The Most Unusual Ways Pi Shows Up In Mathematics | Can You Explain These?
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- Опубликовано: 3 окт 2024
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3Blue1Brown Collision Video: • The most unexpected an...
List of Equations Involving Pi: en.wikipedia.o...
Buffon's Needle: • Surprising π from prob...
Stirling's Formula: terrytao.wordp...
Stirling's Formula Modification (Gosper's Approximation): math.stackexch...
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Made a follow up video to this one about e (euler's number)!
Check it out here: ruclips.net/video/AAir4vcxRPU/видео.html
What about golden ratio?
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.
I’m pretty sure pi=e tho
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.
@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 😁
9:07 "The answer is π - no, I'm just kidding...." :D
9:38 Not so complicated to understand why pi shows up here. Coins are round.
🎼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
thats it, all problems are solved. physics is finished, everyone. we can all play video games now
@@pepe6666 whats wrong with the penis in your avatar?
It’s cylindrical
@@AlgyCuber how is it not the penis of slavery ?
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!
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.
@@Josho4096 You r not completely right..
@@Josho4096 PBS Infinite Series has a video on how Fermat might have thought he’d proven his Last Theorem
@@Josho4096 "recently" in the maths world maybe but not actually that recent, Sir Andrew Wiles' proof was written in 1993.
9:07 I was like WTF?? you got me! :)
I found this stupidly funny.
YOU EFFIN GOT ME TOO
I was laughing so hard... I can't believe I fell for that haha
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.
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*
"The answer is pi" lmao that got me.
J P in which universe?
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.
8:00 "I'll leave that as an exercise for the viewer"
I love you buddy
A video on e would definitely be interesting!
Yep, I've always thought of e as more mysterious!
e
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.
I huess so
”e”
9:06 "The answer is pi. Nah, I'm just kidding" made me laugh as fuck
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!
7:54 Making excuses, Fermat's way.
I don't even need to prove them, I already have the proof ready, but unfortunately this comment section is too small for it.
Pi is quite sneaky, get's where it has no business to be. Great video
7:59 i would sue my professor if he said that lol
If you're in engineering, you better have a good lawyer.
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.
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.
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.
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.
Math is the best subject
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
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.
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.
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”
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.
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,
About fermi paradox plz. Lots of support from india . Thank you
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
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?
yeah, they do. You can see that someone tried it in the comments on that video
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.
The integral from negative infinity to infinity of (cos (x^2)/x^2 + 1) dx =pi / e
MIND - BLOWN
4:22 Blackpenredpen has found a brilliant way to prove this!
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..."
Hey! Can you give me the reference to that 4m+1 and 4m-1 thing? I am really curious why that worked.
Yeah I can't find anything on this
It's in Euler's Introductio, p21: eulerarchive.maa.org/docs/originals/E101capitel14.15.pdf
Bruhh when you said “the answer is pi, nah I’m just kidding” 😂😭
Hattachai noam I was shook 😂
I literally rofled xDDD
7:56 - You're not Fermat reincarnated, are you?
Liked your little nod to Fermat there.
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
Very interesting and worthwhile video.
Happy Pi day, Zach Star!
😂😂😂 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
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! :)
Im glad there are guys like you . Pi is amazing..
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.
Little more deep into pi. Thanks brother.
This channel is awesome, keep up the good work!
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.
3:45 would you care to explain why you tried 1/3 of all numbers?
Good thing you said there were a total of 314 needles
Because 3.14
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
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.
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.
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.
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.
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?
now that you've talk about pi, and about to talk about e, you definitely should talk about Euler equation
I LOVE IT! Pi and e are both wonderfully NATURAL numbers - they're EVERYWHERE in nature! Who wouldn't love that? tavi.
"It's crazy to see just how often circles sneak their way into mathematics!"
7:53 okay!
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.
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"
That paper sounds fun
Take a drink every time he namedrops 3Blue1Brown, just goes to show how useful of a resource that channel is
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.
That Pierre de Fermat reference at the 8 minute mark is just gold.
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.
So when are you gonna posts video about euler's number can't wait
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
Thanks! This was a really interesting video!!
I've heard a different description of Buffons needle where the lines are equal to one needle length apart
Mind blowing video!
Crazy!!! Thank you! 👍
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.
about the column buckling - the value of the critical force comes after many simplifications and assumptions
Nice Fermat reference @8:00.
That coin flip example was so cool!
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?
This channel impressed me.. that's why I'm pressing subscribe
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.
Please do the video for 'e'!
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.
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.
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 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)
Hey, I'm kinda torn between majoring in computer engineering and computer science. Which do you think is a better choice?
Both are literally fucking same
This is the most input at the fastest rate my brain has ever not understood.
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?
I would guess it has it in memory pre-calculated ?
@@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.
@@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.
All mathematicians are made but Ramanujan was born.
what does this have to do with the video
what does thst have to do with the video
what does thst have to do with the video
What does this have to do with the video
could make a video about the history of pi, how it's been calculated, why it will never end?!!
You never expect the Spienish Inquisition.
4:34 Today I found out RUclipsrs "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.
Amazing and quite simple explanations. Math is a language on itself to say thw least.
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 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
i clicked this thinking it was a skit and now i learned something wtf
5:07 Those nail though
I am going to love physics and mathematics!!! Everything is included in them!!!
Circles are cyclical. So it would follow that sequences reflect pi?
Strange how un-random random is? Fascinating!
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.
Great content , could you please do same video on e and phi (golden ratio)
Abdul Aziz dude. It’s “phi”
Thank you @@samk6042 .
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?”
You know, I've just got through the whole video, and I haven't understood a single thing besides "Pi can be everywhere."
Great vid man
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.
It's in Euler's Introductio, p21: eulerarchive.maa.org/docs/originals/E101capitel14.15.pdf
@@tedbo1819 Can you find anything in English?
@@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.
I bet pi shows up everywhere because existence consists of symmetric cycles in cycles in cycles.... infinitely. I could really use a team of mathematicians to help determine this. ^^
Great video
Illuminati replaces triangle with circle