost daily basis now
I love how mathematicians discovered the rarest group of numbers and decided to call them 'normal numbers'.
12:34 not really the rarest but yeah... still a strange name to choose for this kind of like obscure category
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.
they describe the normal - 1 tree 2 monkeys 6 bananas (thats the logic).
@@d3xCl34n *w h a t*
@@d3xCl34n banana monkey brain neuron activation
this video should be called 'None of the Numbers'
Or Parker All of the Numbers...
Infinitely few of the numbers?
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
"Almost all" numbers are transcendental
'Some of the Numbers'
I love how the code on the laptop animation actually does compute pi when you run it! Attention to detail!
So does the recipe.
And written in Python, making the whole thing a play-on-words. I love numberphile.
@@rocketlawnchair9352 python most likely because Matt knows and uses python to play around and research videos
5:05
by just looking to the right you'll be surprised that the 3.14... gives exactly that away
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!
but we don't lol
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.
KrossoverGod why is there a r in it
@@hyungilkoo9340 There's no r in it.
KrossoverGod yes there is there’s also an e in it
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.
"non-terminating decimal expansions with no repetitions"
That sounds like a description of the irrational numbers.
@@isavenewspapers8890 Irrationals definitely are like that but there are rationalsk like 1/3 which have an infinite decimal expansion.
@@eguineldo "with no repetitions"
@@isavenewspapers8890 Apologies, I guess I didn't read your comment very thoroughly. Then I would agree
@@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.
|*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!
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
"Ratio" came first too! :)
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
@@tcoren1 Yeah the Greek term is "alogos" for irrational or unknowable. "ir/ratio" is Latin and was the translation used by later Renaissance mathematicians
Is the "golden ratio" rational or irrational? That was the first question that came to my right after he said that
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.
:O
Random Decimals: 2.817316571046953926392639363856293619263625287483748846362515375828402010164936492638262748392
And so on.
I didn't fully understand what computable and an noncomputable numbers are. Can some one clearly explain? :/
@@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.
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.
Can you give an example of a non-definable number? ;)
@@Liggliluff Wait a minute...
@@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
@@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)
@@trifonmag4205 Are you actually talking about the number defined as "I'm pointing at it right now"?
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.
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).
I am amazed with all the science channels, the breakthrough in physics, medicine etc are almost daily basis now
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.
@@auscaliber1 But we assume axioms which we deem useful and then derive true statements using logic from them
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.
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.
Artificial numbers heh
Ceski.
@@guillaumelagueyte1019 so...
Numbers?
Literally all numbers.
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. ^_^
@@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.
We need a video on non-computable numbers! (please)
By uploading, through a computer, it would become... Computable?
Gibran A ...Mind-blown.
For example: busy beaver numbers and Rayo number
Does not fempute, does not fempute.
@@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.
Ah yes the normal numbers. Their only weakness is against fighting type numbers.
Don't forget ghost-type numbers too.
Normal numbers got nothing on steel type numbers
This is just one of those videos you have to watch every year.
How can one not love Matt Parker?
At times, he is a little bit too unprecise. But thats the price for being popular anong non-mathematicians.
@@TheOneMaddin Imprecise*
Matt Parker loves himself so much that the rest of us don't need to.
@@Triumvirate888 GOT EM 😂
Also, you okay buddy? Sounds like you think loving yourself is a bad thing.
Numberphile: "ALL The Numbers!"
Me: *heavy breathing* (Gets un-countably infinitely excited)
Ω level of excited?
Hey v sauce Michael here
"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.
if you say so🙃
sauce?
@@JGHFunRun ketchup
It's amazing how we will only ever know 0% of all numbers no matter how hard we try.
not exactly ) but an infinetely close number to it
infinitesimal% of the numbers
@@nathantempest9175 The only real number "infinitely close" to 0 is 0.
@@Cowtymsmiesznego Maybe he uses hyperreals.
We have discovered infitecimal% of them
I found hundreds of uncomputable numbers in my calculus homework
just wait when you get to differential equations, no numbers whatsoever, just uncomputable letters and variables
@@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?
Uncomputable teacher xD
@@dlevi67 I agree with you
@@saetainlatin Then wait till you get to partial differential equations
I like that you put 22/7 which is of course Parker Pi :)
355/113
333/106 is Parker 355/113
@@martinepstein9826 Spoken like a true numberphile.
22/7 has been pi longer than he's been alive.
It was what we used in school before they taught us decimals.
@@Tfin Unusual curriculum where they teach pupils fractions and long division before decimals...
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.
“Grease a circular tin.”
I love it!
I love that he snickered during the -1/12 :^)
If you don't get it google zeta function regularization.
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.
Reminds me of Philemon. Cool story.
I want to read some, link plz
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.
R/thathappened
awesome!
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
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.
it was proven that chaitin's constants are normal in 1994
Citation needed
@@cj719521 wikipedia 4Head
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
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)
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.
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!!
LOL @ lay down for a bit and digest this stuff.
This is my favorite numberphile video. Keep coming back to this.
I like the " so its tike the least efficent way to do this"
reaction
-> His mimik and voice for " it is"
9:28 "That's an N, it's just climbing under the A" a.k.a. _Parker spelling_
When Parker said this is beyond me... wow :D
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".
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.
@@heimdall1973 but it gets the point across, it's not an intended pun because it's technically wrong.
In fact, as he explained later - almost all numbers DO go in there
"up until now we have found none of the numbers" - Absolutely love that line!
Can we get Algebraic Parker Number?
I can't believe I met you here!
Almost but not quite
Fauzan D. Rywannis Probabilistically it’s 0
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?
0:40
"Circular Thingys"
10/10 best description
it's so much better when you realize that particular diagram is neither a venn diagram nor euler diagram
Matt Parker managed to spoil even our understanding of numbers!
Thank you very much.
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.
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
I am amazed with all the science channels, the breakthrough in physics, medicine etc are almost daily basis now
Perhaps I have been watching too much Great British Baking Show, but I quite liked the Pi Recipe at 6:05
Let's just admire the genius of the recipe at 6:04 😁
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.
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.
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.
"We gonna talk about ALL the numbers!!!!"
(except the negatives)
In other words, all the parker numbers
Negative numbers have committed the unforgivable crime of being boring.
@@helloofthebeach but without them, we have no fun with complex numbers
PLEASE DO A VIDEO ON UNCOMPUTABLE NUMBERS!!!
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?
@@heimdall1973 yes
@@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...
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.
@@felixmerz6229 Thanks. I'll look into it. I never tried python before but it looks simple enough.
Just saying, watching Turing and Champernowne both mentioned in the same video is quite satisfactory
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.
7:35 When even Matt doesn't understand it, how can we simple mortals understand it? But great video to show us a glimpse of it.
God, Matt Parker is truly the best.
He is isn't he? Man is full of joy and brightens my day to see this video, thank you Matt.
Aidan Worthington nice Feynman pic but mine’s better.
@@joryjones6808 Thanks bby. But mine is the best
u missed a comma after "Parker"
"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
3:18 I never knew 1873 was transcendental
For me, everything outside of the "Rational numbers" circle might as well be labelled "Here be dragons" :)
What's wrong with dedekind-completeness and algebraic closure?
Don't have irrational fears. It's not even complex stuff.
Beyond the computable numbers should be labeled "Here be Lovecraftian Elder Gods"
@@Vietcongster Appropriately surreal...
@@Vietcongster Beyond computable numbers and in normal numbers should be labeled "Here be".... I actually don't know.
9:23 That was a real Parker Square of an 'n' :D
8:40 reminds me of the library of babel
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.
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.
If Pi turned out to be "Normal" then would you be able to find Pi within itself? Would Pi be a fractal?
As a layman I'd say no because π would have to be recursive.
@@Kycilak But how can it be recursive if the digits of π itself never repeat and are infinitely many...
@@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.
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.
@@heimdall1973 I agree, all finite sequences would be in there somewhere.
Thank you for refuting the *assumed* normalcy of π; that ALWAYS bothers me!
One thing I like is that although the set of computable numbers is countably infinite, the set itself is not computable _i.e._ there does not exist a finite algorithmic procedure for generating the set of computable numbers.
I am amazed with all the science channels, the breakthrough in physics, medicine etc are almost daily basis now
Thank you. Great you add subtitles. I link to your video in my Number theory page on my site.
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.
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.
Tbh i thought most numberfile viewer have a mathematic background. Everyday, one can learn something new
@@pdawg216 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.
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.
@@pdawg216 I'm not sure that function can be integrated with a Riemann integral
On the Wikipedia entry for Chaitin’s constant it says that it is indeed normal, contradicting what Matt said. What is it then?
That probably means that people think its normal, but we don't know, unless it has a citation.
It means that like the Hitch Hiker's Guide to the Galaxy, Wikipedia is often incorrect but is the most used encyclopedia because it is cheap (free).
@@piguyalamode164 It seems that there is a proof in "Borel Normality and Algorithmic
Randomness" by Cristian Calude, 1994.
[citation needed]
@@pi314159265358978 Always fun to see youtubers you know in comment sections of something completely different
6:20 By “most” he means “100%”. The ones inside that outermost circle make up the remaining 0%.
But that 0% is actually not 0, but an infinitesimal.
@@Owen_loves_Butters there are no infinitesimals in the real number line
When Matt says "I'm read the paper.... it's beyond me" @ 7:49 i knew that topic is truly hard
Get it, 1873, 1882 and 1934 are transcendental.
Also 139, 1826, 1837, 1852.
It's amazing. Basically every number is an infinite series of digits that follow no underlying rule
Rule #1: Follow no rules
This video puts things to a whole new level
I'd like to introduce the Pitman constant... It's like Champernown's constant except, instead of starting with the beginning of the number line, my constant starts with the end and goes back.
It has the unique quality of being the only number with an infinite number of digits which we know the last uncountable number of digits for.
"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".
TheWinter e is the first non-artificial number to be proven to be transcendental, is what he meant, and this much is true.
..and outside all those groups? The Parker Numbers.
A perfect example of how I can take a joke too literally... XD
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.
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.
really appreciate the content! i have a lot of passion for math and when i’m home sick (which happens a lot because of my weak immune system) this fills the hole that my alg 2 class does
I think we can still easily make an artificial normal, uncomputable number, by defining a new number to be the interlacing of the digits of an uncomputable number with the sequence of whole numbers.
Actually e wasn't the first to be proved transcendental, some weird decimals were.
Those weird decimals are called Liouville numbers.
Like the first normal numbers, the first transcendental numbers were specifically designed to be transcendental.
@@prakashlikhitkar Yep
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.
The username makes this better
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.
nivolord Matt Parker is wrong. Chaitin's constant is in fact normal. This is a well-known fact in mathematical computer science.
@@angelmendez-rivera351 Ah, thank you! Seemed odd there wasn't an example of such a number.
A slight correction; e was not the first number proven to be transcendental. It was Liouville's number in 1851. It is 0.1100010000000000000000010... (the nth digit is 1 if n=k! where k = 1, 2, 3, ... and 0 otherwise, so there is a 1 in the 1st, 2nd, 6th, 24th, etc. digit to the right of the decimal point). But it is true that other than numbers specifically constructed to be transcendental (like Liouville's number) e was the first number to be proven transcendental.
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.
This was a Parker Square of a video for not including the negatives
They were just on the back of the page
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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.
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.
Clearly recommended on my feed to scramble my brain. Cheers....
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.
The set you are talking about is more commonly known as the Definable Numbers.
Sounds like the computable numbers.
Alan Tennant No, because computable numbers deal with algorithms, not definitions.
6:41
And you may ask yourself, "well, how did I get here?!"
I would like to point out that the HUngarian guys' name, who contributed to the Copeland-Erdős constant, was indeed Erdős, and not Erdos or Erdós, or something lik that, because those names connot exist in our language. Respectfully.
Brilliant video. Thank you
I am amazed with all the science channels, the breakthrough in physics, medicine etc are almost daily basis now
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?
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.
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.
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.
@@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.
@@qorilla Hmmm, I guess you're right. In the limit the proportion of Chaitin digits goes to zero.
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
Still one of my favourite videos on this channel! :)
I am amazed with all the science channels, the breakthrough in physics, medicine etc are almost daily basis now
Ha, love it: "Up until now, we have found _none_ of the numbers."