Now let's continue on with more information about pi.
One of the basic facts about pi is that it is a transcendental number.
I'd like to say a few words about what that means.
The most basic numbers are the integers; we get those by counting objects one after another. Now, there are negative integers, and yet you can't hold -3 pebbles in your hand; extending the number system in this direction leads to the imaginary numbers, which raise a different kind of question, so I will put that aside here.
Instead, the next step is the rational numbers; that is, fractions that have an integer as both the numerator and the denominator. When, instead of counting objects, we measure out liquids, or we measure out lengths, or set up systems of units to use in these measurements, we usually use fractions like this.
Since integers can go as big as we like, we can get as close as we like to any desired volume, weight, or distance with a rational number. So from one kind of practical viewpoint, the rational numbers are all we need.
However, the relationships and formulas that are found in mathematics are useful for many practical purposes, and to derive them, we have to be able to think of numbers and points and lines and shapes in an abstract manner, dealing with their exact values and positions, without concern for the limitations of the real world that often mean that a good approximation is as good as we'll either need or be able to get.
One of the simplest numbers that isn't rational is the square root of two - the length of the diagonal of a square of length one.
It's easy enough, starting with a line of unit length, to draw a square with sides all of that length, in a finite number of steps, using a straightedge and compass strictly in the conventional manner. And that lets you then draw the diagonal and copy it.
So the square root of two is what is called a constructible number. These numbers are those which can be produced from integers by addition, subtraction, multiplication, division, and taking square roots. This includes fourth roots or eighth roots or sixteenth roots, because one can take square roots repeatedly.
One thing that can't be done in a finite number of steps, using a straightedge and compass strictly in the conventional manner, is to trisect the angle. But if you do something sneaky, for example, holding your compass against the straightedge to indicate a length along it, while you move the two tools into position, then you can trisect the angle: Archimedes first found this construction, and I remember seeing an elaborate version of that construction featured in the letters column of Mechanix Illustrated.
But while ingenuity like that of the television character MacGyver certainly has its uses, it's also useful to have a category for the numbers that can be made with straightedge and compass in a pedestrian fashion.
One of the other famous problems that was not solvable with straightedge and compasses was the "Duplication of the Cube", that is, given a line of unit length, drawing a line that is the cube root of two times as long.
So the next level above constructible numbers are those numbers that are expressible by radicals. The numbers that can be produced from integers by addition, subtraction, multiplication, division, and taking the n-th root for any integer n are also a group of numbers that are closed under that set of operations.
If you try to solve a quadratic equation, you can use the formula that the equation a*x^2 + b*x + c = 0 is true if x equals either (-b+sqrt(b*b-4*a*c))/(2*a) or (-b-sqrt(b*b-4*a*c))/(2*a). So the roots of a quadratic equation are numbers that are expressible by radicals.
Formulas were also found, but more complicated ones, to solve the cubic equation, where x^3 is also present, and the quartic equation, which also has x^4, too.
But a famous event in the history of mathematics was the proof that the quintic equation didn't always have solutions that were expressible by radicals. x^5 = 7 has the fifth root of 7 as its solution, so some quintic equations do have such solutions - but the equation x^5 + x + a = 0 does not.
Numbers which are the solutions of equations of the form a*(x^n) + b*(x^(n-1)) + c*(x^(n-2)) + ... z = 0, starting with any n, however large, where all the coefficients a, b, c... are rational numbers, are called algebraic numbers.
But even the algebraic numbers don't cover all the possible values for such things as a distance along a line. The circumference of a circle of unit diameter, which is pi, the number we are talking about, is one of those numbers that isn't algebraic, and that means it is called a transcendental number. The first proof that pi was transcendental was due to Lindemann in 1882.
As noted, faster methods of calculating pi were discovered later.
The following series for 1/pi was discovered by the enigmatic Srinavasa Ramanujan:
infinity
_________
\
2 * sqrt(2) \ (4*i)! (1103 + 26390 * i)
----------- * > -------------- * --------------------
9801 / 4 (4*i) (4*i)
/________ (i!) *4 99
i = 0
Yes, 4^(4*i) and 99^(4*i) could be combined to form 396^(4*i); also, 9801 is the square of 99. so this formula can appear in other forms. It was used by R. William Gosper to calculate pi to over 17.5 million digits in 1985. A different series belonging to this class was used by the Chudnovsky brothers, and it has since been used in other calculations for pi:
infinity
__________
\
\ i (6*i)! 13591409 + 545140134 * i
12 * > (-1) * --------------- * --------------------------
/ 3 ((3*i) + (3/2))
/_________ (i!) * (3*i)! 640320
i = 0
again, this is a series for 1/pi.
The denominator of the second factor involves operations such as taking the cube and the square root of 640320 that don't have to be repeated for each term of the sum. Since 640320 is 320 * 2001, 320 being 64 * 5 and 2001 being 3 * 23 * 29, this leads to:
infinity
__________
\
1 \ i (6*i)! 13591409 + 545140134 * i
---------------------- * > (-1) * --------------- * --------------------------
426880 * sqrt(10005) / 3 i
/_________ (i!) * (3*i)! 262537412640768000
i = 0
and it also appears in this form or similar forms in addition to the original form which is somewhat shorter to write, if not to calculate.
How these series are derived involves a number of topics of mathematical interest, as noted in the Wikipedia article on Ramanujan-Sato series.
Series for 1/pi of the type discussed above yield a large number of digits of pi at each step; but since the same number of digits of pi are produced at each step, it might be possible to equal their performance with a suitably-designed arctangent relation in which all the arguments to the arctangent function are sufficiently small numbers.
Recurrence relations are now known, however, that double, triple, quadruple, or more, the number of digits calculated at each step.
The first such relation to be discovered was found by Eugene Salamin and Richard Brent in 1976; they both discovered it independently around the same time.
To find pi by this method, follow this recurrence relation:
a(0) = 1 b(0) = 1/sqrt(2) s(0) = 1/2 a(n+1) = (a(n)+b(n))/2 b(n+1) = sqrt(a(n)*b(n)) s(n+1) = s(n) - (2^(n+1)) * (a(n+1)^2 - b(n+1)^2)
and after iterating it as many times as required to obtain the precision sought, the approximate value of pi will be 2 * (a(N)^2)/s(N).
This recurrence relation includes the process of taking the arithmetic-geometric mean of a and b. While this method of calculating pi was only discovered in 1976, a recurrence relation involving the arithmetic-geometric mean was long known and used for calculating complete elliptic integrals.
For that purpose, this recurrence relation is applied:
a(0) = 1 b(1) = cos(t) a(n+1) = (a(n)+b(n))/2 b(n+1) = sqrt(a(n)*b(n))
and then K(t) is approximated by pi/(2 * a(N)), where N is the number of iterations taken to achieve the desired precision.
E(t) can also be determined, from the formula
infinity
________
K(t) - E(t) 1 \ i
------------- = --- * (sin(t) + > 2 * c(i)^2 )
K(t) 2 /_______
i = 1
where c(i) is defined as (1/2) * (a(i) - b(i)).
Sometimes, the form of the expression is simplified by defining c(0) as sin(x), although that doesn't correspond to the definition of c(i) for other values of i, so that the sum can start from i=0 without an additional term.
In these formulas, t represents the modular angle, often noted by the Greek letter alpha; the argument of the elliptic integral may also be the parameter, noted by the letter m. They are related by m = (sin(t))^2.
Later, a number of improved recurrence relations for pi with even faster convergence were developed by Jonathan and Peter Borwein and others; they developed algorithms which tripled, quadrupled, quintupled, multiplied by seven, and multiplied by nine, the number of correct digits at each iteration. Here is the one that multiplies the number of correct digits by nine at each step:
a(0) = 1/3 r(0) = (sqrt(3) - 1)/2 s(0) = (1 - (r(0)^3))^(1/3) t(n+1) = 1 + 2*r(n) u(n+1) = (9 * r(n) * (1 + r(n) + r(n)^2))^(1/3) v(n+1) = t(n+1)^2 + t(n+1)*u(n+1) + u(n+1)^2 w(n+1) = 27 * (1 + s(n) + s(n)^2)/v(n+1) a(n+1) = w(n+1)*a(n) + ((3^(2*n - 1)) * (1 - w(n+1))) s(n+1) = (1 - r(n))^3 / ((t(n+1) + 2 * u(n+1)) * v(n+1)) r(n+1) = (1 - s(n+1)^3)^(1/3)
The result of this computation is that a(N) is the approximation to 1/pi that improves at this large rate.
Here are the other relations of this type that are known:
the one that triples the number of correct digits at each step is:
a(0) = 1/3 s(0) = (sqrt(3) - 1)/2 r(n+1) = 3/(1 + 2 * ((1 - s(n)^3)^(1/3))) s(n+1) = r(n+1)/2 a(n+1) = (r(n+1)^2 * a(n)) - 3^n * (r(n+1)^2 - 1)
Here, a(N) is the improving approximation to 1/pi.
the one that quadruples the number of correct digits at each step is:
a(0) = 6 - 4*sqrt(2) y(0) = sqrt(2) - 1 y(n+1) = (1 - (1 - y(n)^4)^(1/4))/(1 + (1 - y(n)^4)^(1/4)) a(n+1) = a(n) * (1 + y(n+1))^4 - 2^(2*n + 3) * y(n+1) * (1 + y(n+1) + y(n+1)^2)
Once again, a(N) is the improving approximation to 1/pi.
the one that quintuples the number of correct digits at each step is:
a(0) = 1/2 s(0) = 5 * (sqrt(5) - 2) x(n+1) = (5/s(n)) - 1 y(n+1) = (x(n+1) - 1)^2 + 7 z(n+1) = ((1/2) * x(n+1) * ( y(n+1) + sqrt( y(n+1)^2 - 4*(x(n+1)^3) ))^(1/5) a(n+1) = s(n)^2 * a(n) - (5^n) * ( (s(n)^2 - 5)/2) + sqrt( s(n) * (s(n)^2 - 2*s(n) + 5) ) s(n+1) = 25/( (z(n+1) + (x(n+1)/z(n+1)) + 1)^2 * s(n) )
and, again, a(N) converges to 1/pi.
The one that multiplies the number of correct digits by seven at each step also requires evaluating a trigonometric function to the same high precision as is desired for the approximation to pi, and thus only provides a gain in efficiency if a similarly fast algorithm for calculating the cosine is known. Another algorithm which also multiplies the number of correct digits by seven at each step required a trigonometric function to be evaluated in every step.
Thus, the quintic and nonic algorithms appear to be the most rapid ones available, although there may also be one that multiplies the number of correct digits by sixteen at each step.
The algorithm developed by Brent and Salamin was the one used in 2009 for a calculation of pi to over 2.5 billion digits by a team led by Daisuke Takashi on the T2K Open Supercomputer. This was the last record-setting calculation of pi done on a large, expensive computer: subsequent ones have been done by private enthusiasts on commodity hardware.
The first of these was one to almost 2.7 billion digits by Fabrice Bellard later in 2009. All subsequent ones, in addition to using commodity hardware, also used the y-cruncher software written by Alexander Yee. Surprisingly, although this meant the calculations took longer, many of them were done using the Chudnovsky series, with a second faster iterative calculation being performed only for verification, instead of using two different iterative calculations.
While deriving these fast methods of calculating the value of pi is an impressive accomplishment concerning the computation of pi, this is not the most famous accomplisment of the Borwein brothers in connection with calculating pi.
Incidentally, their father, David Borwein, is also a noted mathematician. He was born in Kaunas, Lithuania, in 1924, but was taken by his parents to South Africa in 1930. In 1948, he moved to London, where his graduate studies took place; from 1963, when he initially came to Canada as a visiting professor, he lived in Canada where he held a number of distinguished posts, including that of the head of the mathematics department of the University of Western Ontario and that of the President of the Canadian Mathematical Society. His own mathematical work dealt with classical analysis, including such things as the summability of integrals.
Jonathan Borwein passed away unexpectedly and suddenly in 2016 at the age of 65 of natural causes, and was survived by his father as well as his brother Peter.
Jonathan and Peter Borwein, together with Simon Plouffe, also developed this series for pi:
infinity __________ \ / \ \ 1 | 4 2 1 1 | > ----- * | --------------- - --------------- - --------------- - --------------- | / i | ((8 * i) + 1) ((8 * i) + 4) ((8 * i) + 5) ((8 * i) + 6) | /_________ 16 \ / i = 0
For i=2, for example, the term would be 1/256 * (4/17 - 2/20 - 1/21 - 1/22).
Now, neither 4/17 nor 1/21 nor 1/22 are fractions which have a terminating representation in hexadecimal notation any more than they would in decimal notation. (For that matter, 2/20, or 1/10, terminates in decimal notation, but it doesn't terminate either in hexadecimal notation.)
However, it is still possible to use this series in an efficient procedure for calculating hexadecimal digits of pi at an arbitrary position without needing to also calculate the digits which come before them, even if how to do this is not trivial.
An improved formula of this type was subsequently developed by Fabrice Bellard:
infinity
__________
\ i / \
1 \ (-1) | 256 64 4 4 1 32 1 |
---- * > ------- * | ---------------- - ---------------- - ---------------- - ---------------- + ---------------- - --------------- - --------------- |
64 / i | ((10 * i) + 1) ((10 * i) + 3) ((10 * i) + 5) ((10 * i) + 7) ((10 * i) + 9) ((4 * i) + 1) ((4 * i) + 3) |
/_________ 1024 \ /
i = 0
Initially, when the first of these series was discovered, there was optimism that a similar formula for calculating high-order decimal digits of pi directly might also be found.
Later, however, a proof was found that a formula of this particular type could not exist for bases other than powers of two because these formulas are related to arctangent relations for pi, such as the identity of Machin noted above, which do not exist in a suitable form for other bases.
Still later, though, a formula of a different type was found that allowed direct calculation of high-order decimal digits of pi. The first such one was found by Simon Plouffe; however, it was quite slow, as the time it would take is proportional to the cube of the position of the digit to be found.
Improved algorithms for high-order decimal digits of pi have since been found, but they are still too slow to be competitive with calculating the entire number. However, they are still of theoretical importance, as it might be possible to gain insights from them for a proof, one way or another, addressing the still unsolved question of whether the digits of pi behave statistically like a sequence of random digits: whether or not pi is normal.
As an alternative way of calculating pi, it occured to me that if there was a high-speed arithmetic-geometric mean algorithm for calculating the arctangent, one could use a Machin-type arctangent relation in connection with that, instead of with the Gregory series.
The first such method I found described in an early paper on the subject of efficient algorithms for the elementary functions required the use of an algorithm for calculating the logarithm that required the value of pi to the full length of the numbers being worked with, but I later found a web page with one without this limitation:
a(0) = sqrt(1 + x^2) b(0) = 1 a(n+1) = (1/2) * (a(n)+b(n)) b(n+1) = sqrt( a(n+1) * b(n) )
with the arctangent of x being approximated by x/(a(N) * sqrt(1+x*x)), where N is the number of iterations carried out, originally given in Numerical Methods that Work by Forman S. Acton, as cited on the Wolfram Research web site.
Machin-like identies have been found that converge even more quickly than the ones given above as having been used in practice. One that I saw which would converge particularly quickly, even using the Gregory series, was this one:
atn(1) = 183*atn(1/239) + 32*atn(1/1023) - 68*atn(1/5832) + 12*atn(1/113021) - 100*atn(1/6826318) - 12*atn(1/33366019650) + 12*atn(1/43599522992503626068)
so there is an arctangent identity with atn(1/239) as the slowest-converging term; this identity was derived by Hwang Chien-Lih in 1997, but he has since derived even better ones, the best one I have seen anywhere having been found by him in 2004:
atn(1) = 36462*atn(1/390112) + 135908*atn(1/485298) + 274509*atn(1/683982) - 39581*atn(1/1984933) + 178477*atn(1/2478328) - 114569*atn(1/3449051) - 146571*atn(1/18975991) + 61914*atn(1/22709274) - 69044*atn(1/24208144) - 89431*atn(1/201229582) - 43938*atn(1/2189376182)
at least, that was the best one I had seen anywhere, when I came across it on Wikipedia, but then I found his web site on the subject, which includes even better ones, such as:
atn(1) = 20435891*atn(1/28841295) + 6458959*atn(1/133311327) + 10571216*atn(1/152087733) - 190529*atn(1/159358932) + 5127580*atn(1/213495433) + 7798670*atn(1/227661182) - 3154610*atn(1/278263393) - 14603096*atn(1/284862638) - 5959396*atn(1/355671793) - 10585611*atn(1/727507932) - 4139726*atn(1/934981432) + 1560722*atn(1/15234751332) - 2675312*atn(1/106334643058) + 6769449*atn(1/138873731225) + 2229180*atn(1/293153860797) - 2967682*atn(1/169838669284032)
Since the successive terms in Gregory's series for the arctangent of x involve x, x^3, x^5, and so on, being multiplied by x^2 each time, having x as one over an eight-digit number means that each step involves multiplying by one over a sixteen digit number, so, although there are more series to sum, this is enough to make Gregory's series comparable to Chudnovsky's series, if still not necessarily quite as fast.
Also, incidentally, since
atn(1) = 183*atn(1/239) + 32*atn(1/1023) - 68*atn(1/5832) + 12*atn(1/113021) - 100*atn(1/6826318) - 12*atn(1/33366019650) + 12*atn(1/43599522992503626068)
and
atn(1) = 4*atn(1/5) - atn(1/239)
we can calculate atn(1/5) as an intermediate result:
atn(1/5) = 46*atn(1/239) + 8*atn(1/1023) - 17*atn(1/5832) + 3*atn(1/113021) - 25*atn(1/6826318) - 3*atn(1/33366019650) + 3*atn(1/43599522992503626068)
which is not surprising, as one expects that these formulas are obtained by repeated application of the tangent addition formula.
Similarly, since
atn(1) = 12*atn(1/18) + 8*atn(1/57) - 5*atn(1/239)
and
atn(1) = 4*atn(1/5) - atn(1/239)
we can also say:
atn(1/5) = 3*atn(1/18) + 2*atn(1/57) - atn(1/239)
Thanks to another identity by Störmer,
atn(1) = 44*atn(1/57) + 7*atn(1/239) - 12*atn(1/682) + 24*atn(1/12943)
knowing also that
atn(1) = 12*atn(1/18) + 8*atn(1/57) - 5*atn(1/239)
we can deduce that
atn(1/18) = 3*atn(1/57) + atn(1/239) - atn(1/682) + 2*atn(1/12943)
and so we have component relations that can be used to trace a line of descent for that identity from Machin's original formula.
Hunting around, I've found another related identity on the web:
atn(1) = 88*atn(1/172) + 51*atn(1/239) + 32*atn(1/682) + 44*atn(1/5357) + 68*atn(1/12943)
so now we can find an expression for atn(1/57) in terms of arctangents of smaller quantities.
However, there's another formula that starts with atn(1/239), unlike the one I first encountered, that is related (it may be due to Jörg Arndt):
atn(1) = 183*atn(1/239) + 44*atn(1/515) - 56*atn(1/682) + 88*atn(1/6050) + 24*atn(1/12943) - 88*atn(6826318)
With that and
atn(1) = 44*atn(1/57) + 7*atn(1/239) - 12*atn(1/682) + 24*atn(1/12943)
we can deduce
atn(1/57) = 4*atn(1/239) + atn(1/515) - atn(1/682) + 2*atn(1/6050) - 2*atn(6826318)
Thus, we now have a series of formulas starting with Machin's original formula, and ending with one the slowest-converging term of which is atn(1/239), where we can follow how the slowest-converging term is replaced with faster-converging ones every step of the way:
atn(1) = 4*atn(1/5) - atn(1/239)
+ 4 * 0 = - atn(1/5) + 3*atn(1/18) + 2*atn(1/57) - atn(1/239)
-----------------------------------------------------------------------
atn(1) = 12*atn(1/18) + 8*atn(1/57) - 5*atn(1/239)
+ 12 * 0 = - atn(1/18) + 3*atn(1/57) + atn(1/239) - atn(1/682) + 2*atn(1/12943)
-------------------------------------------------------------------------------------------------------------------------------------------
atn(1) = 44*atn(1/57) + 7*atn(1/239) - 12*atn(1/682) + 24*atn(1/12943)
+ 44 * 0 = - atn(1/57) + 4*atn(1/239) + atn(1/515) - atn(1/682) + 2*atn(1/6050) - 2*atn(6826318)
-------------------------------------------------------------------------------------------------------------------------------------------------------------
atn(1) = 183*atn(1/239) + 44*atn(1/515) - 56*atn(1/682) + 88*atn(1/6050) + 24*atn(1/12943) - 88*atn(6826318)
Yet another possible arctangent relation, closely related to the one examined here, is:
atn(1) = 183*atn(1/239) - 12*atn(1/682) + 44*atn(1/1240) + 24*atn(1/12943) - 44*atn(1/2485057) - 88*atn(6826318)
which can be considered slightly improved, since now when atn(1/239) is removed at the next step, the slowest-converging term would be atn(1/682) instead of atn(1/515). This relation was developed by Michael Roby Westerfield, a collaborator with Hwang Chien-Lih.
Here is a record of the notable errors in the computation of pi:
The value of pi given by de Lagny to 126 places in 1719 had a 7 instead of an 8 in the 113th place, giving 32723 instead of 32823 in positions 111 to 115.
The value of pi given by Georg Freyherrn von Vega (or Jurij Vega) in 1789 was:
3.14159 26535 89793 23846 26433 83279 50288 41971 69399 37510 58209 74944 59230 78164 06286 20899 86280 34825 34211 70679 82148 08651 32823 06647 09384 44767 21386 11733 138
in 1794, he corrected his error. He is best known for having calculated a set of 10-figure logarithm tables which were of inestimable value to generations of mathematicians and scientists.
The value of pi given by William Rutherford in 1851 was:
3.14159 26535 89793 23846 26433 83279 50288 41971 69399 37510 58209 74944 59230 78164 06286 20899 86280 34825 34211 70679 82148 08651 32823 06647 09384 46095 50582 23172 53594 08128 48473 78139 20386 33830 21574 73996 00825 93125 91294 01832 80651 744
This value was first corrected by Zacharias Dase; later, William Rutherford calculated pi to more places, without error.
The 707 digits computed by William Shanks, of which the first 527 were correct, which appeared in countless sources, sometimes with misprints of various types, from 1873 onwards, were:
3.14159 26535 89793 23846 26433 83279 50288 41971 69399 37510 58209 74944 59230 78164 06286 20899 86280 34825 34211 70679 82148 08651 32823 06647 09384 46095 50582 23172 53594 08128 48111 74502 84102 70193 85211 05559 64462 29489 54930 38196 44288 10975 66593 34461 28475 64823 37867 83165 27120 19091 45648 56692 34603 48610 45432 66482 13393 60726 02491 41273 72458 70066 06315 58817 48815 20920 96282 92540 91715 36436 78925 90360 01133 05305 48820 46652 13841 46951 94151 16094 33057 27036 57595 91953 09218 61173 81932 61179 31051 18548 07446 23799 62749 56735 18857 52724 89122 79381 83011 94912 98336 73362 44065 66430 86021 39501 60924 48077 23094 36285 53096 62027 55693 97986 95022 24749 96206 07497 03041 23668 86199 51100 89202 38377 02131 41694 11902 98858 25446 81639 79990 46597 00081 70029 63123 77381 34208 41307 91451 18398 05709 85
In the December 18, 1873 issue of the Proceedings of the Royal Society of London, in which typographical errors in the original publication of Shanks' value were noted, these digits appeared, except that the 679th digit was a 7 instead of a 1, giving 77387 for the 675th through 679 digits.
That digit, not among the typographical errors noted, however, appeared as a 1 in the original publication of Shanks' value in the May 15, 1873 issue of that journal, and thus the most reliable reproductions of Shanks' value give 1 as the digit in that position.
William Shanks also gave the values of the arctangents of 1/5 and of 1/239 which he calculated in order to obtain pi, thus helping those who would seek to locate the error in his calculatiion.
His value of the arctangent of 1/5, with the digits in error shown in bold, was:
0.19739 55598 49880 75837 00497 65194 79029 34475 85103 78785 21015 17688 94024 10339 69978 24378 57326 97828 03728 80441 12628 11807 36913 60104 45647 98867 94239 35574 75654 95216 30327 00522 10747 00156 45015 56006 12861 85526 63325 73186 92806 64389 68061 89528 40582 59311 24251 61329 73139 93397 11323 35378 21796 08417 66483 10525 47303 96657 25650 48887 81553 09384 29057 93116 95934 19285 18063 64919 69751 94017 08560 94952 73686 73738 50840 08123 67856 14800 93298 22514 02324 66755 49211 02670 45743 78815 47483 90799 78985 02007 52236 96837 96139 22783 54193 25572 23284 13846 47744 13529 09705 46512 24383 02697 56051 83776 17781 64242 33783 03370 18192 64880 28277 68611 91509 85606 75901 21359 85563 63034 32100 56649 97826 76360 88711 52327 56610 84900 93773 38023 19504 70687 65729 38513 59243 19759 37947 36057 50636 20935 07853 2833
which value includes a correction to the 75th digit, which was originally printed in error as a 7 instead of an 8 (69977 instead of 69978).
and his value of the arctangent of 1/239 was:
0.00418 40760 02074 72386 45382 14959 28545 27410 48065 30763 19508 27019 61288 71817 78341 42289 32737 82605 81362 29094 54975 45066 64448 63756 05245 83947 89311 86505 89221 28833 09280 08462 71962 33077 33759 47634 60331 84734 14570 33198 60154 54814 80599 24498 30211 46039 12539 49527 60779 68815 58881 27339 78533 46518 04574 25481 35867 46447 51979 10232 83097 70020 64652 82763 46532 96910 48183 86543 56078 91959 14512 32220 94463 68627 66155 20831 67964 26465 74655 11032 51034 35262 82445 12693 55670 49968 44452 47904 33177 28393 07086 31401 93869 51950 37058 64107 70855 85540 45223 55388 14237 67708 36515 69182 52702 00229 30895 44950 04358 54409 34496 44014 24187 24950 92283 86239 54553 33565 16494 21220 06852 38821 94006 45849 29313 23886 73467 64889 18731 81682 83021 21101 37897 11546 96191 84692 18237 33903 04682 04140 79985 6684
In 1853, he had Contributions to Mathematics, Consisting Chiefly of the Rectification of the Circle to 607 Places of Decimals published. It included arctan(1/5) and arctan(1/239) to 609 places, which both were largely in agreement with the later 707-place values, that of arctan(1/5) to 601 of its 609 places, but that of arctan(1/239) agreed in its entirety.
Since his value of arctan(1/5) was only correct to 529 places, that portion of his error in the 707-digit value had already happened in that calculation; the value of arctan(1/5), used in the 707-digit calculation of pi was correct to 591 places, so that part of the error had also already happened.
The book also included the individual terms of the arctangent series that he calculated, but these were only given to 527 places, and so they do not give an opportunity to check the part of his calculation that was in error. An offer was made in the text to provide supplementary sheets giving the details of the latter portion of the calculation, but as this offer appears not to have been taken up, the error in the 607-digit calculation of pi, also present in the 707-digit calculation of pi, was not noted until much later, and so there was an opportunity that was missed.
The value of pi to 808 digits as first given by D. F. Ferguson in March, 1947, had several individual digits in error starting from the 723rd digit, but then continued on with digits that were correct.
The correct value of pi to 810 digits is:
3.14159 26535 89793 23846 26433 83279 50288 41971 69399 37510 58209 74944 59230 78164 06286 20899 86280 34825 34211 70679 82148 08651 32823 06647 09384 46095 50582 23172 53594 08128 48111 74502 84102 70193 85211 05559 64462 29489 54930 38196 44288 10975 66593 34461 28475 64823 37867 83165 27120 19091 45648 56692 34603 48610 45432 66482 13393 60726 02491 41273 72458 70066 06315 58817 48815 20920 96282 92540 91715 36436 78925 90360 01133 05305 48820 46652 13841 46951 94151 16094 33057 27036 57595 91953 09218 61173 81932 61179 31051 18548 07446 23799 62749 56735 18857 52724 89122 79381 83011 94912 98336 73362 44065 66430 86021 39494 63952 24737 19070 21798 60943 70277 05392 17176 29317 67523 84674 81846 76694 05132 00056 81271 45263 56082 77857 71342 15778 96091 73637 17872 14684 40901 22495 34301 46549 58537 10507 92279 68925 89235 42019 95611 21290 21960 86403 44181 59813 62977 47713 09960 51870 72113 49999 99837 29780 49951 05973 17328 16096 31859 50244 59455
and that initially given by D. F. Ferguson with the incorrect digits shown in bold and underlined as well, to make these intermittent incorrect digits easier to pick out, is:
3.14159 26535 89793 23846 26433 83279 50288 41971 69399 37510 58209 74944 59230 78164 06286 20899 86280 34825 34211 70679 82148 08651 32823 06647 09384 46095 50582 23172 53594 08128 48111 74502 84102 70193 85211 05559 64462 29489 54930 38196 44288 10975 66593 34461 28475 64823 37867 83165 27120 19091 45648 56692 34603 48610 45432 66482 13393 60726 02491 41273 72458 70066 06315 58817 48815 20920 96282 92540 91715 36436 78925 90360 01133 05305 48820 46652 13841 46951 94151 16094 33057 27036 57595 91953 09218 61173 81932 61179 31051 18548 07446 23799 62749 56735 18857 52724 89122 79381 83011 94912 98336 73362 44065 66430 86021 39494 63952 24737 19070 21798 60943 70277 05392 17176 29317 67523 84674 81846 76694 05132 00056 81271 45263 56082 77857 71342 15778 96091 73637 17872 14684 40901 22495 34301 46549 58537 10507 92279 68925 89235 42019 95611 21290 21960 86355 44191 19716 02977 46113 09960 51870 72113 49999 99837 29780 49951 05973 17328 16096 31867 50244 59455
The last two digits of these 810 digits were in parentheses, indicating that their value was uncertain; these appeared in March 1947, and he corrected himself by September 1947.