How is pi calculated to trillions of digits?
ฝัง
- เผยแพร่เมื่อ 1 มิ.ย. 2024
- You know 3.14159265... but how do we calculate these digits in the first place? Measure it? Well, definitely not up to trillions of digits of accuracy, and Archimedes' method is insanely slow for trillions of digits to be calculated. Here is a simple explanation of Machin-like algorithms, which only requires high school knowledge of complex numbers.
*REFERENCES AND FURTHER READINGS*:
1. A comprehensive history of pi and its computation: www.cecm.sfu.ca/~jborwein/pi-s...
2a. A brief history of pi (up to 1996): www.davidhbailey.com//dhbpape...
2b. A more layman version of the above: en.wikipedia.org/wiki/Chronol...
3. Machin-like formula: en.wikipedia.org/wiki/Machin-...
4a. BBP algorithm easy derivation paper: www.jstor.org/stable/3620961?...
4b. The above adapted into an easy calculus exercise: drive.google.com/open?id=1yuH...
5a. Official Google cloud article on breaking record of pi calculation: cloud.google.com/blog/product...
5b. A personal blog for the same event (with a bit more technical details): www.numberworld.org/blogs/2019...
6. Proof of Chudnovsky formula: arxiv.org/pdf/1809.00533.pdf (if you are really into this monstrous formula)
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I put a lot more effort into making this video, particularly on the research part, as you can tell when you read the description, and you should read links in the description if you are interested - you will learn a lot more:)
pi = 4
Put some effort into vocal warmups, vocal training, accent reduction and enunciation as well. This is a video, not a blog post.
On Jan 29th, three weeks after this video was posted, the record for π was broken. It is now 50 trillion digits. Congrats on 1k subscribers! *edit* I subscribed too.
It is now 62.8T
Now 100 trilion
Now infinite😂
Jan 29 is my birthday!
-Take the first three odd integers: 1,3,5
-Double them thusly: 113355
-Divide the last three by the first three thusly: 355/113
There ya go, Pi accurate to 6 decimal places!
Explain
@@tabathataylor2009 I suspect it's just a happy coincidence
@@Devesteter252101 happy little accident
@@ThorOdinson1269 :)
Kinda fun but it's probably a lot easier to just memorize the first 6 decimal places than memorizing this algorithm
This was simple, short and informative. And incredibly helpful.
Thanks!
For what? Im my college days, 3.142 was adequate
for all our needs.
I am looking for a video on pi for an elementary algebra student. This is too complex for him right now, but I found it fascinating. Beautifully done!
Thanks for the appreciation! Indeed, I made this video with the target audience of having roughly high-school math level, so it might be too complex for an elementary algebra student :)
You deserve way more attention! Great Video!
Fun, simple and informative!
This is concise and on-point. Good job
Thank you so much!
You are the one that i want to be my mathematics teacher. Most interesting vedios on youtube is only your one's. Plz always upload vedios similar to this..I gain much knowledge than my traditional
Mathematic books.
Fun fact that I just learned from Vsauce: if we ever calculate the last digit of pi, we can come to the conclusion that we live in a simulation.
@@micahmeneyerji well no, calculating the last digit of pi would require proving that pi is rational, from which point we could conclude the zeros go on forever - but this is not possible since pi is irrational, so I'm not sure what either of you are saying
In which video does he say that? Can you please share?
The last digit of pi is 0.
@@NullGodGaming There is no last digit for pi. Don't blabber without knowing.
Great videos bro...I hope your subscribers increase to the number of decimals in π(pi)!😃
Sure
Interesting and deep explanation. Thanks a million. Then maybe you will understand. Please consider it. We have some tricky formulas to use to calculate the value of Pi. In my opinion, the value of Pi can be calculated in a much easier way. We have the formula Pi = Phi^2 + RC . Phi is the golden number, RC (royal egyptian cubit) is Pi/6 so we have Pi = Phi^2 + Pi/6 and hence Pi - Pi/6 = Phi^2 and then 5*Pi = 6*(Phi^2) so Pi = 6*( Phi^2)/5 We know that we can get a good approximation of Phi by dividing the largest possible values of the Fibonacci series. The values of the sequence can be calculated using the formula: f(n)=f(n-2)+f(n-1) so let's take this notation let's substitute it to calculate Phi, so Phi = f(n) / f(n-1 ). I substitute these values into the formula and we have Pi = 6 * ( f(n) / f(n-1) )^2) / 5 So this is another and rather new formula for calculating the value of Pi. Do you understand? There was no such formula before to calculate the value of Pi. Maybe you should nominate me for the Abel Prize? 😁 Maybe aliens will come to shake my hand after all... 😜
This may be a silly question but I am truly trying to understand. How do we know that these equations work when we don't know the exact numbers pi consist of? Is it declared true based off of the validity of the properties of the equation rather than the end number?
to be more precise: we don't need the decimal representation of a number to study it. we need only a definition. if we define pi to be half the radius of the unit circle, and use definitions of trig functions to study how they are compatible with pi (cf. this video), we can find formulas for pi in terms of trig functions whose validity follow by definition.
Mathemaniac answered this in another comment: th-cam.com/video/xowA7Z5aee0/w-d-xo.html&lc=UgxteG5ca6zOgaQLhl54AaABAg.9JB6SiCr9z_9JBuUgLuUW7
I have a question....
How do they verify that each digit of the trillion digits are correct?
It is checked against another computer algorithm which can compute a specific digit (not in base 10 though) without having to compute the previous digit at all, called the BBP formula, linked in the description (4a / 4b), and since the two algorithms agree, it gives high confidence that the digits are indeed correct.
whenever my little sister walks into my room and wants to watch youtube with me i put this on and i kid you not she leaves after 4 seconds
you are a great youtuber ı was thinking making a video like this but the differnce is ı write program for that and was gonna intorduce it
it can calculate 30 decimal places and it can kept going but when time passes program begun to beign really slow
it uses nilkantha method and write in c++ ı didnt make the vid yet but i will soon.
the fourth special product u wrote is wrong. It should be (5,1)^4 * (2,2)^(-1) = (239,1)
So they spend millions on computers that can calculate Pi
to trillions of decimal places. For what?
Please send me some more link to explore Pie
machin method seems so much simpler... why is it that the chudnovsky method is typially used? it seems like it'd take much longer to compute..
Intuitively that's true, but the main thing is that while the arctangent series seems to be simpler, it is relatively slower when it comes to the convergence of the series - essentially there is also a "trade-off": computational cost of each term in the series vs how fast the series converges. Chudnovsky algorithm seems to have higher computational cost of each term, but the convergence is VERY rapid.
But how did they implement the algorithm with programming language like C, with limited precision?
The usual trick would be to multiply by a power of 10, then chop off the leading digits before carrying on with the implementation.
See more here:
latkin.org/blog/2012/03/20/how-to-calculate-1-million-digits-of-pi/
rosettacode.org/wiki/Pi
They use the precision they need, 50 trillion digits if that's what they need. It's probably quite slow.
C doesn't have limited precision, the normal numeric types of C do. You would use an arbitrary precision library, which does math similar to how humans do. Much slower but more accurate
Where can I watch this "year's worth of videos" ?
I always thought I knew something about math until I watched this
What is the 1000000 digit value of pi or how to find it
You might need to really compute all the million of digits, and find the million-th digit of pi. And this video shows the general principle of how you can do it.
IF you are don't mind switching to hexadecimal, you can actually extract the millionth digit without having to calculate the previous digits first - see the BBP algorithm as mentioned in the last part of the video.
Ok, so basically I'm very smol
What about Brent-Salamin?
It was not an explanation
they draw big circle and measure it with ruler
Best I can do is 3.14
The record now is 108 trillion
My head hurts.
Not how, but why? Why start and end a circle with 360 degrees, and use an infinite, irrational number to define a ratio of finite numbers. Ratio is a portion right? 360 zeros ÷2÷2 is still zero isn't it?. Alpha and Omega are different symbols, not 0 and 0.
ℼ Should be redefined to be ℼ = 6.28...
CODE FOUND IN PI -@
Why not create a cryptocurrency PI? Want a piece of pi? Where the value is pi and I.T. is more valuable the other way around. Less is more.You help calculate pi and get some of I.T. for your input.
I've thought about this before, but there's little practicality to computing Pi to such ridiculous lengths.
If would just be a proof of work algorithm with some novelty on top
0:29
then Do it!!
I'd watch a year worth of math videos
th-cam.com/video/NJDEnifDeyg/w-d-xo.html
If math algorithms do the same damage as social media algorithms, then what? Do they?
3.678926102736761212982838282837732728290183736638328838202828382728639273882726382783278282728273837383783738363864740303828627263737383637646475846663738393737837383739990020272622662727373735366837353926282682727272627278262
Smh it hard
I think people that use commas in digits, to facilitate the reading, are ignorant.
What do you think about that?
21.9911/7
3 atan(1/4) + atan(1/20) + atan(1/1985)
Now 100 digits
Pi day my bday!!!
Meh i came up with a simpler algorithm without using anything but the Pythagorean theorem
Are you a PhD in mathematics
I want to know your degree
Happy Pi day!
(2,1).(3,1)=(5,5)
i like chocate pie
Now its 105 trillion yay.pi❤❤
Uncle Roger?
X+x=
5Head
No is 62 800 000 000 000 it’s not 30 000 000 000 000
the record changed since this video was posted bruh
very interesting and great coincidence:
Pi * 10 ^13
I am in 10th 😑
if I try with 22/7 ,it will ends by giving repeated value of 3.142857........,i want the real value through which we can get infinite value
@
You lost after hello
Who cares... In computer programming lingo... Everything past the second decimal floating point value is usually ignorable bull...
They compute each decimal separately than keeping it together.
Very confusing explanation
And these records are totally useless
Pi = 3.1415926535897......
The maximum number of digits found out after decimal position are 3.1415926535897 * 10^13
Very Interesting Maths
Great Coincidence
Magic with the Numbers
Actually a few weeks after this upload, 50 trillion digits of pi were calculated, breaking a new record, which is sad in a way that there is no such "coincidence" anymore, but also good that we have better technology!
@@mathemaniac which is useless since the number is infinite and we don't need 50 trillion digits of pi (nasa uses 15 digits for example)
@@mathemaniac 3.1415... gogolplex digits or infinite digits perhaps with quantum computing. Even if we can't have all the digits "printed" or stored, ability to retrieve any digit of pi to any precision would be the same as "infinite" digits.
you talk like tom holland but with a lithp