Introduction

◼
  • Happy Pi Day! Today (3/14) we celebrate the most famous mathematical constant: π ≈ 3.141592653589793…
    • ◼
  • π is irrational and transcendental, appears in circles, waves, probability, physics, and even random walks.
    • ◼
  • Wolfram Language (with its built-in π constant, excellent rational support, symbolic programming, and vast collection of Number Theory functions) makes experimenting with π especially easy and enjoyable.
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  • In this notebook we explore a selection of formulas and algorithms.

    • 1. Continued fraction approximation

    The built-in Wolfram Language (WL) constant Pi (or π ) can be computed to an arbitrary high precision:
    In[]:=
    N[Pi,60]
    Out[]=
    3.14159265358979323846264338327950288419716939937510582097494
    Let us compare it with a continued fraction approximation. For example, using the (first) sequence line of On-line Encyclopedia of Integer Sequences (OEIS) A001203 produces π with precision 100:
    In[]:=
    s={3,7,15,1,292,1,1,1,2,1,3,1,14,2,1,1,2,2,2,2,1,84,2,1,1,15,3,13,1,4,2,6,6,99,1,2,2,6,3,5,1,1,6,8,1,7,1,2,3,7,1,2,1,1,12,1,1,1,3,1,1,8,1,1,2,1,6,1,1,5,2,2,3,1,2,4,4,16,1,161,45,1,22,1,2,2,1,4,1,2,24,1,2,1,3,1,2,1};​​N[FromContinuedFraction[s],120]
    Out[]=
    3.14159265358979323846264338327950288419716939937510582097494459230781640628620899862803482534211706808656553677697242141
    Here we verify the precision using Wolfram Language:
    In[]:=
    N[Pi,200]-%
    Out[]=
    -1.0441745026369011477×
    -100
    10
    More details can be found in Wolfram MathWorld page "Pi Continued Fraction" , [EW1].

    2. Continued fraction terms plots

    It is interesting to consider the plotting the terms of continued fraction terms of \pi .
    First we ingest the more "pi-terms" from OEIS A001203 (20k terms):
    In[]:=
    terms=Partition[ToExpression/@StringSplit[Import["https://oeis.org/A001203/b001203.txt"],Whitespace],2]〚All,2〛;​​Length[terms]
    Out[]=
    20000
    Here is the summary:
    In[]:=
    ResourceFunction["RecordsSummary"][terms]
    Out[]=
    
    1 column 1
    1st Qu
    1
    Min
    1
    Median
    2
    3rd Qu
    5
    Mean
    12.6809
    Max
    20776
    
    Here is an array plot of the first 128 terms of the continued fraction approximating \pi :
    In[]:=
    mat=IntegerDigits[terms〚1;;128〛,2];​​maxDigits=Max[Length/@mat];​​mat=Map[Join[Table[0,maxDigits-Length[#]],#]&,mat];
    In[]:=
    ArrayPlot[Transpose[mat],Mesh->True,ImageSize->1000]
    Out[]=
    Next, we show the Pareto principle manifestation of for the continued fraction terms. First we observe that the terms a distribution similar to Benford's law :
    In[]:=
    tallyPi=KeySort[AssociationThread@@Transpose[Tally[terms]]]〚1;;16〛/Length[terms];
    In[]:=
    termsB=RandomVariate[BenfordDistribution[10],2000];​​tallyB=KeySort[AssociationThread@@Transpose[Tally[termsB]]]/Length[termsB];
    In[]:=
    data={tallyPi,tallyB};​​data=KeySort/@Map[KeyMap[ToExpression,#]&,data];​​data2=First/@Values@Merge[{data〚1〛,Join[AssociationThread[Keys[data〚1〛],Null],data〚2〛]},List];
    BarChartdata2,
    
    Out[]=
    π
    Belford's law
    Here is the Pareto principle plot -- ≈5% of the unique term values correspond to ≈80% of the terms:
    ResourceFunction["ParetoPrinciplePlot"]terms,
    
    Out[]=

    3. Classic Infinite Series

    Many ways to express π as an infinite sum — some converge slowly, others surprisingly fast.
    Leibniz–Gregory series (1671/ Madhava earlier)
    π4
    ∞
    ∑
    n=0
    n
    (-1)
    2n+1
    =41-
    1
    3
    +
    1
    5
    -
    1
    7
    +⋯
    WL implementation:
    In[]:=
    PiLeibniz[n_Integer]:=Block{},​​4*TotalTable
    i
    (-1)
    (2i+1),{i,0,n-1}​​
    In[]:=
    N[PiLeibniz[1000],20]
    Out[]=
    3.1405926538397929260
    Verify with Wolfram Language (again):
    In[]:=
    N[Pi,100]-N[PiLeibniz[1000],1000]
    Out[]=
    0.000999999750000312499046880410106913745780068595381331607486808765908475046461390362862493334446861
    Nilakantha series (faster convergence):
    π=3+
    ∞
    ∑
    n=1
    1+n
    (-1)
    ·4
    2n(1+2n)(2+2n)
    WL:
    In[]:=
    PiNilakantha[n_Integer]:=Block{},​​3+4*TotalTable
    (i+1)
    (-1)
    (2i(2i+1)(2i+2)),{i,1,n}​​
    In[]:=
    N[PiNilakantha[1000],20]
    Out[]=
    3.1415926533405420519
    In[]:=
    N[Pi,40]-N[PiNilakantha[1000],40]
    Out[]=
    2.49251186562514647026299317045×
    -10
    10

    3. Beautiful Products

    Wallis product (1655) — elegant infinite product:
    π
    2
    =
    ∞
    ∏
    n=1
    2
    (2n)
    (2n-1)(2n+1)
    =
    2×2×4×4×6×6⋯
    3×3×5×5×7
    WL running product:
    p=2.0;​​Do[​​p*=(2n)^2/((2n-1)(2n+1));​​If[Divisible[n,100],​​Print[Row[{n," → ",p/π," relative error"}]]​​],​​{n,1,1000}]

    4. Very Fast Modern Series — Chudnovsky Algorithm

    One of the fastest-converging series used in record computations:
    Each term adds roughly 14 correct digits.

    5. Spigot Algorithms — Digits “Drip” One by One

    Spigot algorithms compute decimal digits using only integer arithmetic — no floating-point errors accumulate.
    The classic Rabinowitz–Wagon spigot (based on a transformed Wallis product) produces base-10 digits sequentially.
    Simple (but bounded) version outline in WL:

    6. BBP Formula — Hex Digits Without Predecessors

    Bailey–Borwein–Plouffe (1995) formula lets you compute the n-th hexadecimal digit of π directly (without earlier digits):
    Very popular for distributed π-hunting projects. The best known digit-extraction algorithm .
    WL snippet for partial sum (base 16 sense):

    7. (Instead of) Conclusion

    ◼
  • π contains (almost surely) every finite sequence of digits — your birthday appears infinitely often.
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  • The Feynman point : six consecutive 9s starting at digit 762.
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  • Memorization world record > 100,000 digits.
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  • π appears in the normal distribution, quantum mechanics, random walks, Buffon's needle problem (probability ≈ 2/π).
    • Let us plot a “random walk” using the terms of continued fraction of Pi -- the 20k or OEIS A001203 -- to determine directions:
    Here is a bubble chart based on the path points and term values:
    Remark: The bubble sizes indicate that some of terms are fairly large, but the majority of them are relatively small.
    Here is a 3D point plot with of moving average of the 3D path modified to have the logarithms of the term values:

    References

    [EW1] Eric Weisstein, "Pi Continued Fraction" , Wolfram MathWorld .

    CITE THIS NOTEBOOK

    Pi Day 2026: formulas, series, and plots for π​
    by Anton Antonov​
    Wolfram Community, STAFF PICKS, March 14, 2026
    ​https://community.wolfram.com/groups/-/m/t/3661100