Irrational Mathematical Constants

An irrational number is a nonalgebraic number that consists of an infinite series of nonrepeating digits. This page shows how to generate decimal approximations to selected irrational mathematical constants using Maxima. Maxima is a free open-source computer-algebra software, from which commercial softwares such as Wolfram Mathematica and Maple are based.


Archimedean Constant (π)

The Archimedean constant pi (π) represents the ratio of the circumference to the diameter of a Euclidean circle. You can generate the decimal expansion of the Archimedan constant using Maxima on Ubuntu Debian Linux (Terminal command-line input below).

  • sudo apt-get install wxmaxima
  • maxima
  • bfloat(%pi),fpprec:2^8+1;

This will generate the first 257 (eighth power of two, or 28 = 256, significant) decimal digits of the Archimedean constant. The 256-digit approximation is shown below.

π = 3.141592653589793238462643383279502884197169399375105820974944592307816406286208998628034825342117067982148086513282306647093844609550582231725359408128481117450284102701938521105559644622948954930381964428810975665933446128475648233786783165271201909145649

The Archimedean constant is displayed here in hexadecimal color (hexcolor) #3243f7 ("Archimedean blue"), which corresponds to the hexadecimal value 3.243F716 = 3.1415910.

You can download the decimal approximation of the Archimedean constant to the fourth superpower of two (42 = 65536) digits here as plaintext (TXT), a LibreOffice Writer document (ODT), or in portable document format (PDF), or generate it yourself using the Maxima syntax below (rounding off the last digit).

  • bfloat(%pi),fpprec:2^2^2^2+1;

Practical Archimedean Digits

If you are using the Archimedean constant just to measure circumferences and diameters, then you actually do not need so many digits. The largest measurable circumference at the current time is the current circumference of our (expanding) Local Universe in Planck lengths, which is 2π × (~4.3 × 1026 meters)/(~1.61623 × 10−35 meters) = ~1.7 × 1062 (twice the Archimedean constant times the current radius of our Local Universe divided by the Planck length, or about 170 novemdecillion). Therefore, any number of decimal digits of the Archimedean constant greater than 63 will provide a precision of less than one Planck length to the current circumference of our Local Universe (which is physically impossible, since measurements below the Planck scale are physically meaningless). However, as our universe expands, more and more digits of the Archimedean constant will be required to maintain a maximum (theoretical) precision of one Planck length. The Archimedean constant to sixty-three decimal digits is provided below, which corresponds to a value one digit greater than the ratio of the number of Planck lengths in circumference (~1.7 × 1062) to the number of Planck lengths in diameter (~5.3 × 1061, or about 53 novemdecillion) of our Local Universe at this time.

π = 3.14159265358979323846264338327950288419716939937510582097494459

History

This table charts the number of decimal digits of the Archimedean constant known to humanity over time. The oldest recorded instance of the Archimedean constant (circa 1900 years before the Common Era, or BCE) is from Babylonia (Akkadian 𒆍𒀭𒊏𒆠/bābili), where it was approximated as the number three and written in Akkadian (𒀝𒅗𒁺𒌑/akkadû) cuneiform (𒄖𒋧/mihiştu) as "𒁹𒁹𒁹" (šalāš). The Archimedean constant gets its name from Archimedes (Classical Greek Ἀρχιμήδης/Arkhimḗdēs), who was the first human in recorded history to compute more than two digits of the constant (circa 250 BCE). The symbol "π" (pi) derives from the use of the Greek letter pi to represent perimeter, and its first recorded use to represent the ratio of the perimeter (circumference) to the diameter of a circle was in the year 1706 of the Common Era (CE) by William Jones.

year known digits
≥2000 BCE ≥(160 = 1), greater than or equal to one
≥1424 CE ≥(161 = 10), greater than or equal to ten
≥1853 CE ≥(162 = 100), greater than or equal to one hundred
≥1957 CE ≥(163 = 1000), greater than or equal to one thousand
≥1961 ≥(164 = 105), greater than or equal to ten thousand
≥1981 ≥(165 = 106), greater than or equal to one million
≥1985 ≥(166 = 107), greater than or equal to ten million
≥1989 ≥(167 = 109), greater than or equal to one billion
≥1995 ≥(168 = 109)
≥1999 ≥(169 = 1011), greater than or equal to one hundred billion
≥2002 ≥(1610 = 1012), greater than or equal to one trillion
≥2016 ≥(1611 = 1013), greater than or equal to ten trillion
[graph: known decimal digits of the Archimedean constant over time]
Graph showing the number of decimal digits of the Archimedean constant known to humanity over time. (Image from Wikimedia.)

You can download the data here as a LibreOffice Calc document (ODS).


Napier Constant (e)

The Napier constant (e) forms the base of natural logarithms. To generate the decimal expansion of the Napier constant, use the Maxima input below.

  • bfloat(%e),fpprec:2^8+1;

The decimal approximation of the Napier constant to 28 = 256 digits is reproduced below.

e = 2.718281828459045235360287471352662497757247093699959574966967627724076630353547594571382178525166427427466391932003059921817413596629043572900334295260595630738132328627943490763233829880753195251019011573834187930702154089149934884167509244761460668082265

The Napier constant is displayed here in hexcolor #2b7e15 ("Napier green"), which corresponds to the hexadecimal value 2.B7E1516 = 2.7182810.

The decimal approximation of the Napier constant to 42 = 65536 digits is also available to download here as TXT, ODT, or PDF, or you can generate it yourself using the Maxima syntax below (rounding off the last digit).

  • bfloat(%e),fpprec:2^2^2^2+1;

History

Below are the number of digits of the Napier constant known to humanity over time. The Napier constant was actually first computed by Jacob Bernoulli (in 1690) and not by John Napier. It is called the Napier constant because it is the base of natural logarithms, and logarithms were formalized by John Napier (in 1614). The symbol "e" was coined by Leonhard Euler (in 1731), and stands for "exponential," since the Napier constant is the base of the natural exponential function ex.

year known digits
≥1690 ≥(160 = 1)
≥1748 ≥(161 = 10)
≥1884 ≥(162 = 100)
≥1961 ≥(164 = 105)
≥2009 ≥(168 = 1010)
≥2010 ≥(169 = 1012)
≥2015 ≥(1610 = 1012)
[image: nautilus shell with logarithmic spiral shape outlined]
The Napier constant is an important component of logarithmic spirals, shown here with a nautilus shell. (Image from Wikimedia.)

Additional Constants

Pythagorean Constant (√2)

The square root of two, named after the Pythagorean theorem of Euclid (Classical Greek Εὐκλείδης/Eukleídēs, circa 300 BCE).

bfloat(sqrt(2)),fpprec:2^8+1;

(2) = 1.414213562373095048801688724209698078569671875376948073176679737990732478462107038850387534327641572735013846230912297024924836055850737212644121497099935831413222665927505592755799950501152782060571470109559971605970274534596862014728517418640889198609552

Shown here in hexcolor #16a09e ("Pythagorean teal"), from 1.6A09E16 = 1.4142110.

Theodorean Constant (√3)

The square root of three, named after the Theodorean spiral of Theodorus (Classical Greek Θεόδωρος/Theódōros, circa 450 BCE).

bfloat(sqrt(3)),fpprec:2^8+1;

(3) = 1.732050807568877293527446341505872366942805253810380628055806979451933016908800037081146186757248575675626141415406703029969945094998952478811655512094373648528093231902305582067974820101084674923265015312343266903322886650672254668921837971227047131660368

Shown here in hexcolor #1bb67b ("Theodorean lime"), from 1.BB67B16 = 1.7320510.

[image: Theodorean spiral]
The Pythagorean and Theodorean constants form the hypotenuses of right triangles, shown here in a Theodorean spiral. (Image from Wikimedia.)