The Most Unusual Ways Pi Shows Up In Mathematics | Can You Explain These?

[@zachstar]

Made a follow up video to this one about e (euler's number)! Check it out here: kzbin.info/www/bejne/d3LMo2esmN2Fhrc

[@varunkumar-lb3wq]

What about golden ratio?

[@thelivingglitch307]

hmmm, at the end of the day a real number is a value. Values can be shown as any combination of smaller values so long as it equates. So you can find any number inside any other number, it just so happens that pi is one of the easiest to find after investing much time into a function. Funny enough in math vocab pi is not a real number.

[@samr6408]

I’m pretty sure pi=e tho

[@MartinCHorowitz]

Big fan of euler's formula which has e, pi and the square root of -1. Has a practical purpose in analysis of AC Circuits.

[@robdeskrd]

@Zach Star π as a ratio has a very strong influence over the shape material objects have and the motion of objects as the move through dimensional space because 1) gravity is an attraction toward the center of object, 2) the conservation of angular momentum as these tend to make things round and make those things go around each other in round ways and with so many spheres and circles π got it's fingers into a LOT of pies 😁

[@DexM47]

9:38 Not so complicated to understand why pi shows up here. Coins are round.

[@Abdega]

🎼I like 'em round, and big And when I'm throwin' a gig I just can't help myself, I'm actin' like an animal Now here's my scandal

[@pepe6666]

thats it, all problems are solved. physics is finished, everyone. we can all play video games now

[@dicktracy790]

@@pepe6666 whats wrong with the penis in your avatar?

[@MrScateboy]

It’s cylindrical

[@henryphelps3590]

@@AlgyCuber how is it not the penis of slavery ?

[@noobkilla3]

7:53 "There's plenty of complex looking formulas that I could totally prove geometrically, but unfortunately I just can't fit them on the margin of the screen, so I'll leave that as an exercise for the viewer" Lmao, a new age take on a classic old quote by Fermat!

[@Josho4096]

HA! I also caught on to that joke lol. If anyone is interest Fermat's Last Theorem which was proved quite recently only, had everyone frigging going crazy that Fermat had proved it but didnt have enough space on the margin of his book or w/e. Which I believe he was just trolling everyone lol. FLT is one of the simplest, innocent looking problems, but it was hard as fck for centuries for anyone to prove. Basically, you guys know that a^2 + b^2 = c^2 is possible right? Like in Pythagorean theorem. 4 squared plus 3 squared equals 5 squared, 16+9=25. But it was proven that is impossible for that to apply to any other factor higher than 2. A^n + B^n = C^n cannot exist if n >2. Google it lol.

[@cypherx7247]

@@Josho4096 You r not completely right..

[@goldenwarrior1186]

@@Josho4096 PBS Infinite Series has a video on how Fermat might have thought he’d proven his Last Theorem

[@jjpswfc]

@@Josho4096 "recently" in the maths world maybe but not actually that recent, Sir Andrew Wiles' proof was written in 1993.

[@martinconrad9260]

9:07 "The answer is π - no, I'm just kidding...." :D

[@jcnot9712]

A video on e would definitely be interesting!

[@Roosyer]

Yep, I've always thought of e as more mysterious!

[@Mot-dh5sx]

e

[@lesnyk255]

Start drawing numbers at random from the uniform distribution (0,1) and adding them together. Keep track of the number of draws D it takes for the sum to exceed 2. When it does, start over & repeat. After many many such trials, the average value of all those D's will approach e. You can very easily write a program to verify this.

[@erikgrady2452]

I huess so

[@mollof7893]

”e”

[@smittyflufferson1299]

8:00 "I'll leave that as an exercise for the viewer" I love you buddy

[@JP-qg7bu]

"The answer is pi" lmao that got me.

[@HighestRank]

J P in which universe?

[@yosefmacgruber1920]

On a calculator that does symbolic solutions, such as my TI-89, the answer to π is simply π. What else would be it be? It is already fully simplified. π+π would be 2π. Oh, did you want a decimal approximation? Well it will do that also, if you bother to specify that that is what you actually want.

[@Roosyer]

9:07 I was like WTF?? you got me! :)

[@shen144]

I found this stupidly funny.

[@GeyzsonKristoffer]

YOU EFFIN GOT ME TOO

[@techsupport2173]

I was laughing so hard... I can't believe I fell for that haha

[@danielyuan9862]

I was like, "It's 0, right? It's completely symmetric around 0." and when he said that it was pi, I was unsure on whether I understood the question.

[@romano-britishmedli7407]

Yup. I was gotten too! ^^' Me: "It should be 0, right; because the probability for any coin flip for n->infinity = 0.5, right?" Zach Star: "The answer is π!" Me: "Wait, WUT?!? O_o" Zach Star: "Just kidding, it's 0!" Me: *takes deep relieved breath/sigh*

[@Carcharoth313]

7:54 Making excuses, Fermat's way.

[@edgeisloveedgeislife5439]

I don't even need to prove them, I already have the proof ready, but unfortunately this comment section is too small for it.

[@JacobGoodman]

3:31 The Sterling Approximation is just the first term in the aymptotic approximation of the gamma function (at integer values). It can be derived with Laplace's Method, which is absolutely one of my favorite pieces of mathematics.

[@SiddharthKulkarniN]

Pi is quite sneaky, get's where it has no business to be. Great video

[@hirvonenjoonas7281]

Stirling's approximation comes by using Laplace's method, which is basically approximating an integral as an integral of gaussian function. That is why there is a pi.

[@Roosyer]

I truly hope you continue motivated to keep making these type of videos. They are so interesting and extremely well presented! Thank you dude!

[@Sailor-Dave]

Pi shows up in the twists and turns of rivers and streams; the jagged outlines of a coastline; and in the distribution and density of plant/tree species in a forest. Plus, it's delicious!

[@StarryNightGazing]

a lot of these (esp. series and integration problems) can easily be explained by thinking that the truest domain of definition of those functions is the complex plane, which works really well with closed loop integrals which in turn almost every time give out pi when integrating around a singularity,

[@vtvtify]

The integral from negative infinity to infinity of (cos (x^2)/x^2 + 1) dx =pi / e MIND - BLOWN

[@cogitoergosum2846]

Your videos are at the same time thrilling and educative. You have shown the real use of the materials that we learn. As I have seen almost all of your videos I think that you should start a series of videos about math topics from algebra to calculus, topology and all other things and how they come up in our day to day life industries and so on. Yes you had done it in your mathematics of engineering and other few videos. So you could give this a thought. And once again thank you for making such awesome videos keep making them. Love you Majorprep

[@aaronkornacki5644]

You should definitely do a video on e, I think it can be even more interesting than pi simply because it shows up in just as many seemingly random places but isn’t as easily understood as “well it just has something to do with circles”

[@zachstar]

Yeah after going through college I thought it was so weird since (at least for me) we start off learning how e has to do with continuous compounding interest. Then all of a sudden I was using it in quantum mechanics and signal analysis.

[@atriacharya2967]

The fact that this channel has few subscribers indicates that there are far too few curious science students. Keep it up, bro. You're excellent.

[@notnilc2107]

There are plenty of students interested in science (sci show is huge). The problem is getting them interested in math, because it's not really the kind of topic you can glance/generalise over, since most of the interesting stuff requires you to think.

[@atriacharya2967]

Yes bro, not only think, but if you're following him, that proof of pi required me to write down some steps to clear my doubts.

[@inflammatorycommentswithno2407]

Math is the best subject

[@neuroxik]

9:06 "The answer is pi. Nah, I'm just kidding" made me laugh as fuck

[@umairjibran7]

why am I watching this? oh yeah tomorrow is my Computer Architecture exam so yeah that's the best time to watch videos not related to EXAM

[@miguelcontreras5782]

8:00 “So I’ll leave that as an exercise for the viewer” got me triggered

[@JhoCarGo]

7:57 "I understood that reference" ;)

[@beenaalavudheen4343]

7:59 i would sue my professor if he said that lol

[@mienaikoe]

If you're in engineering, you better have a good lawyer.

[@bob-ny6kn]

A professor threatened me, in public, with a physical beating, because I proved his method and solution wrong. Just do what your prof wants and regurgitate what they say for their self-absorbed reading pleasure. Nobody cares how school went in the real world.

[@heisenberg6912]

About fermi paradox plz. Lots of support from india . Thank you

[@xyyxyxxy1616]

9:09 - I literally just closed the tab when I heard that, and just when I heard you say you were kidding from the corner of my ear, I came back.

[@wrstark]

At 11:30 "randomly pick any two numbers" implicitly uses a uniform probability measure on N (= the set of all positive integers) to pick the numbers, but this is nonsense. A uniform probability on N cannot exist. P(m)=P(n) for all integers m

[@nexninja1479]

At 12:00 A simple interpretation jumps at the viewer when the constant is used with the equations of force, both for charges and magnets. You will notice a (4pir^2) in the denominator which implies that the effect of a charge gets diluted to the surface area of a sphere of radius r when measuring the force at a distance r. This makes Intuitive sense because we can then imagine the force travelling outwards in all directions and getting more spread out and diluted as we observe it over time . What does it mean that the "force travels outwards in all directions from a charged particle" is something I do not know , and furthermore an Interesting observation is the lack of this surface area term in the effects of gravity and it's constant.

[@rudrapratapsingh8904]

Hey! Can you give me the reference to that 4m+1 and 4m-1 thing? I am really curious why that worked.

[@pryan22]

Yeah I can't find anything on this

[@tedbo1819]

It's in Euler's Introductio, p21: eulerarchive.maa.org/docs/originals/E101capitel14.15.pdf

[@robertschlesinger1342]

Very interesting and worthwhile video.

[@justsomeguy5628]

6:33 The explanation for this actually comes back to a very obscure fact- this is how you can pratically calculate integrals. I do forget the exact way to use this in general, or what it's called. This is also why you can get e and pi from primes. The fact that the set of numbers used are prime isn't important, as any set of numbers meeting some very basic conditions will work, but primes are a good canidate, and can occasionally illuminate a deeper connection that primes have with the integral.

[@sidkemp4672]

Intriguing video that could get one learning about pi for a lifetime. Thanks! With regard to the 3-blue-1-brown generation of digits of pi, has anyone checked to see, for example, if two blocks with a ration of 1:8^n would generate pi's digits in base 8? Would the same be any other base? How would this work for binary, which displays pi in ones and zeroes?

[@aguyontheinternet8436]

yeah, they do. You can see that someone tried it in the comments on that video

[@vishalmishra3046]

Area of circle of radius Davg = 2n = (π)(Davg)^2 So, Davg = sqrt(2*n/π). π shows up here because of the circle representing the average of absolute value of the distance from the origin. n spots fall into the left semi-circle and n spots fall into the right semi-circle. Hence the total area is 2n and radius is Davg.

[@nizamieminov3648]

I am a structural engineer and I use Pi in formulas to calculate critical load for thin frame members in compression and I havent yet understood what Pi has to do with the critical load. I am talking about the Euler's critical load equation Fcr = (Pi^2 * E * I) / (k^2 / L^2)

[@jackhack1972]

Im glad there are guys like you . Pi is amazing..

[@ЧингизНабиев-э2г]

4:13 the Stirling formula we know is just a part of Stirling series(approximation is just first member of a series), but the one with (2n + 1/3)pi is the second member of a series, It’s called series, but it’s not an infinite sum, but rather infinite product

[@StuffBudDuz]

Liked your little nod to Fermat there.

[@mathevethugo9544]

11:25 I think that "picking two numbers at random" is not correct, what was meant is to pick to numbers at random in the set [1,n] the probability tends to 6/(pi)^2 when n tends to infinity (wich is not the same)

[@kishorekumarsathishkumar1562]

3:45 would you care to explain why you tried 1/3 of all numbers?

[@TheJagjr4450]

If there are higher life forms any where in the universe, they will have a constant for Pi, there is no scenario where the relation of a diam to circumference does not involve a constant representing something which converted is our number 3.14159... which allows a place to begin conversing.

[@theflaggeddragon9472]

For 6:35, the key words your looking for are Dirichlet's L-functions, Dirichlet characters, and Dirichlet density. I'll try and come up with a derivation, but be warned, it's going to require some analytic number theory, thus some complex analysis.

[@tedbo1819]

It's in Euler's Introductio, p21: eulerarchive.maa.org/docs/originals/E101capitel14.15.pdf

[@pryan22]

@@tedbo1819 Can you find anything in English?

[@tedbo1819]

@@pryan22 I couldn't find anything free. Someone called J.D. Blanton translated it for Springer, calling it 'Introduction to Analysis of the Infinite', but I doubt you'd want to buy the whole book for just this result.

[@janv.8538]

4:22 Blackpenredpen has found a brilliant way to prove this!

[@user-nj1qc7uc9c]

with the equations used to calculate pi, how do they find out how many digits is accurate? do they just have to see which digits aren't changing?

[@devanathanb8713]

Little more deep into pi. Thanks brother.

[@CCABPSacsach]

Good thing you said there were a total of 314 needles Because 3.14

[@justsomeguy5628]

When it comes to approximations, pi, e, or other constants showing up are not trivial to find, but often aren't unique values that can be put in. When ranges are of small numbers, often times pi or e will fall within the range. Especially due to the use of the natural logarithim or just logarithms in general, e will often be what is found. However, a lot of it also comes down to people messing around with equations using whatever numbers or constants, and with the fame of e and pi, that leads people to experiment more with them. The fact that they are such common outputs however shows their true beauty.

[@gonderage]

Happy Pi day, Zach Star!

[@mr.cheese5697]

9:55 this is probably coincidence, because if you plug in instead of π 3 or 3.141 it won't feel really different because it's not approximation isn't a single point, it's set of points.

[@jimi02468]

When I plug in the formula at 7:14 to a calculator software to the precision of 12000 digits of pi, it takes about 2000 times longer to calculate it with the formula than just using the pi button. What is the program doing differently when just using the built-in pi function?

[@k0lpA]

I would guess it has it in memory pre-calculated ?

[@jimi02468]

@@k0lpA Not it doesn't because it takes a while to get something like a million digits with the pi button. You can get any number of digits if you just wait long enough. But the pi-function does it much faster than typing in any of the pi formulas and using those.

[@k0lpA]

@@jimi02468 hum, then maybe the calculator uses a native function when using the pi button and is interpreting the formulas you type in on the fly.

[@complex314i]

Recipricalled squares adding up to (1/6)pi^2 is hard. However, the alternating sum of the recipricalled odds is pi/4 is easy because. That sum is the power series of arctangent at 1. And arctan(1) = pi/4.

[@jewelsonguinabo5472]

This channel impressed me.. that's why I'm pressing subscribe

[@mnada72]

I believe pi appears everywhere because solutions envolve complex plane or complex analysis and rotation is essential transformation there or also because of polar coordinates and/or using trigonometry in the solutions.

[@bosfbanksy7673]

How about in Stirlings approximation, try adding 1/Pi instead of 1/3. I haven't tried it but it might get a lot closer, or maybe even exact.

[@zachstar]

I really liked that idea but I just tried it and at least for n =10 and n=50, the 1/3 was a better approximation.

[@superdau]

I was wondering that as well since pi equals 3 most of the time anyway ;) . Astonished even more that it's not the better fit.

[@blackfalcon594]

The 1/3 doesn't come from being close to pi, but rather being the next term in a Taylor series expansion of a function that comes up in the Laplacian summation. Essentially, the 2 pi n is the most important part of approximating something, and the 1/3 is less important but the next most important, so adding it gives a closer approximation. 1/3 is the exact right thing to add there, though.

[@Kon0925]

Hey, I'm kinda torn between majoring in computer engineering and computer science. Which do you think is a better choice?

[@thewiedzmin6062]

Both are literally fucking same

[@hafizajiaziz8773]

now that you've talk about pi, and about to talk about e, you definitely should talk about Euler equation

[@reda29100]

Q: Considering: physically, we can get an approx. of π up to 3 decimal digits, but further down there we, for arguement sake, cannot be sure of how round the circle gets. And that vid showing how formulas might, at some point or term, fail our expectations. i.e. sum of (1/n)^n, made up example, would mostly be confined between values a and b except for at 7483928th term where it suddenly goes above b and then goes back for the rest of the terms. my question is: given tons of mathematical formulas featuring, say 20 decimal digits of, π, how are we sure those values retrieved are actually the values of π? In other words, a) if all digits of values found by two methods match, then how are we so sure there is not a mismatch at a later digit? b) if we find a mismatch with more than 2 methods, then which method qualifies to be the true method of calculating π, or any constant for that matter?

[@carly09et]

In the real world pi to twenty seven digits are sufficient. Plank's angle will limit the useful precision.

[@danielkoprak4243]

about the column buckling - the value of the critical force comes after many simplifications and assumptions

[@SylkaChan]

I have a paradox explanation request because it really is one and I discovered it. It should be possible to test in real life with animals. -A creature takes a month to lay an egg, and an egg takes three months to hatch -This species is either pregnant or laying an egg all year around -This experiment is to determine the rate of eggs laid and the number of hatchlings -Already hatched eggs count in the rate, if not subtract 1 egg per adding a hatchling 0: 0 eggs 0 hatchlings 1: 1 egg 0 hatchlings 2: 2 eggs 0 hatchlings 3: 3 eggs 0 hatchlings 4: 4 eggs 1 hatchling 5: 5 eggs 2 hatchlings 6: 6 eggs 3 hatchlings etc .... The rate of hatchlings should be one third the rate of eggs laid. Yet, a hatchling must be born for every three eggs laid. Because of this, the number of hatchlings will increase at the same rate as eggs due to the adding number of eggs hatching three months before that egg wad laid. The equation looked like this : Hatchlings = Eggs - 3 after I do my experiment on paper. If I rewrite the equation the other way around, then : Eggs = Hatchlings + 3 if Hatchlings >= 1.

[@sanchezzz69420]

Amazing and quite simple explanations. Math is a language on itself to say thw least.

[@victorpaesplinio2865]

I think that the weird sum made with pluses and minuses that sum up to pi has something to do with the probability of two numbers being coprimes. In fact any prime can be written in the form 4n-1 or 4n+1

[@richarddeese1991]

I LOVE IT! Pi and e are both wonderfully NATURAL numbers - they're EVERYWHERE in nature! Who wouldn't love that? tavi.

[@mendi1122]

Circle is simple manifestation of symmetry and equality, there is no preferable direction, all points are at the same distane from some point, hence, naturally, pi will appear a lot in nature.

[@cros108]

3:50 what if you make it 1/pi instead of 1/3? 1/3 seems pretty arbitrary to me, surely 1/pi would make more sense?

[@modestorosado1338]

That Pierre de Fermat reference at the 8 minute mark is just gold.

[@fascialrufaie]

So when are you gonna posts video about euler's number can't wait

[@zazugee]

the most mindblowing thing i saw, including Pi emerging out of a cartesian latice was in this 3D cellular reversible automata named SALT by Fredkin and miller in their paper "Circular Motion of Strings in Cellular Automata, and Other Surprises"

[@ashar4121]

That paper sounds fun

[@connor5890]

That coin flip example was so cool!

[@ablene_teklie]

I am going to love physics and mathematics!!! Everything is included in them!!!

[@patricksauer6101]

Your videos are really great. Keep up the good work! Mark my words, this channel will be well over one million subs on New Year's Eve! :)

[@gordonis179577]

ROFL! Was that a reference to Fermat's last theorem at 7:54? "I could prove them, but I can't fit them in the margin of the screen..."

[@denelson83]

7:56 - You're not Fermat reincarnated, are you?

[@gheffz]

"It's crazy to see just how often circles sneak their way into mathematics!"

[@Yutaro-Yoshii]

5:15 sounds like it'd be something that's related to gaussian prime 'cause you know primes that fits 4m-1 is also a gaussian prime

[@mostafa9093]

could make a video about the history of pi, how it's been calculated, why it will never end?!!

[@whalingwithishmael7751]

Mind blowing video!

[@NativeVsColonial]

After watching the whole video my brain is telling me *fuck* *this* *shit* *i'm* *out*

[@fosyay1780]

Please do the video for 'e'!

[@PeterAnthonyMartins]

I'm curious, what aproximation from 1/3 would speed up the aproximation to 1? 1/л?

[@gaeb-hd4lf]

This channel is awesome, keep up the good work!

[@e7ebr0w]

I've heard a different description of Buffons needle where the lines are equal to one needle length apart

[@FunnyMemes-dr3se]

Wow! Zach Star is a 3B1B fan?

[@PhilippeCarphin]

Nice Fermat reference @8:00.

[@heinrichkreuser2659]

Thanks! This was a really interesting video!!

[@TheTedandfred]

Take a drink every time he namedrops 3Blue1Brown, just goes to show how useful of a resource that channel is

[@nandakumarcheiro]

Pi may be connected with a convergence and operated by integration and in the absence it becomes a convergence. It also is operated as a locking system when operated as Bosons and fermions as full spin and half spin momentum possibly in lunar solar interactive nodes locked up as Rahu and Kethu share its role in Astrogenetics by initiating a time delay in absorption and delayed emissions. Sankaravelayudhan Nandakumar.

[@nandakumarcheiro]

correction: convergence following a divergence.

[@NigelMartinhome]

Circles are cyclical. So it would follow that sequences reflect pi? Strange how un-random random is? Fascinating!

[@derfunkhaus]

These are all interesting but the one where the primes / prime factors are either of the form (4m+1) or (4m-1) really blows my mind.

[@hattachainoam2066]

Bruhh when you said “the answer is pi, nah I’m just kidding” 😂😭

[@sethgartner5057]

Hattachai noam I was shook 😂

[@LångemansLya]

I literally rofled xDDD

[@bob-ny6kn]

The reason the example results seem to be "phenomena" is because you have ten fingers. Had Nature's dice come up a little different long ago, we might have a thirteen or seven "digit" counting system. Once science and math listen and work toward finding Nature's counting system(s), all mathematical "phenomena" will not be.

[@richardshane2080]

The expression pi Is a spiral That's why It has an endless amount of digits PI it's describing a spiral of a trajectory with spin The basic geometric form of mass is a trajectory with spin that is what pi is describing, the fundamental essence of matter in motion from a singularity emitted from a spiraling Galaxy the most common geometric shape in the universe there for all matter is a trajectory with spin The spin is created from the entropy of the universe, perhaps dark matter is the resistance, the entropy

[@dk-fk4xm]

4:34 Today I found out KZbinrs "do stuff in front of the camera'". Like how can they see what they're doing if the camera is blocking the view? Nope. The camera is in front of them and the video is then flipped. Neat.

[@josephcastellanos8859]

Great video

[@user-wi3yx3gy2o]

If the deviation in a random walk is the square root of pi/2n. And if there are k instances of n per period then. Then, if we denote period “p” the deviation is the square root of 2kp/pi. In other words, d^2=2kp/pi. So we know that the number of trades on the Nasdaq approaches 100,000 per second. There are also about 3300 stocks trading on the Nasdaq. We also know that the stock market is not strictly speaking a random walk, of course, but many have suggested it approximates one. But in this way if we set up a random walk model to model a stock trading on the Nasdaq with the coin flip representing 1-cent up ticks and 1-cent down ticks occurring every 1/(100,000/3300) seconds, then d after 1 7- hour trading day is represented by the random walk model model d=sqrt((2)(100,000/3300)*60^2*7)/pi. This number is 493 cents or $4.93, On average, the deviation should be $4.93 on a random stock on the Nasdaq. Not very accurate. But then since this is admittedly an approximation, we could just reduce k until we get something that is more representative like divide k by 50, for example. But this still fails unless applied to rather short timeframes like an individual trading day. So how useful is such an Idea if it only applies to an assumption of say a notional trading day with only 1/10th of the trades per day, and only in the relatively short term? I really don’t know why anyone ever thought this was a good idea, other than it once was the case that there were once much fewer trades 6 actually on the specialist system, it was more like less then 1 trade per 10-20 seconds or less.

[@josephbrandenburg4373]

Maybe if you used a smaller money value?

[@user-wi3yx3gy2o]

Joseph Brandenburg smaller than 1 cent? I think the real problem is even if you make an adjustment by dividing the whole formula by 50 to get it closer to the actual observed average deviations say on 1 day, which is about 0.5 percent on the average stock, over longer periods the standard deviation of the market is about 20%, and about 40% on individual stocks per year. The median price of a stock on the NASDAQ is about 20 percent. average deviations should therefore be fairly close to 0.40*20, or 8 annually. A random walk which gives 0.5 percent or $0.1 per day extrapolated to 250 or so trading days would be much greater than $8. It would be d=sqrt((2)(100,000/3300)/60^2*7*250)/pi/50, or $124.4.

[@jamesmeza8432]

Great job you should be my math teacher you make everything seem easy

[@benjaminbrady2385]

Can't easily explain it but Sterling's approximation is connected to the ratio of the distances between prime numbers, which for another reason that I can't explain easily - involves pi

[@rebelkassadin]

I see you do a lot of videos on probability theory, and I would like to suggest a very sofisticated and interesting topic I haven't seen any youtuber do. The LaPlace's theory of succession.

[@JulesMoyaert_photo]

Crazy!!! Thank you! 👍

[@michaeljull1175]

You should make a video on the fine arts major, I’m not in fine arts I’m in robotics engineering but I see fine arts people and can’t help but wonder “why?”

[@abdallababikir4473]

I think for the coin flip example, it's impossible to be at 0 if you flip the coin an odd number of times

[@99bits46]

All mathematicians are made but Ramanujan was born.

[@obesecow5706]

what does this have to do with the video

[@stardustreverie6880]

what does thst have to do with the video

[@a.f.nik.4210]

what does thst have to do with the video

[@jaffil4026]

What does this have to do with the video