The pi man

Some weeks ago a reader sent me a note about an old article of mine, written back in 1989. The PDF, he reported, was missing a page. So I went scrounging through some untidy heaps of papers, found a copy, scanned it, and posted a new PDF. But along the way I stopped to reread my own words from 25 years ago.

The piece was a review of the first version of Mathematica, which debuted that summer of 1989. I began my review:

Sometime in the 1850′s William Shanks, a British schoolkeeper and amateur mathematician, began calculating the digits of the decimal expansion of π. Twenty years later, when he published his results in the Proceedings of the Royal Society, he had gotten as far as 707 digits. Today, with the aid of the computer program Mathematica, performing the same calculation requires no more effort than typing the expression:

N[Pi, 707] .

In response to this command, the labor that occupied Shanks for two decades is completed in less than eight seconds. Furthermore, Mathematica produces the correct 707 digits. Shanks made an error in the 528th decimal place that spoiled all the rest of his work.

I remember being wowed by the sub-eight-second running time of the π computation. (My notes say the actual time was 7.97 seconds.) That was with Mathematica 1.0 on a Macintosh SE/30 (16 MHz 32-bit processor, 5 MB of memory). How long would it take with more recent hardware and software? Naturally I had to try the experiment. Wolfram Research has just released version 10.0 of Mathematica. I ran it on the modern equivalent of the SE/30: a middle-of-the-road Mac laptop (2.6 GHz 64-bit processor, 8 GB of memory). The result: 190 microseconds.

I’m wowed again.

We don’t know how long Shanks actually spent noodling away at the digits of π. He did not—despite what I wrote in 1989—work continuously for two full decades. One known data point is that he extended his result from 530 decimal places to 607 places during March and April of 1853. Suppose he gave the task 60 hours a week during those two months. That’s roughly 500 hours for 77 digits, which we might extrapolate to 5,000 hours for 707. The average computation speed is thus 0.15 digits per hour, or 0.000043 digits per second. Compare this performance with later results:

Computer Year Speed
Shanks with a pencil 1873 0.000043
ENIAC with punch cards 1949 0.0079
16 MHz Mathematica 1.0 1989 89
2.6 GHz Mathematica 10.0 2014 3.7 × 106

What I find surprising in these numbers is that the ENIAC was only a few hundred times faster than a Victorian English gentleman wielding pencil and paper, whereas my current laptop is a billion times faster than the ENIAC. Much of this recent speedup is technological—a blessing of Moore’s Law. But algorithmics surely plays a role too.

As I mused about these developments, I realized that I had no idea how Mathematica computes the numerical value of π, or whether the method might have changed between version 1.0 and version 10.0. I also had only a vague notion of how Shanks did his computation. It turns out that both he and the ENIAC programmers used something called Machin’s formula:

\[\frac{\pi}{4} = 4\arctan\left(\frac{1}{5}\right) - \arctan\left(\frac{1}{239}\right)\]

Thus they found the value of π by first finding the value of two arctangents. That’s good to know, but it raises a further question: How do you evaluate the arctangent of 1/5 or 1/239? You can’t just look it up in a table if you need 700+ digits of precision. The solution, I learned, is to evaluate some terms of an infinite series, and then sum them up:

\[\arctan x = \frac{x^{1}}{1} - \frac{x^{3}}{3} + \frac{x^{5}}{5} - \frac{x^{7}}{7} + \cdots = \sum_{k=0}^\infty \frac{(-1)^{k}}{2k+1}x^{2k+1}\]

Mathematically, this is all you really need to know to understand what Shanks was doing in his computation of π. The whole job reduces to elementary arithmetic: Adding and subtracting, multiplying and dividing. But this was arithmetic at scale. I was having a hard time imagining the logistical and mechanical details of working with hundreds of numbers that run to hundreds of decimal places. “More is different” is a slogan from physics, but surely it applies to arithmetic too.

The last time I mentioned Shanks in these pages, I heard from Erwin Engert, who has done some thoughtful analysis of Shanks’s methods (and his errors). The documents on Engert’s web site include records of his own pencil-and-paper computations of π to 20 and 40 decimal places. I’m not brave enough to try that myself, but I thought I might be able to learn something useful by writing programs that follow some of the same steps a human solver might choose. The project would take a day or two, I supposed. Then one thing led to another (as it has a habit of doing).


For many of us, William Shanks is a figure to be pitied or ridiculed. What a sad waste of human potential to spend a major chunk of your lifespan churning out digits that no one would have any use for even if they were correct. What I’ve been doing over the past several weeks may be just as foolish and futile: I have expended a nontrivial chunk of my own lifespan tracking down all things Shanks—reading his publications, casting about for scraps of biographical detail, and most of all trying to reconstruct his work and figure out where his calculations went awry.

The main results of this dubious effort are in my new American Scientist column (HTML, PDF). I have also made available some supplementary material, including an annotated reading list and an IPython notebook with code you are welcome to explore in case you too would like to devote a portion of your life to following the Pied Piper.

In this posting I merely want to tie up a couple of loose ends: one item that wouldn’t fit in the magazine (it would have required a multipage foldout), and another development I stumbled upon after the article was sent to the printer.


In one phase of his computation, Shanks faced the task of adding up 506 numbers, each with 709 decimal places. Half of these terms were positive and half were negative, so the problem would naturally decompose into two parts. But half of the addition problem is still a tableau of 253 × 709 = 179,377 digits. What would such an array of numbers look like, if you wrote them all down on one sheet of paper? I don’t have a sheet big enough to answer that question directly. But here’s a hint—in very tiny type, with scroll bars. These are the 253 positive terms in the series for arctan 1/5. Feel free to add up the columns and write the sum at the bottom.

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Written at normal size, this 253 × 709 array would fill a sheet of paper roughly two meters wide by one meter high. I’m sure that Shanks never actually created such an unwieldy artifact. Instead he must have broken the process up into small sections, with some sort of hierarchical system of subsums copied over from one sheet to another. But I was curious: If he had decided to tackle the summation all in one go, could he have found paper in large enough sizes? The British Association of Paper Historians suggests not: The largest sheets in their list of Old English Paper Sizes are Imperial, Elephant, Double Elephant, Atlas, Columbier, and finally Antiquarian, which is 31 by 53 inches. But another source, paper-sizes.com, includes a sheet called Emperor, which is 4 feet by 6 feet—just about right. Of course even without the Emperor, Shanks could have pasted together smaller sheets. (Scotch tape was still in the future.) I like to imagine him taking over the family dining table with this huge army of numbers in tidy rows and columns.


Shanks’s 707 decimal places of π were accepted as correct for more than 70 years; his value was enshrined in carved wooden numerals on the ceiling of the Palais de la Decouverte in Paris. His errors were not discovered until the 1940s, when D. F. Ferguson, working with a mechanical desk calculator, was making rapid progress on a new computation of the constant. His value of π differed from Shanks’s starting just before decimal place 530. Many subsequent computations confirmed that Ferguson was right and Shanks was wrong.

But where and how and why did he go wrong? If we had all of his working papers, it would be easy to track down the mistakes. In 1853 Shanks published a small book that included the individual terms he summed to calculate arctan 1/5 and arctan 1/239. Unfortunately, the terms are given only up to the 530th decimal place—exactly where the trouble begins. For all the remaining digits—up to 607 in the book, then on to 707 in an 1873 publication—we have only the sums of the terms, that is, the computed values of arctan 1/5 and arctan 1/239, along with the resulting value of π.

In the absence of the working papers, we can never identify Shanks’s errors with complete certainty, but there’s hope of finding some plausible candidates. The basic idea is to take the difference between the true arctan value and the value that Shanks reported, then subtract this difference from each of the individual terms. If we get lucky, an error will show up as some sort of distinctive pattern.

As far as I know, Engert was the first to identify a likely error in this way. Shown below are decimal places 520 through 580 in the true and the Shanks values for arctan 1/5, along with their difference. When we subtract this difference from the true value of term 248 in the arctan series, we get an adjusted value that immediately suggests a clerical blunder. If we omit the 0 at position 530 (marked in red), letting all the following digits slide left one place, the true value of term 248 is converted into the adjusted value. It’s just the kind of slip that might well happen when copying a number from one page to another. This anomaly popped up when I ran a program searching for shifts caused by omitted or repeated digits; then I discovered that Engert had already spotted it some time ago. There is also a single-digit substitution of 2 → 3 (flagged in green) that Engert apparently didn’t notice.

a5 true     0.197 ... 560518377574220877835853152464749330914587633823112490332030
a5 shanks   0.197 ... 560518377617781642423378303370181926488028277686119150985606
diff       -0.000 ... 000000000043560764587525150905432595573440643863006660653576

t248 true   0.000 ... 744466800804828973843058350100603621730382293762575452716297
t248 adj    0.000 ... 744466800848389738430583501006036217303822937625582113369874

The omission at decimal place 530 in term 248 “uncorrects” the value of arctan 1/5 through place 568; then something else goes wrong. In my American Scientist column I suggest the following explanation of this second error. Starting with the “uncorrected” value of arctan 1/5—that is, the true value with the error in term 248 already incorporated—we subtract the Shanks value of arctan 1/5 to get a second difference. Then, subtracting this difference from term 72 in the series, we observe this intriguing pattern:

a5 248ucx   0.197 ... 018192648802827768629156477871020728799805452914758511130462
a5 shanks   0.197 ... 018192648802827768611915098560675901213598556363034321005664
diff        0.000 ... 000000000000000000017241379310344827586206896551724190124797

t72 true    0.000 ... 413793103448275862068965517241379310344827586206896551724137
t72 adj     0.000 ... 413793103448275862051724137931034482758620689655172361599340

We can convert the true term 72 into the adjusted value by excising the five digits marked in red and letting all the rest of the digits shift to the left to fill up the void. Again, one can imagine the omission as a careless error, probably made while transcribing intermediate results for use in a later stage of the computation.

This much of the story is told in my American Scientist column. But something about those numbers bothered me, and I kept poking at them even after the magazine went to press. Here they are again with different highlighting:

t72 true    0.000 ... 413793103448275862068965517241379310344827586206896551724137
diff        0.000 ... 000000000000000000017241379310344827586206896551724190124797
t72 adj     0.000 ... 413793103448275862051724137931034482758620689655172361599340

The very same motif—the 28-digit sequence 1724137931034482758620689655—appears in all three numbers, shifted left or right by a few places. The presence of these digits in t72 true is easy to explain: This sequence is the repeating unit, or repetend, in the decimal representation of term 72, which is equal to \((1/5)^{145}/145\). If you were to compute still more digits of the term, you would find that the same sequence repeats endlessly. But seeing the same sequence turn up in the other two numbers seems a bit surprising.

Try an experiment (or consult my IPython notebook, which includes code for the experiment). Take the sequence 1724137931034482758620689655 and create a cyclic permutation: for any k between 1 and 27, lop off k digits from the front of the sequence and paste them onto the back. Now subtract the permuted sequence from the original one. Notice anything?

Here’s an example, with k = 7:

 1724137931034482758620689655
–9310344827586206896551724137
–7586206896551724137931034482

The digits in the result form another cyclic permutation of the same sequence. That can’t be a coincidence!

And indeed it’s not a coincidence. The sequence 1724137931034482758620689655 is a cyclic permutation of 0344827586206896551724137931, which is the repetend of 1/29. It’s also not a coincidence that 145, the denominator in our term, is equal to 5 × 29. And the secret behind all of this hocus-pocus is that 29 is a “full-repetend prime” in base 10: a prime p whose reciprocal 1/p has a repetend with p–1 digits, the most possible. For all such primes the repetend is a “cyclic number”: multiplying by any integer in 1, 2, 3 … p–1 yields a cyclic permutation of the same digits. And, as observed above, subtracting any two cyclic permutations also yields another cyclic permutation.

I was led to look at term 72 by this note at the end of the 1946 paper in which Ferguson announced the discovery or errors in Shanks’s work:

In calculations made since the preceding part of this Note was written, from the 569th decimal place onwards I have a further disagreement with Shanks’ figures, namely 6896551724137931034482758620 (recurring). These are precisely the figures which should occur at this exact spot (i.e. from the 569th d.p. onwards) in the term 145 which comes in the series for \(\tan^{-1}(1/5)\). This surely cannot be a mere coincidence. It looks very much as if Shanks omitted the whole of this term from the 569th place onwards.

I am deeply puzzled by this statement. The sequence of digits mentioned does indeed appear exactly where Ferguson indicates in term 72. (In Ferguson’s numbering this is term 145; 2 × 72 + 1 = 145.) But truncating that term at position 569 does not have the desired effect of transforming the true value of arctan 1/5 into the value given by Shanks. To accomplish that, as noted above, what’s needed is not truncating the term but shifting the digits left by five places.

At a loss to explain the discrepancy between Ferguson’s observations and my own results, I went looking for other occurrences of the same cyclic number. It has to appear in at least one other place, namely term 14, which has the denominator 2 × 14 + 1 = 29. Here’s what I found:

t14         0.000 ... 4137931034482758620689655172413793103448275862068965517241379310
t14-diff    0.000 ... 4137931034482758620689654999999999999999999999999999999999478062
diff        0.000 ... 0000000000000000000000000172413793103448275862068965517241901247

Hence in term 14 we see the very phenomenon that Ferguson reported in term 72: truncating the term beyond decimal place 568 would recreate Shanks’s erroneous value, through decimal place 601. (The truncation could be expressed by setting the sigits to 5000… or to 4999…; it doesn’t matter which.)

With this discovery we have an alternative theory of how Shanks goofed in the neighborhood of decimal place 569. Instead of skipping five digits while transcribing term 72, he might have neglected the trailing digits of term 14 while summing up the series of terms. Which mechanism is more likely? If we could diagnose the further error beginning at place 602 in the arctan 1/5 series, we might learn the answer to this question. Within the range 569–601, the shift in term 72 and the zeroing of term 14 have identical effects, but in the later digits they would yield different values. Unfortunately, my attempts to find the source of those further errors in the arctan 1/5 computation have come up empty.

There’s also an error in Shanks’s value for arctan 1/239 that remains unaccounted for. And there’s the further mystery of Ferguson’s note. Why did he point to term 72 instead of term 14? To explain away his comment as a result of simple confusion or a memory lapse, we need to assume that he was mistaken both about which term he was looking at and which permutation of the cyclic number he was seeing. That’s hard to credit in someone who was an exceedingly meticulous calculator. I can offer up another hypothesis, though it’s no less farfetched: Maybe Ferguson derived the error term not from arctan 1/5 but from 4 × arctan 1/5, which is how the number enters into Machin’s formula. It turns out that 4 × 1724137931034482758620689655 is the very permutation that appears in term 72: 6896551724137931034482758620.

Even if we solve this mystery, there’s another intriguing conundrum in Ferguson’s observation. I don’t understand how he could have spotted the error at position 569—whatever its cause—without having discovered the site of the earlier problem at position 530. The distinctive repetend patterns appear only after the single-digit omission in term 248 has been “uncorrected.” Ferguson knew there was an error somewhere near decimal place 530, but his publications on the subject (or at least the three known to me) say nothing about the cause of that error.

In his 1853 book on π, Shanks ends his introductory remarks by declaring that he is going to “take leave of the subject for the present.” I’m going to follow his lead and declare that I’ve had enough of Shanks for the present. Maybe, like him, I’ll return to the topic after a suitable interval—say 20 years.

This entry was posted in computing, mathematics.

6 Responses to The pi man

  1. Martin Cohen says:

    Current computations of pi do not use the arctan series. Instead, either highly convergent series based on work of Ramanujan of iterations of type like Gauss’ arithmetic-geometric mean iteration are used.

    There are lots of sites with info about this. Just do a search for “computing pi”.

  2. Chris Hanusa says:

    I’ve read that Computer Algebra systems like Mathematica have many digits of pi recorded in their software since this is a common way in which people compare the speed of said programs. So I would venture to say that Mathematica 10 didn’t do any calculations; it just looked it up your answer in a table.

    • Brian Hayes says:

      Ever since I mentioned Shanks and his 707 digits in a review of Mathematica 1.0, I have wondered about the possibility of such benchmark-rigging.

      Here are some timings recorded in Mathematica 10.0, as the program computed (or not?!) 10^i digits of pi for 0 ≤ i ≤ 8:

      Table[{10^i, Timing[N[Pi, 10^i]][[1]]}, {i, 0, 8}]
        {{1, 0.000053},
         {10, 0.000021},
         {100, 0.000033},
         {1000, 0.000105},
         {10000, 0.001292},
         {100000, 0.026601},
         {1000000, 0.430528},
         {10000000, 6.249267},
         {100000000, 98.862143}}

      For fewer than about 1,000 digits, the results look like noise; there’s no clear correlation between precision and time. That outcome would be consistent with your hypothesis that we’re just doing table lookup in this range. But it’s also consistent with the hypothesis that the computation is so fast that the overall time is dominated by OS and UI events, or other overhead. Without access to the source code, it’s hard to know which.

      We do have the source code for Sage, the open-source mathematics system. Here’s what I get running the equivalent command there:

      for i in range(9):
        [(10**i, 
         '{:.6f}'.format(timeit('n(pi, digits='+str(10**i)+')', 
            seconds=True, number=1)))]
      [(1, '0.000019')]
      [(10, '0.000021')]
      [(100, '0.000024')]
      [(1000, '0.000105')]
      [(10000, '0.004632')]
      [(100000, '0.245613')]
      [(1000000, '3.692356')]
      [(10000000, '55.570486')]
      [(100000000, '1012.650426')]

      The times for large numbers of digits are about an order of magnitude slower, but for fewer than 1,000 digits the results look very similar to those of Mathematica. In Sage I’m pretty sure there’s no precomputed value being looked up. Sage apparently uses the mpmath package for computing constants and elementary functions. The mpmath source code (see libelefun.py) suggests there’s some caching or memoizing of results as they are computed, but not lookup from a pre-baked table.

      As a point of interest, here’s the mpmath docstring on pi computations:

      For computation of pi, we use the Chudnovsky series:
      
                   oo
                   ___        k
            1     \       (-1)  (6 k)! (A + B k)
          ----- =  )     -----------------------
          12 pi   /___               3  3k+3/2
                          (3 k)! (k!)  C
                  k = 0
      
      where A, B, and C are certain integer constants. This series adds roughly
      14 digits per term. Note that C^(3/2) can be extracted so that the
      series contains only rational terms. This makes binary splitting very
      efficient.
      
      The recurrence formulas for the binary splitting were taken from
      ftp://ftp.gmplib.org/pub/src/gmp-chudnovsky.c
      
      Previously, Machin's formula was used at low precision and the AGM iteration
      was used at high precision. However, the Chudnovsky series is essentially as
      fast as the Machin formula at low precision and in practice about 3x faster
      than the AGM at high precision (despite theoretically having a worse
      asymptotic complexity), so there is no reason not to use it in all cases.

      (Disclaimer: I don’t know my way around the Sage or mpmath source files, and I may have misunderstood what’s going on.)

      • Carl Witty says:

        I’m fairly sure that n(pi, digits=…) actually ends up using MPFR (the function mpfr_const_pi(), in MPFR’s const_pi.c).

        MPFR does have a cache to avoid recomputing Pi, but Sage doesn’t use it (the cache doesn’t play well with Control-C). And as far as I can tell the cache starts out empty.

  3. Peter says:

    Thanks for your interesting article on the lifetime labor of a (human) computer. A search led me to Wikipedia’s fascinating list of formulas for computing pi.

  4. Jason says:

    What a great read, thank you for sharing!