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.
1 Million Digits Of Pi Download
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This mathematical monster was discovered by Curtis Cooper at the University of Central Missouri in Warrensburg as part of the Great Internet Mersenne Prime Search (GIMPS), a collaborative effort to find new primes by pooling computing power online. It has 22,338,618 digits in total.
The file(s) provided above are ZIP-formatted archives, which most modern systems can natively unpack. If your computer does not unpack the archive when you double-click it, you may need to use a separate decompression program such as UnZip.
This book was a product of RAND's computing power (and patience). The tables of random numbers in the book have become a standard reference in engineering and econometrics textbooks and have been widely used in gaming and simulations that employ Monte Carlo trials. Still the largest known source of random digits and normal deviates, the work is routinely used by statisticians, physicists, polltakers, market analysts, lottery administrators, and quality control engineers.
The following persons participated in the production, testing, and preparation for publication of the tables of random digits and random normal deviates: Paul Armer, E. C. Bower, Mrs. Bernice Brown, G. W. Brown, Walter Frantz, J. J. Goodpasture, W. F. Gunning, Cecil Hastings, Olaf Helmer, M. L. Juncosa, J. D. Madden, A. M. Mood, R. T. Nash, J. D. Williams. These tables were prepared in connection with analyses done for the United States Air Force.
Early in the course of research at The RAND Corporation a demand arose for random numbers; these were needed to solve problems of various kinds by experimental probability procedures, which have come to be called Monte Carlo methods. Many of the applications required a large supply of random digits or normal deviates of high quality, and the tables presented here were produced to meet those requirements. The numbers have been used extensively by research workers at RAND, and by many others, in the solution of a wide range of problems during the past seven years.
One distinguishing feature of the digit table is its size. On numerous RAND problems the largest existing table of Kendall and Smith, 1939, would have had to be used many times over, with the consequent dangers of introducing unwanted correlations. The feasibility of working with as large a table as the present one resulted from developments in computing machinery which made possible the solving of very complicated distribution problems in a reasonable time by Monte Carlo methods. The tables were constructed primarily for use with punched card machines. With the high-speed electronic computers recently developed, the storage of such tables is usually not practical and, in fact, much larger tables than the present one are often required; these machines have caused research workers to turn to pseudo-random numbers which are computed by simple arithmetic processes directly by the machine as needed. These developments are summarized in Juncosa, 1953; Meyer, Gephart, and Rasmussen, 1954; and Moshman, 1954, where other references may be found. The Monte Carlo Method, 1951; Curtiss, 1949; Kahn and Marshall, 1953; and Kahn, 1956, discuss the uses and applications of the Monte Carlo methods and give references to other applications.
The random digits in this book were produced by rerandomization of a basic table generated by an electronic roulette wheel. Briefly, a random frequency pulse source, providing on the average about 100,000 pulses per second, was gated about once per second by a constant frequency pulse. Pulse standardization circuits passed the pulses through a 5-place binary counter. In principle the machine was a 32-place roulette wheel which made, on the average, about 3000 revolutions per trial and produced one number per second. A binary-to-decimal converter was used which converted 20 of the 32 numbers (the other twelve were discarded) and retained only the final digit of two-digit numbers; this final digit was fed into an IBM punch to produce finally a punched card table of random digits.
Production from the original machine showed statistically significant biases, and the engineers had to make several modifications and refinements of the circuits before production of apparently satisfactory numbers was achieved. The basic table of a million digits was then produced during May and June of 1947. This table was subjected to fairly exhaustive tests and it was found that it still contained small but statistically significant biases. For example, the following table[1] shows the results of three tests (described later) on two blocks of 125,000 digits:
Block 1 was produced immediately after a careful tune-up of the machine; Block 2 was produced after one month of continuous operation without adjustment. Apparently the machine had been running down despite the fact that periodic electronic checks indicated it had remained in good order.
The transformation was expected to, and did, improve the distribution in view of a limit theorem to the effect that sums of random variables modulo 1 have the uniform distribution over the unit interval as their limiting distribution. (See Horton and Smith, 1949, for a version of this theorem for discrete variates.)
These tables were reproduced by photo-offset from pages printed by the IBM model 856 Cardatype. Because of the very nature of the tables, it did not seem necessary to proofread every page of the final manuscript in order to catch random errors of the Cardatype. All pages were scanned for systematic errors, every twentieth page was proofread (starting with page 10 for both the digits and deviates), and every fortieth page (starting with page 5 for both the digits and deviates) was summed and the totals checked against sums obtained from the cards.[2]
Frequency Tests. The table was divided into 1000 blocks of 1000 digits each and the frequency of each digit was recorded for each block. Then for each block a goodness-of-fit 2 was computed with 9 d.f. These 1000 values of 2 provided an empirical fit to the 2 distribution (with 9 d.f.); to test the fit, a goodness-of-fit 2 was computer using 50 class intervals, each of which was expected to contain 2 per cent of the values. (The number of intervals was chosen in accordance with the result of Wald and Mann, 1942.) The value of the test 2 was 54.6 which, for 49 d.f., corresponded to about the 0.45 probability level.
To examine further the frequencies, the digits were tallied in 20 blocks of 50,000 digits each. The results are shown in Table 1 together with the goodness-of-fit 2 for each block. On the total frequencies the 2 (13.316) for 9 d.f. has been partitioned into three components as follows:
There were 200 sets of 1000 poker hands in the table, and for each set a goodness-of-fit 2 was computed with 5 d.f. (the fours and fives were combined). The manner in which these 200 values fit the 2 distribution is shown in Table 2.
The combined frequencies of poker hands in the whole table are shown in Table 3. The largest difference between expected and observed frequencies (for threes) is about 2.25 times its standard deviation, which is roughly at about the 9 or 10 per cent probability level (looking merely at the largest of five independent normal observations).
Also, the frequencies of poker hands were computed for each of ten blocks of 100,000 digits and the mean and standard deviation was computed from the ten values for each kind of hand. The results are shown in Table 4.
Serial and Run Tests. Some further tests were made on the first block of 50,000 digits to look particularly for any evidence of serial association among the digits. The serial test classified every successive pair of digits by each digit of the pair in a ten-by-ten table. The frequencies of the different pairs are given in Table 5, where the first digit of the pair is shown in the left column of the table and the second digit is shown at the top. Thus there were 510 cases in which a zero followed a one. The frequency 2 for the row (or column) totals is 7.56, which is about the 0.60 probability level for 9 d.f.
The deviates were determined by the five-digit numbers on the left-hand half of every page of the digit table. The deviates in the first column correspond page by page with the five-figure digits in the first column of the first 200 pages of the digit table; the deviates in the second column correspond page by page with the first column of the second 200 pages of the digit table. Similarly, the third and fourth columns of deviates were derived from the second column of five-figure digits, etc.
A 2 test of the fit of the entire table of deviates to the normal distribution was performed using 400 class intervals (Mann and Wald, 1942) with roughly 250 expected in each. The 2 value was found to be 346.4, which for 399 d.f. indicates a very close fit; the probability of a larger value of 2 is about 0.97. The detailed data for this test are given in Table 7.
A more refined test of the fit in the tails was made on the deviates exceeding 2.326 in absolute value. Eighty intervals (Mann and Wald, 1942) were used, each with an expectation of approximately 25. The 2 value was 76.26, with 80 d.f.; the probability of a larger value is about 0.61. The details of this test are given in Table 8. 152ee80cbc
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