What a wonderful view into the mind of a mathematical genius. Glorious.

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

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.

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.

Love the way he's bouncing with enthusiasm.

@@Pratanjali64 Are you in isolation?

Same bruv

Same

Don't forget about sixty symbols

S

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.

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.

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.

I hope you’ve heard the podcast interviews over on Numberphile2, including the one with James Maynard.

@@numberphile I'll check it out!

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.

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

he is left handed

@@v3le So am I. Yet my handwriting looks like it was written by satan himself.

@@ReconFX Mine too!

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.

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

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 🤷‍♂️

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.

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.

It's a Parker palindrome

@@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*

Thanks

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.

+

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.

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)

BTW, 3.141593-3.140625 = 0.000968 under (for 3-9/64), while 3.142857-3.141593 = 0.001264 (22/7) over, so for these lower ranges, the shop value is more accurate than 22/7.

@@johncanfield1177 355/113.

yes

Zu Chongzhi is sad.

I was looking to see if someone else caught that.

@@jherbranson same

Magnificently well explained.

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.

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.

tromoff the next 6 digits are easy to memorise too. Think of a triangle with the angles 45, 90, 45. The next 6 digits are 459045

Okay, now just gimme a way to memorise facts about Andrew Jackson and I'm sorted lol

e ~= 49171 / 18089 ~= (1731 - 384 * pi) / 193 Heck, the simple fraction's still nearer lol

Yes, I did notice and it jarred because it blemished an otherwise faultless video.

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!

You again! (I also watch bprp and Dr πm)

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.

No need to read them. I show them more to explain WHY we can’t do the whole proof. ;)

Numberphile hi numberphile