Approximating Irrational Numbers (Duffin-Schaeffer Conjecture) - Numberphile
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- Опубліковано 7 гру 2019
- James Maynard recently co-authored a proof of the Duffin-Schaeffer Conjecture.
More links & stuff in full description below ↓↓↓
More James Maynard on Numberphile: bit.ly/JamesMaynard
On the Duffin-Schaeffer conjecture - by Dimitris Koukoulopoulos and James Maynard - arxiv.org/abs/1907.04593
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Correction of typo: 355/113 is Pi approximation, not 133 - Наука та технологія
My favorite moment in this video is at around 9:49 when James refers to the Duffin-Schaeffer Conjecture and then qualifies that by saying "now a theorem." Bravo, James!
11:23 he does it again
Yes, I can feel how proud a person can say something like that.
"Duffin-Schaeffer conjecture, now a theorem"
guess that is what mathematicians do when they want to show off xD
@@insomnia20422 i think, he was just too humble to call it the manyard-koukoulopoulos theorem - which is what mathematicians are going to call it because theorems are traditionally named after the first person(s) who proved it.
"The Duffin-Schaeffer conjecture is a conjecture (now a theorem) in mathematics" is also the first line in the Wikipedia article
Congratulations to James Maynard for being awarded with Fields Medal in 2022 such a brilliant mathematician
What a wonderful view into the mind of a mathematical genius. Glorious.
This man is a mathematical machine.
i always wondered about how ppl like Leibnitz could make mathematics so accible to others
I'm just nitpicking, but at 4:10, pi is approximately equal to 355/113
Pah. He might have a flair for maths, but can he eat a 32oz steak in one sitting? I think not.
@@boudicawasnotreallyallthat1020 We know not of the eating habits of one such as himself.
@@ethanbottomley-mason8447actually, we can approximate it using the meat function. This chap has a hard limit of an 8oz sirloin.
We need more James Maynard. The guy's a genius and nice to listen to on top of that.
Aleph Null is your first name possibly Seth? Sorry for the strange question.
Need to turn off the image stabilisation on the camera when it's zoomed in close though - the way the camera compensated for his head movements is seasickness inducing.
Best handwriting on numberphile
I'm not convinced. 355/113 is approximately 3.14159292035 which gives an error of less than 1/q^3.2 for pi (q=113).
@@jessstuart7495 Dirichlet's approximation theorem (4:50) says that the difference should be less than 1/q^2. 355/113 exemplifies this since the difference between 355/113 and pi is less than 1/113^2.
Just wonderful - letting Maynard speak 20 minutes about his recent work. He is quite adept at explaining his work in simple terms!
Love the way he's bouncing with enthusiasm.
@@Pratanjali64 Are you in isolation?
Numberphile is my inspiration of my dream being mathematician...
Math and physics is my life.
Same bruv
Same
Don't forget about sixty symbols
S
Even a genius like James has the impostor syndrome of "oh that doesn't count, because I'm cheating." No James, you aren't cheating, you are winning! Well done.
The opposite of dunning kreuger
@@MrMebigfatguy The opposite of Charlie Sheen was my thought.
It's actually more common on smart people.
@@MrMebigfatguy Well, yes, but actually no.
Dave Brosius
It’s still part of the Dunning-Kruger effect.
He just won a fields medal for this amazing work. Amazing.
You can almost imagine his desk chair being a large bouncy ball with how he bobs up and down when he's excited. It's adorable.
Isn't it great that this genuine genius with a mind the size of a planet, is also a really nice guy with no ego?
Genuinely humbling
Scientists and Politicians/Pop Culture Stars are really inversely correlated.
He was literally talking about being scared of making a fool out of himself... can't have that without an ego (like every single person) :p
@@jamirimaj6880 mathematician please
I have spoken to some mathematicians and physicists who I would consider to be extremely smart and they are almost always very nice and humble with no ego at all, you don't need ego when you can actually back it up.
He's so pleasant to listen to and so happy in his work.
I really like the way he explains things.
He makes sure you are thinking with him and assures us of that!
It’s grandtastic!!
I love his handwriting so much
he is left handed
@@v3le So am I. Yet my handwriting looks like it was written by satan himself.
@@ReconFX Mine too!
I'd love to see more in-depth interviews like this one! Great video and very thought provoking.
The fact that such definite structures exist in such abstract and general cases absolutely blows my mind!
I hope you’ve heard the podcast interviews over on Numberphile2, including the one with James Maynard.
@@numberphile I'll check it out!
I love how he bounces around all happy when he's explaining things
Cracking stuff... and in a much smaller way when I'm writing software and I know the result isn't right, I find myself thinking, in the shower or on a walk to the supermarket, "Ah! I should try this or that." The other day, I woke up suddenly at 5am with the solution to a problem (which on hindsight was obvious) - but did write a note - and fell into a lovely deep sleep for a couple of hours after.
Awake you are thinking logically and asleep intuitively. Kind of like the difference between CPU and GPU computation. Logic runs a single process and intuition compares many processes for efficiency.
I do my best thinking while I'm pissing. I solve 90% of my difficult problems staring down at a toilet bowl.
so did you remember the solution in the end? cause I don't remember the dreams very well..
haha, I don't dream at all... I get to a solution after playing a game. So when I'm stuck I'll play COD or Apex or Minecraft and it'll just come to me.
When almost all the books on his bookshelves are Springer-Verlag graduate texts you know you're dealing with someone pretty hardcore.
Nah.
Almost all, or 90%? ;)
In this case, both almost all and almost none because there is a finite amount.
Why almost all then? If you measure it with Lebesgue, then his shelf is almost empty. And with a counting measure, it would not be almost all except if it is indeed all
Or pretty rich
Every engineer knows pi + e = 6
every engineer knows 6=10
but did you know that pi^2 = g ?
+Wecoc1 π = 3 = e
Wecoc1 Every baker knows pi + e = pie
@@cosmicjenny4508 This is a really nice elegant proof. All the other proofs I've seen use sophisticated tools from analytic number theory.
Now if only we could approximate irrational comments.
Replace the last word with three...
Well played.
Perfection.
@andy low Yes, but they would be "rational".
Easy: Select one of "No YOU", "Your mom," or "ur Hitler" and replace entire comment with selection.
OK so he is now a part of the main Numberphile guests, right?
James is a brilliant describer of maths, Will
Such a likable fellow! And Brady- you did a fantastic job interviewing him. You really made him shine.
Possibly the best handwriting of any mathematician I've ever seen!
This guy is so passionate about maths, a joy to watch! Thank you & Bravo for the proof!
at 3:44 the visuals are misleading as π < 22/7 but on the number-line, it is other way around
Spotted that. Got slightly triggered. Checked comments. You made me happy.
Yup - Numberphile likes it when you point out errors, so no big deal. But nice to note it.
I didn’t notice that.
Did anyone notice at 4:10 it would be 355/113 and not 355/133 ?
@@theseeker7194 just noticed and checked in the comments if I was first 🤷♂️
James just oozes enthusiasm for his subject - so inspiring. I've only a vague idea of what he has done but still can feel the excitement.
The last 3 minutes of this interview are absolutely delightful! Wow, what a rush it must be to have such breakthroughs! Thank you for sharing: the inspiration is palpable!
Congratulation for your Fields medal James !
At least one out of e+pi and e•pi is transcendental, but we can’t even prove which one.
Nillie Do you have proof that we can’t prove it?
@@hopp2184 Do you have a proof why we need a prove to prove that we cant prove it?
Hopp
Ok, “can’t” was bad phrasing on my part. Rather, we can prove that at least one of them has to be transcendental very easily, but nobody has been able to prove that either of them is definitely (or definitely not) transcendental.
However for some reason, if someone with the answer put a gun to my head and forced me to choose between algebraic and transcendental for either one, I'd only feel mildly nervous about picking "transcendental."
@UCXvl0QTbElub-bZq_S5gMPw yeah could be both and it's likely, but we currently don't know
Great video. Have this man on as much as possible.
Love how excited and passionate he is about what he does, must be the best job in the world for him
Congratulations to Drs Koukoulopoulos and Maynard. Thank you for advancing human knowledge, and for inspiring others.
Love the James Maynard uploads!
Engineers after watching this video:,
"π=e=3 is good enough"
Play the video on mute, and listen to bumping music. James Maynard's head will bounce to the music regardless of the song.
Something something "Guile's Theme goes with everything"
Disco
15:56 just before the beat drops
Can we prove that, for an arbitrary choice of music...? 😉
The new dancing ninja .gif, finally.
Well done Brady in explaining such a complicated topic with the helpful graphics!
Superb video, I could listen to James Maynard all day, thanks for the video
Probably the most beautiful hand drawn Pi I've ever seen. Now I feel bad for my own numerical calligraphy.
This guy has the best penmanship of anyone I've ever seen on Numberphile.
Fields Medalist, James Maynard! Totally awesome! Keep doing some of the best math in the world!
It’s not everyday you come across a Field Medalist who can explain, in simple terms, their Field Medal-winning work.
This idea and the way of thinking by checking with prime numbers and investigation of geometric base values is brilliant. You can't mess that up :)
I think James is fantastic at explaining things and being open about what it's like solving problems.
This is such a wonderful area of mathematics. I find this stuff endlessly fascinating.
Very good questions, Brady - great interview
Thanks
who's here again after James Maynard has won the 2022 Fields medal for proving this conjecture?
Love this guy! An obviously great mind with great presentation and overflowing with enthusiasm
I like that you can see him stepping down his understanding of the problem to something that I can understand. Great communicator, great mind.
12:56 Note that since ε_i (shown on screen as E_i) was earlier defined to be the actual error bound, the corresponding test would actually be that Σ φ(q_i) ε_i needs to diverge, without dividing by q_i. The division by q_i is done in this article because they were also dividing by q_i in the error bound.
Yes - I worked through the maths using the given formula and ended up disproving Dirichlet's Approximation Theory [which has stood for around 150 years] - so I knew something had gone wrong! As you say, you don't need the extra divide by q_i because this is already incorporated in the given ε_i.
Oops there, guys. The flying Pi bullets -- you had 355/133 where I expect you actually intended 355/113.
Yes I've been waiting for this!!!
This is amazing, congratulations!
My favourite Brady Number is 73857, it's almost palindromic but not quite leading you to ponder on how similar 3 and 5 are (or aren't), yet circumscribing that dilemma between a comforting safety padding of 7s (the most common number people ascribe mystical significance to), all neatly orbiting around a beautiful cubic symmetric 8...
It's a Parker palindrome
1:25
But e is actually really easy to get farther! 2.7 1828 1828 45 90 45. You already know the 2.71828 part; you just need to repeat "1828" for another four digits. Then 45, then twice 45 is 90, then 45 again.
No clue what comes after that, but that much is easy.
I can feel he's very happy about it and proud. Well done. All the best.
I like the enthusiasm in his voice. Because approximation problems are very interesting.
Let’s start calling convergants to the irrational numbers “silver bullets” now!
This guy literally solved a problem i can't even understand properly! >.
Let me work backwards through brute force a really small (but still annoying if you had to figure out by hand) example, and I maybe this will help you (and the 18 other people that also liked your comment) understand better:
Let's try and find the lowest errors that would be acceptable if we wanted to approximate *pi* with the following set of (5) denominators: (q1, q2, q3, q4, q5) = (1, 2, 3, 4, 5). I'll round the maximum error to the nearest integer.
To do this I set up a quick spreadsheet to divide every number from 1 to 30 (because 30 > pi*qmax). I then minus pi from each approximation and see which is closest for each numerator, a. The following a's yield the best approximation with the (unsigned) exact error and the integer rounded error (you can copy much of this into google if you want to check it out):
a1=3, pi-3/1=0.14159... -> 1/(pi-3/1) = 7.0625.... or an maximum error of E1 = 1/7.
a2=6, pi-6/2=0.14159... -> 1/(pi-6/2) = 7.0625..., E2 = 1/7.
a3=9, pi-9/3=0.14159... -> 1/(pi-9/3) = 7.0625..., E3 = 1/7.
a4=13, pi-13/4=0.1084... -> 1/(pi-13/4) = 9.224..., E4 = 1/9.
a5=16, pi-16/5=0.0584... -> 1/(pi-16/5) = 17.121..., E5 = 1/17.
So, if you input into his 'simple formula':
(q1, q2, q3, q4, q5) = (1, 2, 3, 4, 5)
and any set of E's equal to or less than:
1/(E1, E2, E3, E4, E5) = 1/(7, 7, 7, 9, 17)
then you'll get a WORKS, and you can, as shown above, approximate the given irrational to less than some error E associated with each q.
If you wanted to associated the irrational *pi* to a higher precision than E5=1/17, say E5=1/100, the test would FAIL. You can not approximate it that well (or even 1/18th well).
Hope this helps!!
@@kindlin what is the "simple formula", can you explain please? thankyou
@@nenwah3974
That's actually my biggest question from this video. He keeps saying 'its a simple fomula' but then never shows it or talks about it, not even once. I have a feeling that it's not "simple" in the way that most of are thinking. It's probably quite complicated, I mean, listen to this guy, but relative to some of the craziness that modern math has been putting together in the last half century, it might be relatively plug and play. Once you know your q's and a's, just follow the process and out spits your answer.
Other examples of mathematics that become very complicated very fast are anything involving complex integrals, half of the basis of calculus. If the equation isn't very simple or is nonlinear (like the navier stokes equation, look up that cluster of variables if you want your brain to start frying), you'll never get exact answers and can only approximate an answer. Or, modern particle physics with so many mathematical hoops to jump through I don't even know where to begin. So, yes, it might be relatively simple, but obviously not simple enough for a quick youtube vid.
He is so humble..love this genius ❤️❤️
"I get this fear that I'm about to completely embarrass myself by putting a plus instead of a minus somewhere"
This guy knows my exact fear on a math exam
That pi is so perfect at 1:30
Really fantastic questions, Brady, at the end. Was fascinating. The only thing I wished to know more about is what does collaboration on a math problem look like.
Dang, that handwriting and that crisp fresh sharpie are really lovely
I wonder what the border between "yes" and "no" looks like in input space.
@@samgraf7496 I feel like that is the right line of thought. I imagined something akin to a hamiltonian modeling phase space. But I'm already so fuzzy on what exactly that would entail.
Mabye
Sam Graf the rigorus way to define what infentesimals are is by creating a number system where each number is an infinite sequence of rational numbers. I bet you could make that an intuitive way to look at the input space with some minor modifications. The space looks like the number line except you can zoom in at each point of the number line to find a new number line. You can repeat the same thing on the new number line to find an even more zoomed in number line.
Probably incredibly fractal.
@@MrMctastics infinitesimals*
First, within an arbitrarily small approximation
A sound proof is a thing of beauty and a joy forever.
- my high school maths teacher
This was a great video, and I'll add my congratulations to James Maynard!
One glitch that I paused to verify though, is around 4:10, the second "silver bullet".
Should of course, be 355/113 - my favorite approximation.
"First three odd (positive) integers, each duplicated once, arranged as a fraction close to 3"
4:10 That's supposed to say 355/113 not 355/133
yes
Zu Chongzhi is sad.
I was looking to see if someone else caught that.
@@jherbranson same
2:19
*Engineering intensifies*
sin x = x
that's wrong, pi is exactly equal to 3
Pi =~ 4, so lets say it equals 10 to be safe.
@@recklessroges That depends on whether Pi is in the numerator or denominator. You may want Pi = 1 to be safe.
Nice joke you just typed on your extremely cheap electronic device.
Math goals right there...
'Proven by me'!!
Love him
Damn, James been putting in that work!
Little known fact: James Maynard is also famous for providing the video capture used to create the head bob effect in first-person video games.
*THE LEGEND IS BACK*
You again!
(I also watch bprp and Dr πm)
So nice seeing him succeeding at all of those fun sounding maths problems :)
Well done! Awesome work. TY.
At 4mn appears an approximation of PI defiling as 355/133
This is wrong and instead it is 355/113 (you wrote 133 instead of 113)
+
355/133 is also an approximation of pi... it's just a terrible one :)
@@superscatboy LoL
You made this mistake as a test to check if people followed carefully enough... :)
how could he!
Zu Chongzhi would turn in his grave.
4:09 the correct approximation is 355/113, not 133. The animator didn't know the easy way to remember it : take the first 3 odd numbers and double them up, thus: 113355. Split this list in two, and put the first half underneath the second half: 355/113 = 3.1415920... (pi~3.14159265...)
Magnificently well explained.
That first pi (1:28) is a work of art.
Congratulations man you are now officially a great mathematician .
2:20 *laughs in engineer*
1:33 that reverse writing of "4"
that's unnatural
That's how a genius writes 4.
sources?
@@aka5 thats how I write 4. It is easier and looks clearly not like a 9.
@@wierdalien1 I think he meant complete reverse of the usual way of writing a 4 - as in the stroke first, then the "L" *from the bottom to the top* . Dunno if being left-handed matters, but as a right-hander I find it fiddly to write it this way.
Congrats for the article
16:43 best description of every math problem ever.
1:25 the next bit is 1828 again. Should be easy to remember.
And then follow the 3 corners of a right triangle: 45 90 45
@@zockertwins Yeah it's just uncanny tbh
@@alephnull4044 hi infinity
@@hamiltonianpathondodecahed5236 hi
@@zockertwins thank you a lot, I now know 16 digits of e.
and as i live near basel, the town leonhard euler lived, my goal now is to also know 24 digits. (euler knew 24)
3:43 but 22/7 is actually greater than п, so it should be slightly on the right
I was about to write the same, but then I found your comment.
@@espenkristoffersen4887 ...and another year later I went on a quest for a similar comment!
Hi everyone, has anybody an idea how to solve this problem?
Let a(n) be recursively defined by a(1)=0, a(2)=2, a(3)=3 and a(n)=maximum(a(d)•a(n-d)) (where 0
Awesome insight into the mind of a hard working mathematician.
At 4:09 the anmimation shows 355/133 as an approximation for pi. It should be 355/113 (which is amazingly close to the actual value of pi). Did anyone else notice?
Yes, I did notice and it jarred because it blemished an otherwise faultless video.
3:42 ...Brady got it wrong ....... 22/7 will be on the other side of "PI"
I have to say I love this guys mannerisms
Another great way to start off a week.
2:38 The graphics use the ≈ symbol but James uses ≃. Then there's also ≅? What a mess of symbols!
I've gathered through some online research, though potentially incorrectly, that the symbols roughly mean the same thing. Some have specialized uses; ≃ can mean "asymptotically equal to" mathematically, ≈ can mean "homeomorphic to" in topology, and ≅ can mean "isomorphic to" in logic/algebra. Oh what fun :D
Or in nuclear chemistry, the symbol pi means the parity of the nucleus instead of ~= 355/113 lol
Can also depend on author or publisher. If unsure look in the index of the book, there should be a brief definition. Also whether |N includes 0 or not and such.
When
CONJECTURE BECOMES THEOREM ,HAPPINESS BECOMES ECSTASY....CHEERS JAMES MAYNARD🙂
Very impressive - I can follow most of Numberphile examples and have that "Aah got it" moment but this one is way over my head. Guy is a genius.
You can choose an arbitrary number of numbers and fraction.
The reason he gives examples of this squared and Fibonacci or whatever is because you need an infinite series and those usually have functions.
If you write 1/(2^n) for n= 1,2,3,4... that is much easier than just coming up over and over again with a new number.
e is easy-ish to remember. 2.7 1828 1828 459045
The 1828 bit is duplicated, and 45-90-45 are the angles inside an isosceles right triangle
So many yellow books behind him. (That is how we know that he is an actual mathematician)
Congratulations! Need a new interview :D
Congratulations!
I like how he moves his head.
I hate it. Why does he move it *so much* while he's speaking? It makes it needlessly hard to concentrate on the video. Unless you don't look at it, and then you miss the informative graphics. Which don't wobble, so they're nicely watchable.
He could be Japanese.
Rosie Fay
It’s his way of showing enthusiasm.