Pi

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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 π and $T e X$ \pi both use lowercase “pi”, since the capitalized versions Π 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

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 +

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).