In November 2016 I made a data set with one trillion decimal digits available for download. The digits are saved as zip-compressed text files in chuncks of one hundred billion digits. Each file has a size of 43.7 Gigabytes. In total you have to download more than 430 Gigabytes.
Since February 2019 you can download all decimal digits from my Google drive account at DECTRIS. These are the original files from my world record computation in the compressed format of y-cruncher. You need to compile the DigitViewer provided by A. Yee in order to read these files. Each file contains 200 billion digits and has a size of 78 Gigabytes. The total size is 8.8 Terabytes. If you encounter permission issues, this could be due to restrictions imposed by your domain administrator. I have verified that the files can be accessed when not being logged in with a google account.
Random walk based on pi: pi2e.ch/blog/2016/07/11/the-pi-protein/3D scroll with the decimal digits of pi: pi2e.ch/blog/2017/07/14/the-pi-scroll/Pi art by M. Krzywinski: mkweb.bcgsc.ca/pi/art/Pi graphics by Cristian Vasile: fineartamerica.com/profiles/cristian-vasile
Visualize transitions from one digit to the next in Pi using D3.js. Each transition draws a line from a digit in the circle to another digit. Once you have a large amount of digits you can see which digits transition more often and which digits are more common.
For Google Cloud Next 2019 in San Francisco, a team created the Pi Experiment Showcase, and uses the Pi API & generative algorithm to generate unique art pieces for each of the 31.4 trillion digits of Pi.
We have the same function deployed in three regions (US, Belgium, Japan) that fetches digits from a Cloud Storage bucket in the US multi-region. The API endpoint, api.pi.delivery is exposed via a Global HTTP(s) Load Balancer. The authorative DNS servers are provided by Cloud DNS.
In 2020, we updated the Pi API to return 50 trillion digits, and added Machine Learning models to generate even better Pi music. The numbers were kindly donated by Timothy Mullican, former world record holder. We also had an opportunity to interview Timothy for Pi Day 2020. Check out this video if you want to hear two world record breakers discuss about pi!
In 2019, the Pi digits world record was broken by Emma Haruka Iwao. Google Cloud Experiment for Pi Day was created by Mathias Paumgarten. The Pi API was updated to serve all 31.4+ trillion digits by Ray Tsang.
In 2016, we calculated 500 billion digits of Pi and made it searchable. The indexing peaked at 2-million writes per second. You can learn more how we did that in the blog Calculating and searching 500 billion digits of Pi, by Francesc Campoy, Jen Tong, Ray Tsang, Sara Robinson.
Whether you want to very accurately calculate the area of a circle, paint the digits of Pi on your room, face, a t-shirt, or your baby brother, or memorize digits of Pi to impress your friends...
Alternately, you could download a program to compute pi and compute them yourself. Alexander Yee's y-cruncher for Windows and Linux is the fastest program out there. On a fast computer, it can compute 1 billion digits in perhaps 10 minutes. If you prefer the open source route, check out how to compute Pi using GMP, one of the most popular open-source math libraries.
Last year I celebrated Pi Day by using SAS to explore properties of the continued fraction expansion of pi. This year, I will examine statistical properties of the first 10 million digits of pi. In particular, I will show that the digits of pi exhibit statistical properties that are inherent in a random sequence of integers.
Editor's note (17Mar2015): Ken Kleinman remarked on the similarity of the analysis in this article to his own analysis from 2010. I was reading his blog regularly in 2010, so I should have cited him for the chi-square and Durbin-Watson analyses in this post. My apologies to Ken and Nick Horton for this oversight!Reading 10 million digits of piI have no desire to type in 10 million digits, so I will use SAS to read a text file at a Princeton University URL. The following statements use the FILENAME statement to point to the URL:
You can run many statistical tests on these numbers. It is conjectured that the digits of pi are randomly uniformly distributed in the sense that the digits 0 through 9 appear equally often, as do pairs of digits, trios of digits, and so forth.
The frequency analysis of the first 10 million digits shows that each digit appears about one million times. A chi-square test indicates that the digits appear to be uniformly distributed. If you turn on ODS graphics, PROC FREQ also produces a deviation plot that shows that the deviations from uniformity are tiny.
That's a pretty cool triangular distribution! I won't bore you with mathematical details, but this shape arises when you examine the difference between two independent discrete uniform random variables, which suggests that the even digits of pi are independent of the odd digits of pi.
In fact, more is true. You can run a formal test to check for autocorrelation in the sequence of numbers. The Durbin-Watson statistic, which is available in PROC REG and PROC AUTOREG, has a value near 2 if a series of values has no autocorrelation. The following call to PROC AUTOREG requests the Durbin-Watson statistic for first-order through fifth-order autocorrelation for the first one million digits of pi. The results show that there is no detectable autocorrelation through fifth order. To the Durban-Watson test, the digits of pi are indistinguishable from a random sequence:
Researchers have rundozens of statistical tests for randomness on the digits of pi. They all reach the same conclusion. Statistically speaking, the digits of pi seems to be the realization of a process that spits out digits uniformly at random.
Nevertheless, mathematicians have not yet been able to prove that the digits of pi are random.One of the leading researchers in the quest commented that if they are random then you can find in the sequence (appropriately converted into letters) the "entire works of Shakespeare" or any other message that you can imagine (Bailey and Borwein, 2013).For example, if I assign numeric values to the letters of "Pi Day" (P=16, I=9, D=4, A=1, Y=25), then the sequence "1694125" should appear somewhere in the decimal expansion of pi.I wrote a SAS program to search the decimal expansion of pi for the seven-digit "Pi Day" sequence. Here's what I found:
There it is! The numeric representation of "Pi Day" appears near the 4.7 millionth decimal place of pi. Other "messages" might not appear in the first 10 million digits, but this one did. Finding Shakespearian sonnets and plays will probably require computing more digits of pi than the current world record.
The digits of pi pass every test for randomness, yet pi is a precise mathematical value that describes the relationship between the circumference of a circle and its diameter. This dichotomy between "very random" and "very structured" is fascinating! Happy Pi Day to everyone!
Interesting! There something I do not understand though...If the digits of PI have a uniform distribution based on chi-square 2.78 at 10mill, how come if you calculate chi-square further, at around position 86mill you will get a chi-sqare of 8.9 suggesting it is not uniform.There is an even smaller chi-square at around 12million (~0.9) so I find it strange that the chi-square would then increase.
1. If it *weren't* random--if it really did have a pattern--would it then necessarily be rational?2. Is it possible that pi actually DOES have a pattern that reveals itself, say, after 31.41592653589 trillion digits?
Yes, I should have deduced that a pattern of digits can exist without repetition, and thereby still represent an irrational number. But it is irking me that numbers like pi and e might be infinite random sequences of digits, with no underlying structure or pattern...because at least intuitively that seems to imply that the natural phenomena they so beautifully represent are somehow intrinsically random as well. Silly idea, but maybe it's related to the probabilistic nature of quantum physics: If there WERE a pattern (like 0.1011011101111...) then we would KNOW the number even though we couldn't write it all out or represent it as a ratio. But if there is NO pattern or structure, then we can never really pin it down...like an intrinsic uncertainty. We can get closer and closer but never completely represent it. No..no, I'm wrong again, I'm always wrong, because pi and e are easily represented with tidy infinite sums. Aha, so infinite sums can therefore be random number generators?! I see I'm a little too existential, and not quantitative enough, about all this.
Interesting article! When you say, "Finding Shakespearian sonnets and plays will probably require computing more digits of pi than the current world record," that is a vast understatement. To find an entire Shakespeare play will almost certainly require the computation of more digits, at a rate a trillion times the speed of today's computers, than is possible given the age of the universe.
Digits 1-9 show up by the 13th decimal place of pi, but the first 0 does not appear until the 32nd decimal place. For the first 13 places a new digit shows up every digit or to), then 20 places go by before the first 0 appears. That seems very odd if the digits are randomly distributed. But I guess that's the nature of true randomness. Its results are far less uniform that we expect!
For the digits 0-9, imagine rolling a ten-sided die. How many times do you expect to roll it before all 10 faces have appeared? The answer is 10/10 + 10/9 + 10/8 + ... + 10/1 = 29.3 rolls. So it is not odd at all that you have to look at 32 decimal places in pi until all digits appear! 2b1af7f3a8