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Pi
The transcendental number π, also called Archimedes’ constant, is
- the ratio of the circumference of a circle over its diameter;
- the ratio of the area of a disk over the square of its radius;
- the ratio of the area of an ellipse over the product of the lengths of its semi-major and semi-minor axes;
- the smallest positive real number root of the power series
(all integer multiples of π being solutions).∑ ∞n = 0
x 2 n + 1( − 1) n (2 n + 1)!
Swiss scientist Johann Heinrich Lambert in 1761 proved that π is irrational. (Lambert’s proof exploited a continued fraction representation of the tangent function.) French mathematician Adrien-Marie Legendre proved in 1794 that π 2 is also irrational. In 1882, German mathematician Ferdinand von Lindemann proved that π is transcendental, confirming a conjecture made by both Legendre and Euler.
In OEIS sequence entries, π is written “Pi” (and likewise in Mathematica’s InputForm), but “pi” occurs quite often in math discussion forums. The HTML character entity π
and TeX \pi
both use lowercase “pi”, since the capitalized versions Π
and \Pi
give the capital letter Π instead.
Contents
- 1 π
- 2 1 / π
- 3 2 π
- 4 π / 2
- 5 2 / π
- 6 4 π
- 7 π / 4
- 8 4 / π
- 9 π2
- 10 π3
- 11 Approximations
- 12 Almost integers related to π
- 13 See also
π
Decimal expansion of π
The decimal expansion of π is
- π = 3.1415926535897932384626433832795028841971693993751058209749445923078164...
giving the sequence of decimal digits (A000796)
- {3, 1, 4, 1, 5, 9, 2, 6, 5, 3, 5, 8, 9, 7, 9, 3, 2, 3, 8, 4, 6, 2, 6, 4, 3, 3, 8, 3, 2, 7, 9, 5, 0, 2, 8, 8, 4, 1, 9, 7, 1, 6, 9, 3, 9, 9, 3, 7, 5, 1, 0, 5, 8, 2, 0, 9, 7, 4, 9, 4, 4, 5, 9, 2, 3, ...}
The three significant decimal digits centered at 360
Look at the b-file: http://oeis.org/A000796/b000796.txt
1 3 2 1 3 4 4 1 5 5 (...) 355 2 356 5 357 9 358 0 <-- To top it off, no nonzero decimal digit dared to precede or follow it! 359 3 \ 360 6 > <-- Amazingly, the three decimal [significant] digits centered at 360 are 360, and 360 degrees = 2*pi radians! 361 0 / 362 0 <-- To top it off, no nonzero decimal digit dared to precede or follow it! 363 1 364 1 365 3 (...)
The digit 9
The digit 9 appears 6 times in a row starting 762 digits after the decimal point, v.g.
3. |
1415926535897932384626433832795028841971693993751058209749445923078164062862089986280348253421170679 |
8214808651328230664709384460955058223172535940812848111745028410270193852110555964462294895493038196 |
4428810975665933446128475648233786783165271201909145648566923460348610454326648213393607260249141273 |
7245870066063155881748815209209628292540917153643678925903600113305305488204665213841469519415116094 |
3305727036575959195309218611738193261179310511854807446237996274956735188575272489122793818301194912 |
9833673362440656643086021394946395224737190702179860943702770539217176293176752384674818467669405132 |
0005681271452635608277857713427577896091736371787214684409012249534301465495853710507922796892589235 |
4201995611212902196086403441815981362977477130996051870721134999999... |
Base b expansion of π
Binary expansion of π
Binary expansion of π is
- π = 11.001001000011111101101010100010001000010110100011000010001101001100010011000110011000101000101110000000110111000001110011010001001010010000001... 2
A004601 Expansion of π in base 2 (or, binary expansion of π).
- {1, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 0, 0, 0, ...}
Base √ 2 expansion of π
The base√ 2 |
- π = 1000.00010001000000000000010010000000000100001000010000000010000001... √ 2
- {1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, ...}
Base ϕ expansion of π
The base ϕ expansion of π is
- π = 100.0100101010010001010101000001010...ϕ ,
where ϕ is the golden ratio.
ϕ =
|
- {1, 0, 0, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, ...}
Continued fraction expansion of π
The simple continued fraction expansion of π is
π = 3 +
|
giving the sequence of partial quotients (A001203)
- {3, 7, 15, 1, 292, 1, 1, 1, 2, 1, 3, 1, 14, 2, 1, 1, 2, 2, 2, 2, 1, 84, 2, 1, 1, 15, 3, 13, 1, 4, 2, 6, 6, 99, 1, 2, 2, 6, 3, 5, 1, 1, 6, 8, 1, 7, 1, 2, 3, 7, 1, 2, 1, 1, 12, 1, 1, 1, 3, 1, 1, 8, 1, 1, 2, 1, 6, 1, 1, 5, ...}
1 / π
Decimal expansion of 1 / π
The decimal expansion of 1 / π is
-
= 0.3183098861837906715377675267450287240689192914809128974953...1 π
giving the sequence of decimal digits (A049541)
- {3, 1, 8, 3, 0, 9, 8, 8, 6, 1, 8, 3, 7, 9, 0, 6, 7, 1, 5, 3, 7, 7, 6, 7, 5, 2, 6, 7, 4, 5, 0, 2, 8, 7, 2, 4, 0, 6, 8, 9, 1, 9, 2, 9, 1, 4, 8, 0, 9, 1, 2, 8, 9, 7, 4, 9, 5, 3, 3, ...}
2 π
2 π is
- the ratio of the circumference of a circle over its radius;
- a full circle arc (corresponds to an angle of 2 π radians).
Decimal expansion of 2 π
The decimal expansion of 2 π is
- τ := 2 π = 6.283185307179586476925286766559005768394338798750211641949889184615632...
giving the sequence of decimal digits (A019692)
- {6, 2, 8, 3, 1, 8, 5, 3, 0, 7, 1, 7, 9, 5, 8, 6, 4, 7, 6, 9, 2, 5, 2, 8, 6, 7, 6, 6, 5, 5, 9, 0, 0, 5, 7, 6, 8, 3, 9, 4, 3, 3, 8, 7, 9, 8, 7, 5, 0, 2, 1, 1, 6, 4, 1, 9, 4, 9, 8, 8, 9, 1, 8, 4, 6, 1, 5, 6, 3, 2, 8, 1, 2, 5, 7, 2, 4, 1, 7, ...}
Continued fraction expansion of 2 π
The simple continued fraction expansion of 2 π is
τ := 2 π = 6 +
|
giving the sequence of partial quotients (A058291)
- {6, 3, 1, 1, 7, 2, 146, 3, 6, 1, 1, 2, 7, 5, 5, 1, 4, 1, 2, 42, 5, 31, 1, 1, 1, 6, 2, 2, 4, 3, 12, 49, 1, 5, 1, 12, 1, 1, 1, 2, 3, 1, 2, 1, 1, 3, 1, 16, 2, 1, 1, 15, 2, 3, 6, 3, 8, ...}
π / 2
Decimal expansion of π / 2
The decimal expansion of π / 2 is
-
= 1.570796326794896619231321691639751442098584699687552910487472296153908...π 2
giving the sequence of decimal digits (A019669)
- {1, 5, 7, 0, 7, 9, 6, 3, 2, 6, 7, 9, 4, 8, 9, 6, 6, 1, 9, 2, 3, 1, 3, 2, 1, 6, 9, 1, 6, 3, 9, 7, 5, 1, 4, 4, 2, 0, 9, 8, 5, 8, 4, 6, 9, 9, 6, 8, 7, 5, 5, 2, 9, 1, 0, 4, 8, 7, 4, ...}
2 / π
2/π is known as Buffon’s constant.
Decimal expansion of 2 / π
The decimal expansion of 2 / π is
-
= 0.6366197723675813430755350534900574481378385829618257949906...2 π
giving the sequence of decimal digits (A060294)
- {6, 3, 6, 6, 1, 9, 7, 7, 2, 3, 6, 7, 5, 8, 1, 3, 4, 3, 0, 7, 5, 5, 3, 5, 0, 5, 3, 4, 9, 0, 0, 5, 7, 4, 4, 8, 1, 3, 7, 8, 3, 8, 5, 8, 2, 9, 6, 1, 8, 2, 5, 7, 9, 4, 9, 9, 0, 6, 6, ...}
4 π
4 π is the ratio of the area of a sphere over the square of its radius.
Decimal expansion of 4 π
The decimal expansion of 4 π is
- 4 π = 12.566370614359172953850573533118...
2 π |
5 |
- {1, 2, 5, 6, 6, 3, 7, 0, 6, 1, 4, 3, 5, 9, 1, 7, 2, 9, 5, 3, 8, 5, 0, 5, 7, 3, 5, 3, 3, 1, 1, 8, 0, 1, 1, 5, 3, 6, 7, 8, 8, 6, 7, 7, 5, 9, 7, 5, 0, 0, 4, 2, 3, 2, 8, 3, 8, 9, 9, 7, ...}
π / 4
Decimal expansion of π / 4
The decimal expansion of π / 4 is
-
= 0.7853981633974483096156608458198757210492923498437764552437361480769541015715522496...π 4
giving the sequence of decimal digits (A003881)
- {7, 8, 5, 3, 9, 8, 1, 6, 3, 3, 9, 7, 4, 4, 8, 3, 0, 9, 6, 1, 5, 6, 6, 0, 8, 4, 5, 8, 1, 9, 8, 7, 5, 7, 2, 1, 0, 4, 9, 2, 9, 2, 3, 4, 9, 8, 4, 3, 7, 7, 6, 4, 5, 5, 2, 4, 3, 7, 3, ...}
4 / π
Decimal expansion of 4 / π
The decimal expansion of 4 / π is
-
= 1.2732395447351626861510701069801...4 π
giving the sequence of decimal digits (A088538)
- {1, 2, 7, 3, 2, 3, 9, 5, 4, 4, 7, 3, 5, 1, 6, 2, 6, 8, 6, 1, 5, 1, 0, 7, 0, 1, 0, 6, 9, 8, 0, 1, 1, 4, 8, 9, 6, 2, 7, 5, 6, 7, 7, 1, 6, 5, 9, 2, 3, 6, 5, 1, 5, 8, 9, 9, 8, 1, 3, 3, 8, 7, 5, 2, 4, ...}
Generalized continued fraction expansions for 4 / π
A nice generalized continued fraction expansion for 4 / π is
|
giving the sequence of odd numbers (square gnomonic numbers) interleaved with the square numbers (A079097)
- {1, 1, 3, 4, 5, 9, 7, 16, 9, 25, 11, 36, 13, 49, 15, 64, 17, 81, 19, 100, 21, 121, 23, 144, 25, 169, 27, 196, 29, 225, 31, 256, 33, 289, 35, 324, 37, 361, 39, 400, 41, 441, 43, ...}
Generalized continued fraction from n-gonal gnomonic numbers and their corresponding n-gonal numbers
Generalized continued fraction from triangular gnomonic numbers and their corresponding triangular numbers
One might wonder what the generalized continued fraction from natural numbers (triangular gnomonic numbers) and triangular numbers begets?
? = 1 +
|
giving the sequence of natural numbers (triangular gnomonic numbers) interleaved with the triangular numbers (A160791)
- {1, 1, 2, 3, 3, 6, 4, 10, 5, 15, 6, 21, 7, 28, 8, 36, 9, 45, 10, 55, 11, 66, 12, 78, 13, 91, 14, 105, 15, 120, 16, 136, 17, 153, 18, 171, 19, 190, 20, 210, 21, 231, 22, 253, 23, 276, 24, 300, 25, 325, 26, 351, 27, ...}
Generalized continued fraction from pentagonal gnomonic numbers and their corresponding pentagonal numbers
One might wonder what the generalized continued fraction from pentagonal gnomonic numbers and pentagonal numbers begets?
? = 1 +
|
giving the sequence of pentagonal gnomonic numbers interleaved with the pentagonal numbers (A??????)
- {1, 1, 4, 5, 7, 12, 10, 22, 13, 35, 16, 51, 19, 70, 22, 92, 25, 117, 28, ...}
π2
Decimal expansion of π 2
The decimal expansion of π 2 is
- π 2 = 9.869604401089358618834490999876151135313699407240790626413349376220044...
giving the sequence of decimal digits (A002388)
- {9, 8, 6, 9, 6, 0, 4, 4, 0, 1, 0, 8, 9, 3, 5, 8, 6, 1, 8, 8, 3, 4, 4, 9, 0, 9, 9, 9, 8, 7, 6, 1, 5, 1, 1, 3, 5, 3, 1, 3, 6, 9, 9, 4, 0, 7, 2, 4, 0, 7, 9, 0, 6, 2, 6, 4, 1, 3, 3, ...}
π3
Decimal expansion of π 3
The decimal expansion of π 3 is
- π 3 = 31.00627668029982017547631506710139520222528856588510769414453810380639...
giving the sequence of decimal integers (A091925)
- {3, 1, 0, 0, 6, 2, 7, 6, 6, 8, 0, 2, 9, 9, 8, 2, 0, 1, 7, 5, 4, 7, 6, 3, 1, 5, 0, 6, 7, 1, 0, 1, 3, 9, 5, 2, 0, 2, 2, 2, 5, 2, 8, 8, 5, 6, 5, 8, 8, 5, 1, 0, 7, 6, 9, 4, 1, 4, 4, 5, 3, 8, 1, 0, 3, ...}
Approximations
Approximations of π
- √ 9.87654321= 3.14269680529319... (1.00035146240304... × π)
while
- √ 9.87= 3.141655614481... (1.00002004107411... × π)
A slightly better approximation is
-
ϕ 2 = 3.14164078649987... (1.000015321181... × π)6 5
where ϕ is the Golden ratio.
Approximations of 2π
-
21 ⋅
= 6.283185 (0.99999999618119... × 2 π)299199 999999
where
-
= 0.299199299199 999999
An almost integer (which is almost 355) from the convergents of the continued fraction expansion of π is
-
113 π = 354.9999698556466359462787023105838259142801421293869577701...
π |
e |
-
e π − π = 19.99909997918947576726644298466904449606893684322510617247...
Almost integer π 3
π 3 is somewhat close to an integer (the first 2 digits after the decimal point are 0).
- π 3 = 31.00627668029982017547631506710139520222528856588510769414453810380639...
Also, observe that the decimal expansion of π 3 − 31 is nearly
-
= 0.0062831853071...2 π 10 3
See also
π in other integer bases
In the following, the OEIS denotes π as Pi.
- A004601 Binary expansion of Pi.
- A004602 Expansion of Pi in base 3.
- A004603 Expansion of Pi in base 4.
- A004604 Expansion of Pi in base 5.
- A004605 Expansion of Pi in base 6.
- A004606 Expansion of Pi in base 7.
- A006941 Expansion of Pi in base 8.
- A004608 Expansion of Pi in base 9.
- A000796 Decimal expansion of Pi.
- A068436 Expansion of Pi in base 11.
- A068437 Expansion of Pi in base 12.
- A068438 Expansion of Pi in base 13.
- A068439 Expansion of Pi in base 14.
- A068440 Expansion of Pi in base 15.
- A062964 Pi in hexadecimal.
- A060707 Base-60 (Babylonian or sexagesimal) expansion of Pi.