Made a follow up video to this one about e (euler's number)! Check it out here: kzfaq.info/get/bejne/d6eZpZenyN62gIk.html

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!

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

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

"The answer is pi" lmao that got me.

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

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

A video on e would definitely be interesting!

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.

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

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

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

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.

7:54 Making excuses, Fermat's way.

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

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.

7:57 "I understood that reference" ;)

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

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

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”

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!

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.

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

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

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

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,

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.

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?

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

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! :)

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

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

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

Happy Pi day, Zach Star!

Very interesting and worthwhile video.

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.

Liked your little nod to Fermat there.

Little more deep into pi. Thanks brother.

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

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

This channel is awesome, keep up the good work!

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

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

All mathematicians are made but Ramanujan was born.

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.

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

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

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..."

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.

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

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"

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?

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

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

That coin flip example was so cool!

😂😂😂 your introduction I was thinking “I know this voice, why do I know this voice? I’m getting a weird satan playing video games vibe

7:53 okay!

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

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?

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.

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

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)

Please do the video for 'e'!

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.

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

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

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?

Thanks! This was a really interesting video!!

Mind blowing video!

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.

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

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

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

Nice Fermat reference @8:00.

I'm a game dev teacher, and usually, when some student ask me about 'why pi?', I usually answer that pi is not related just to circles or circular things. Pi is more likely to be related to distance, and since circles are just a perfect representation between a point and another point on an edge, over all angles, AND almost everything could be plained as a 2D (or 3D, or 4D, or nD), almost everything could be related with pi by just measuring its distances and possible positions in a circle. Most of them usually gets really surprised on how pitagoras, baskhara and pi gets (or can get) related on almost everything, spetially in game dev

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.

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

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.

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

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

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.

Illuminati replaces triangle with circle

Crazy!!! Thank you! 👍

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.

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)

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.

4:34 Today I found out KZfaqrs "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.

This is the most input at the fastest rate my brain has ever not understood.

Great video

5:07 Those nail though

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

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

this is exactly what I wonder when studying physics And about that e, I genuinely have no idea how it works

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

3:53 what about + 1/pi

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?”

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.

Great vid man

7:51 damn just slipping a quick math reference huh? I won’t explain the reference since I need to go to the toilet. I’ll explain it after.

very good, thanks and please make a video about Phi ( Golden Ratio ) and Silver Ratio.