Pi Digits

The digits of begin with the decimal expansion

(OEIS A000796). The following table summarizes selected historical record computations of the digits of through 2013.

The calculation of the digits of has occupied mathematicians since the day of the Rhind papyrus (1500 BC). Ludolph van Ceulen spent much of his life calculating to 35 decimal places. His result was memorialized after his death by the bounds

The inscription placed the two pi approximations inside and around a circle.

The diagram above reconstructs the circular arrangement of the two bounds. The original stone was lost by about 1800. An anonymous contemporary report (2000) records the installation of a replica in the Pieterskerk in Leiden on July 5, 2000.

The restored van Ceulen memorial is shown above (Boon 2009). Wells (1986, p. 48) discusses a number of other calculations.

The calculation of also figures in the Season 2 Star Trek episode "Wolf in the Fold" (1967), in which Captain Kirk and Mr. Spock force an evil entity (composed of pure energy and which feeds on fear) out of the starship Enterprise's computer by commanding the computer to "compute to the last digit the value of pi," thus sending the computer into an infinite loop.

Al-Kāshi of Samarkand obtained the sexagesimal approximation

using -gons, a value accurate to 17 decimal places (Borwein and Bailey 2003, p. 107). The sexagesimal digits of begin 6, 16, 59, 28, 1, 34, 51, 46, 14, 49, 55, ... (OEIS A091649).

The binary representation of the decimal digits of (top figure) and decimal representation (bottom figure) of are illustrated above.

A plot of the first 1600 decimal digits of (mod 2) is shown above (left figure), with the corresponding plot for 22/7 shown at right. Here, white indicates an even digit and black an odd digit (Pickover 2002, p. 285).

Spigot (Rabinowitz and Wagon 1995, Arndt and Haenel 2001, Borwein and Bailey 2003, pp. 140-141) and base 16 digit-extraction algorithms (the BBP formula) are known for . A remarkable recursive formula conjectured to give the th hexadecimal digit of is given by , where is the floor function,

is the fractional part and (Borwein and Bailey 2003, Ch. 4, Bailey et al. 2007, pp. 22-23).

The limit pi formulas

and

where is a Bernoulli number, give rapidly converging approximations to (Plouffe 2022). In particular, letting

a candidate for the th digit to the right of the decimal point of for is

where is the integer part and is the fractional part. To certify the digit, an error bound must exclude crossing a decimal truncation boundary; a small absolute error alone does not guarantee this. Similar approximations can be obtained using

and

where is an Euler number (Plouffe 2022). Plouffe uses the first approximation for octal (base 8) digits and the second for decimal digits, subject to the same truncation-boundary requirement. Related limits and formulas can also be obtained for (Plouffe 2022).

Including reported probable primes, pi-primes, i.e., -constant primes, occur at 1, 2, 6, 38, 16208, 47577, 78073, 613373, ... (OEIS A060421) digits, counting the initial 3.

The beast number 666 appears in at decimals 2440, 3151, 4000, 4435, 5403, 6840, ... (OEIS A083625). The first occurrences of just consecutive 6s are 7, 117, 2440, 21880, 48439, 252499, 8209165, 55616210, 45681781, ... (OEIS A096760), while (or more) consecutive 6s first occur at 7, 117, 2440, 21880, 48439, 252499, 8209165, 45681781, 45681781, ... (OEIS A050285).

The digits 314159 appear at positions 176451, 1259351, 1761051, 6467324, 6518294, 9753731, 9973760, ... (correcting Pickover 1995).

The sequence 0123456789 occurs beginning at digits , , , , , and (OEIS A101815, cf. Wells 1986, pp. 51-52).

The sequence 9876543210 occurs beginning at digits , , , , and (OEIS A101816).

The sequence 27182818284 (the first few digits of e) occurs beginning at digit (see also Pickover's sequence).

There are also interesting patterns for . 0123456789 occurs at , 9876543210 occurs at and , and 999999999999 occurs at of .

The starting positions of the first occurrence of , 1, 2, ... in the decimal expansion of (including the initial 3 and counting it as the first digit) are 33, 2, 7, 1, 3, 5, 8, 14, ... (OEIS A032445).

Scanning the decimal expansion of until all -digit strings have occurred, including strings with leading zeros, the last 1-, 2-, ... digit strings appearing are 0, 68, 483, 6716, 33394, 569540, ... (OEIS A032510), which end at digits 33, 607, 8556, 99850, 1369565, ... (OEIS A080597).

A curiosity relating to the beast number 666 involves adding the first three sextads of . First, note that

The numeral 1588419 can be read by concatenating the number 15 of skipped decimal places with the following digits 88419:

(pers. comm., P. Olivera, Aug. 11, 2005, Olivera).

It is not known if is normal (Wagon 1985, Bailey and Crandall 2001), although the first 30 million digits have approximately equal digit frequencies (Bailey 1988).

The following frequency distribution of decimal digits is found for the first digits of (Kanada 2003). It shows no statistically significant departure from a uniform distribution.

The following table gives the first few positions at which a digit occurs times. The sequence 1, 135, 1698, 54525, 24466, 252499, 3346228, 46663520, 564665206, ... (OEIS A061073) given by the diagonal (plus any terms of the form 10 10's etc.) is known as the Earls sequence (Pickover 2002, p. 339). The sequence 999999 occurs at decimal 762 (which is sometimes called the Feynman point, Wells 1986, p. 51) and continues as 9999998, which is the largest value of any seven digits in the first million decimals.

Knuth (2025, p. 18) notes a "miraculous" occurrence of the first 30 digits of in the graceful pi-way, namely a particular graceful labeling of the contiguous USA graph.