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# Pi

(Redirected from 2 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
 ∞

 n  = 0
 ( − 1) n (2 n  +  1)!
x 2 n  + 1
(all integer multiples of π being solutions).

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 &pi; and TeX \pi both use lowercase “pi”, since the capitalized versions &Pi; and \Pi give the capital letter Π instead.

## 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√  2
expansion of π is
π  =  1000.00010001000000000000010010000000000100001000010000000010000001...
2  2

A238897 π in the base √  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.

A102243 Expansion of π in golden base (i.e., in irrational base
ϕ =
 1 + 2√  5 2
).
{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 +
1
7 +
1
15 +
1
1 +
1
292 +
1
1 +
1
1 +
 1 ⋱

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

 1 π
=  0.3183098861837906715377675267450287240689192914809128974953...

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

### 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 +
1
3 +
1
1 +
1
1 +
1
7 +
1
2 +
1
146 +
 1 ⋱

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

 π 2
=  1.570796326794896619231321691639751442098584699687552910487472296153908...

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

 2 π
=  0.6366197723675813430755350534900574481378385829618257949906...

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...
giving the sequence of decimal digits (A019694 Decimal expansion of
 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

 π 4
=  0.7853981633974483096156608458198757210492923498437764552437361480769541015715522496...

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

 4 π
=  1.2732395447351626861510701069801...

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

 4 π
=  1 +
1
3 +
4
5 +
9
7 +
 16 ⋱

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 +
1
2 +
3
3 +
6
4 +
 10 ⋱

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 +
1
4 +
5
7 +
12
10 +
 22 ⋱

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

 2√  9.87654321  =  3.14269680529319... (1.00035146240304... × π)

while

 2√  9.87  =  3.141655614481... (1.00002004107411... × π)

A slightly better approximation is

 6 5
ϕ 2  =  3.14164078649987... (1.000015321181... × π)

where ϕ is the Golden ratio.

### Approximations of 2π

21 ⋅
 299199 999999
=  6.283185 (0.99999999618119... × 2 π)

where

 299199 999999
=  0.299199

## Almost integers related to π

An almost integer (which is almost 355) from the convergents of the continued fraction expansion of π is

 113 π  =  354.9999698556466359462787023105838259142801421293869577701...
An almost integer (which is almost 20) with both
 π
and
 e
is
 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

 2 π 10 3
=  0.0062831853071...