the story of pi
by lazarus mudehwe

undoubtedly, pi is one of the most famous and most remarkable numbers you have ever met.  the number, which is the ratio of circumference of a circle to its diameter, has a long story about its value.

even nowadays supercomputers are used to try and find its decimal expansion to as many places as possible.  for pi is one of those numbers that cannot be evaluated exactly as a decimal --- it is in that class of numbers called irrationals.

the hunt for pi began in egypt and in babylon about two thousand years before christ.  the egyptians obtained the value (4/3)^4 and the babylonians the value 3 1/8 for pi .  about the same time, the indians used the square root of 10 for pi.  these approximations to pi had an error only as from the second decimal place.

the next indication of the value of pi occurs in the bible.  it is found in 1 kings chapter 7 verse 23, where using the authorized version, it is written "... and he made a molten sea, ten cubits from one brim to the other: it was round about ... and a line of thirty cubits did compass it round about."  thus their value of pi was approximately 3.  even though this is not as accurate as values obtained by the egyptians, babylonians and indians, it was good enough for measurements needed at that time.

jewish rabbinical tradition asserts that there is a much more accurate approximation for pi hidden in the original hebrew text of the said verse and 2 chronicles 4:2.  in english, the word 'round' is used in both verses.  but in the original hebrew, the words meaning 'round' are different. now, in hebrew, etters of the alphabet represent numbers.  thus the two words represent two numbers. and - wait for this - the ratio of the two numbers represents a very accurate continued fraction representation of pi! question is, is that a coincidence or ...

another major step towards a more accurate value of pi was taken when the great archimedes put his mind to the problem about 250 years before christ.  he developed a method (using inscribed and circumscribed 6-, 12-, 48-, 96-gons) for calculating better and better approximations to the value of pi, and found that 3 10/71 < pi < 3 10/70.  

today we often use the latter value 22/7 for work which does not require great accuracy.  we use it so often that some people think it is the exact value of pi!

the last approximation is so good (9dp) that my ancient casio calculator tells me it's the same as pi! (sadly, many people would believe my calculator).

finding info on the web is one of the easiest tasks in existence.  steve berlin has a nice article, and this site offers software that can be used to get pi to plenty decimal places.  want to change the value of pi ?  sorry, the voting is over, but the results are here.  i could go on and on, but instead i'll just leave you with the albany pi club which has several links, including the brilliant uselessness of pi page and the recently started joy of pi page.

ancient pi ():  knowers of the universe
by charles william johnson

any practical attempt to divide the diameter of a circle into its own circumference can only meet with failure.  such a procedure is entirely theoretical in nature.  dividing unlikes, a straight line (the diameter of a circle) into a curved line (the circumference of a circle) can only be met with frustration.  

the kind of frustration that is portrayed throughout history in humankind's attempt to measure the incommensurable.  no matter how hard one may try, even with the assistance of contemporary electronic computers, bending either the straight line or the curved line, alters the nature of the problem and yields an impossibility.  as soon as one of the lines is bent the results are tainted.  

then, there is the question of the very thickness of the lines being measured in length.  whether one measures the inner part of the curved line of the circumference or the outer edge makes a great deal of difference; especially, when one is attempting to achieve an exactness in the concept of pi () to hundreds or even thousands of decimal places.

if we realize that the measurement of the ratio between the diameter and the circumference of a circle is entirely theoretical and speculative, then we may also realize that the result shall always represent an approximation.  in fact, the very fact that pi is always expressed in terms of an unending fraction (with mathematicians searching it to the nth number of decimal places), should cause us to accept the idea that pi can only be an approximation. (as lambert illustrated in 1767, " is not a rational number, i.e., it cannot be expressed as a ratio of two integers"; beckmann, p.100.)

once we realize that pi represents a fractional expression in numbers, it were as though either nature itself were wrong, or the numbers must surely be able to be manipulated to render whole numbers.  the ancients sought to work with whole numbers.  however, once we realize that the ancient reckoning system may have been based upon the concept of a floating decimal place, then we should understand that all numbers, in fact, may be visualized as whole numbers. the cut-off point becomes one of arbitrary choice at times.  

with regard to the concept of pi , contemporary mathematicians have not decided to accept that arbitrary cut-off point, and continue to search for the unending decimal expression of pi.  

at one time, not too long ago, pi was simply represented to be 3.1416 ; and, in a practical sense, it served all purposes of constructing things out of matter and energy.  

today, the unending expression of pi to hundreds of thousands of decimal places serves no practical purpose that we know of, at least, other than that of an unending contest to discover the ultimate expression of pi.  one has only to admire the relation of the diameter of any circle to its circumference to note that particular expression.

throughout history, the expression of pi has taken on many variations. petr beckmann (cfr., a history of (pi), golem, 1971), offers an exemplary analysis of the concept throughout history. the babylonians 3 1/8; the egyptians 4(8/9) ²; siddhantas, 3.1416; brahmagupta, 3.162277; chinese, 3.1724; liu hui, 3.141024 < < 3.142704; liu hui, 3.14159; tsu chung-chih, 3.1415926 < < 3.1415927; archimedes, 3.14084 < < 3.142858 (3 1/7); heron, 3.1738; ptolemy, 3.14167; fibonacci, = 864:275 = 3.141818; vi�te, 3.141592635 < < 3.1415926537; and, finally in the computer language of fortan: 3.14159265358979324. 

again, citing beckmann (p.101):  "there is no practical or scientific value in knowing more than the 17 decimal places used in the foregoing, already somewhat artificial, application".

nonetheless, in 1844, johann martin zacharias dase calculated to 200 decimal places, with the first zero appearing at the 32nd decimal place ---meaning, possibly that the exercise should have ended there. it has not; just as pi is an unending fraction, so is the human practice of finding the number of unending decimal places in pi.

decimal hunting games aside, the practical uses of knowing pi (the ratio of the diameter of a circle to its circumference) even as an approximation has infinite applications in astronomy. and, the ancients were on the whole astronomers; knowers of the universe.  this ratio becomes significant in calculating the movements of the planets and the stars; in computing their coming and going in the sky.  once more, since we are dealing with movement, the movement of the planetary bodies and the stars, we are always speaking about approximations; even in and especially so in astronomy.  therefore, the approximations to pi serve a purpose in knowing the approximate movements of the planets.  such are the problems concerning the measurement of moving bodies. as soon as they have been measured, they have already moved from that measurement.

when we observe the measurements offered by tsu chung-chih given above, it becomes obvious that ancient approximations were at times far ahead of latter day computations.  and, then there is the problem that one may obtain pi to the nth decimal place, but such decimal expressions are beyond the human capacity to measure or even observe matter-energy to such a minute degree.

in our analyses, we cannot cite any specific ancient documents for the computation of pi among the ancients.  yet, the historically significant numbers that do exist within the ancient reckoning systems may reveal some partial aspects of the computations themselves.  no matter which contemporary studies we examine, pi is always given in relation to the number ca. 3.1-something, as a guidepost. yet, it may be the case that the ancients conceived of pi in relation to the number of divisions that made up the circle; the number of degrees or segments contained therein.

the concept of pi refers to the constant ratio of the diameter: circumference of any circle; irrespective of the number of degrees contained within that circle.  historically, the babylonians came to use the number 360 for the divisional segments within a circle, and we have employed that same number ever since.  the abstracted universal circle, then, would have a constant diameter of one (1.0), and the length of its circumference would be pi () of that: 3.1-something (whichever one might choose). hence, diameter is 1.0 in length; while, circumference is 3.141592654 (for example) in length.

now, if we consider the circumference to be divided into 360 degrees (or segments; angular divisions with lines cutting through the center of the circle as we know them), then using the contemporary figure for pi (3.141592654), the length of the circumference could be 360 units, while the length of the diameter would be 114.591559 (i.e., 360 /).

now, let us suppose that the circle is divided into 260 degrees (something that we are unaccustomed to considering, in fact). if we employ the same length of the diameter of the previous example (114.591559), then the relational figure for pi for a 260-degree circle would be: 2.268928028.  with that something very intriguing develops.  within ancient nineveh, there exists an historically significant cited as 2268.  

one could imagine that the 2268 fractal number may relate to the concept of proportion (i.e., pi ) regarding a 260-division circle.  the number 260 is relevant because during ancient times there existed in various cultures a calendar based on a 260c day-count.  furthermore, the great cycle of the sun, known as precession, also involves a fractal of 260 (i.e., 26000 years). now, were we to consider the nineveh number for representing pi on a 260-degree circle, then the constant value for the diameter would then be 114.638448 (i.e., 260 / 2.268).

throughout history, an inexact representation of pi has always been cited as that of 3 1/7 (or, 3.142857); a reciprocal of seven number.  however, when we consider that the length of the diameter of a 360-degree circle yields a number that approximates a reciprocal of seven number (114.591559), we can consider the possibility of employing 114.285714 in its place.

the use of the reciprocal of seven number (114.284714) for the length of the 360 and 260 circle would offer the following values for pi, respectively:
.

note that the 3.15 number offers a mediatio/duplatio series based on the 63c, which was significant in ancient reckoning systems:  315, 630, 1260, 2520, 5040, 10080, 20160, 40320, 80640, 161280, 322560, 645120,1290240, 2580480 (a precession number/fractal); and, 63, 126, 189, 252, 315, 378, 441, 504, 567 (kemi), 630, 693 (sothic), 756 (giza), 819 (k'awil; maya), 882, 945, 1008, 1071, 1134 (nineveh, 2 x 1134 = 2268), 1197, etc.

note that the 2275 fractal number is relevant for the computational series within the ancient reckoning system of the 364c day-count:  2275, 4550, 9100, 18200, 36400, etc. also, note that the difference between the nineveh 2268c and the pi-like number 2275 is seven (2275 - 2268 = 7); which could be easily translated from one series to the other by remainder math based on multiples of seven.

many of the distinctive historically significant numbers of the ancient reckoning system reflect a relationship based on the reciprocal of seven.  consider the maya long count period number of 1872000, which has received so much speculation regarding its beginning and ending date.  also, consider the period called the k'awil of the maya cited as consisting of 819c days.

now, notice the number that obtains from the division resulting from half of the long count period figure by the k'awil: 936 / 819 = 1.142857143.  the same figure obtains regarding the constant length of a diameter of a circle based on a pi-like number in relation to the reciprocal of seven as explained earlier.

other relationships obtain regarding similar historically significant numbers from other systems.  the great pyramid entails the number 756c as its baseline.  also, there exists the 432c number/fractal associated with the consecration.  if we double the 432 figure and divide by the 756c, the same result obtains: 864 / 756 = 1.142857143.

consider: 360 x .864 = 311.04 (31104 being an historically significant number for china and mesoamerica).

the significance of seven and its reciprocal becomes obvious throughout the historically significant numbers/fractals.  even the obvious relationship, of the 364c day-count of ancient mesoamerica, which was employed for computations, reveals a direct basis of seven: 364 / 7 = 52. immediately, one will recognize the 52c that is so well-known in ancient mesoamerica as the calendar round (52 years times 365 days = 18980 days; and 52 years times 360 days = 18720! days).  and, the ancient kemi appear to have employed a 54c in its place: 7 times 54 = 378 (2 x 378 = 756; or, 7 x 108 = 756).  no matter where one turns, the number seven and its reciprocal make their appearance. the reasoning behind this procedure may be rather obvious, although we have not discerned it previously.

the number 1.142857143 concerns the ratio 8/7ths.  the aztec calendar appears to be based upon a spatial division that reflects the logic of 7:8 or 8:7, depending upon the rings and segments to be considered (cfr., earth/matrix no.88). if one were attempting to consider the diameter of the solar system, or the universe, knowing that these events consist of imaginary circles (ellipses), then the use of the unit 1.0 for the length of their respective diameters would not be of much value.  and, furthermore, if the ancients had employed the contemporary (and possibly past) concept of pi (based on a close approximation to 3.141592654, give or take a fraction), then the numbers would have been unmanageable and not very attractive.

the apparent relational aspects of the many different historical numbers found in the many distinctive ancient reckoning systems suggest a common origin and reasoning. if the length of the diameter of the solar system or the universe were assigned a value consisting of the reciprocal of seven (i.e., 1.142857143), then this would be the next best thing to working with whole numbers for computing the time cycles of the movement of the planetary bodies and the stars. 

furthermore, knowing the actual measurement of pi (the exact proportion of the diameter: circumference ratio) could be compensated with remainder math adjustments quite easily. consider the following computations:

one of the most interesting relationships of this nature concerns the 2268c nineveh count:
.

scholars consider the figure of 3 1/7ths to have been an erroneous computation for pi.  yet, we have never really known how the ancients computed their mathematics.  the few documents that remain (such as the rhind document of the ancient kemi) concern everyday matters; not the mathematics and geometry of the study of the universe.  

by employing the reciprocal of seven in the computations, which is what an initial analysis of the historically significant numbers reveals, the ancients may have been seeking an easier method for arriving at their knowledge of the universe than what is offered by the precise unending fractional expression of pi , the proportion of the diameter to the circumference of a circle.  this may be further understood when we realize that the comings and goings of the planetary bodies and the stars throughout the universe do not travel on perfect pi-like circles.

the ancients may have employed distinct constant fractals/numbers for adjustments in their computations: the length of the diameter may have been based on 114.2857, 114.591559, 114.638448; etc; the distance of the circumference ay have been related to the 260c, 360c, 378c, 936c, etc.; and, the pi ratio (proportion) of the diameter: circumference may have been 2.268, 3.15, 3.1416, 3.142857, 819, etc. 

the distinctive historically significant numbers reflect different aspects of the computations and their corresponding adjustments.  from this dynamic perspective, the historically significant numbers may be communicating to us a much more precise knowledge of astronomy and mathematical and geometrical computations than we have been willing to concede to the ancients.

useless facts about pi

euler's relation

exp(z)=sum(0->infinity)z^n/n!
z=iy then exp(iy)=sum(0->infinity)(iy)^n/n!
=sum(0->infinity)[(iy)^(2n)/(2n)!+(iy)^(2n+1)/(2n+1)!]
=sum(0->infinity)(-1)^n[y^(2n)/(2n)!+iy^(2n+1)/(2n+1)!]
which, because they have identical power series, is
cos(y)+i*sin(y)
setting y=pi we have
exp(i*pi)=cos(pi)+isin(pi)=-1

irrationality

suppose pi is rational; then there exists a,b natural numbers pi = a/b.
let f be in q[x] f=x^n(ax-b)^n/n! (pi = a/b, remember?).
suppose we define g in q[x] as sum(0->n)(-1)^n(d/dx)^(2n)f.
it should be obvious that d^2g/(dx)^2 is f-g.
getting trickier, we see that
(d/dx)[(dg/dx)sin(x)-g*cos(x)] is [d^2g/(dx)^2+g]sin(x) = f*sin(x).
integrating f*sin(x) from zero to pi with respect to x yields g(0)+g(pi).

at zero and pi, the first n derivatives of f vanish.  but then the coefficients become integers by an easy result on products of n consecutive integers being divisible by n!. so our integral is an integer.  but the maximum of f on (0,pi) is (a*pi)^n/n! and the max of sin(x) is 1.  both f and sin are bounded by zero from below.  thus the integral is bounded strictly between zero and an arbitratily small number, resulting in a contradiction. therefore pi cannot be rational.

transcendence

there is a polynomial the set of whose roots is t, and it is in q[x]. it is obviously product(q in t)(x-q). it has rational coefficients because a permutation of the roots merely causes permutation of the various sums of the roots which comprise t, making our candidate polynomial symmetric, and thus in q[x] (it's fixed under those automorphisms, so it must be).  we can easily make a polynomial in z[x] out of this, and i call it g.  (divide by the lead coef to get the old one back.) in the spirit of the first proof, let's define h = a^s*x^(p-1)*g^p/(p-1)! for some p, with a the leading coef of g, and

s = degree of g * p - 1. again like the first proof we define
f = sum(0->s+p)(d/dx)^n h
since deg f = s+p, (d/dx)[exp(-x)f]=-exp(-x)h
now integrating from zero to z, we have exp(-z)f(z)-f(0) = i, or
f(z)-exp(z)f(0) = exp(z)i.
letting z assume values in t and summing we have
sum(q in t)[f(q)-exp(q)f(0)] = sum(q in t)f(q) + f(0)
recalling one of the aforementioned products.

the sum of the f(q) is an integer, as all derivatives in the defining sum of order less than p vanish, and the remaining terms have the product of enough integers from differentiation to make them integral, canceling the (p-1)!.

furthermore, the expression is symmetric under permutation of the elements of t (or for that matter, s) so it is in z.
f(0) is obviously in z, for until the terms in the sum defining it have been differentiated enough times, they vanish, and once they no longer vanish, they have been differentiated enough to cancel out their denominator. that would make our integral integral. (nice pun, eh?) we have 

sum(q in t)[exp(q)integral(0 to q)exp(-z)f(z)dz]
supposing b bounds |a*z*g| in the disk |z| less than max(q in t)|q| and c be the greatest bound on the |exp(q-z)g| (q in t) in that same disk, the terms beneath the integral signs in the sum are all bounded by a^(m-1)*c*b^(p-1)/(p-1)!, which may be made arbitrarily small for p sufficiently large. thus the integral in question cannot be an integer, we have a contradiction, and thus pi cannot be algebraic. therefore it is transcendental.
(psst! think about what might happen if p were prime)

quadrature of the circle with ruler and compass

straightedge and compass constructions amount to solving at best quadratic polynomials.  reduction of order establishes this for the intersection of two circles thus drawn, and the other cases are trivial.  since such constructions amount to solving polynomials over q, and pi is transcendental, pi cannot be constructed.  a square with area equal to a circle requires the construction of the length sqrt(pi), also an impossibility given the above.

delving into galois theory (not even very deeply), one may determine further the nature of constructible numbers. in particular, one finds that the only constructible nth roots of unity must have n divisible only by two and fermat primes. this has obvious implications regarding the construction of regular polygons.  the preceding observations imply that all constructible numbers must have degree a power of two, and the degree of a minimal polynomial for a root of unity is phi(n) where phi is the number of relatively prime natural numbers less than n, so one might simply observe this from the properties of phi. namely,

phi(n)/n = prod(p|n)(1-1/p) (in q!)so that
phi[2^k*prod(p|n,p!=2)p] = 2^(k-1)prod(p|n,p!=2)(p-1)

and, of course, in the case of fermat primes, p-1 is a power of two.  one notes that in doubling the cube and trisecting arbitrary angles, that with the exception of a small number of particular angles, one is required to solve cubic equations, or equivalently, construct numbers with degree three over q.

the content of n-spheres

a history of pi

and he made a molten sea, ten cubits from the one brim to the other: it was round all about, and his height was five cubits: and a line of thirty cubits did compass it about. (i kings 7, 23)

= 3. not a very accurate value of course and not even very accurate in its day, for the egyptian and mesopotamian values of 25/8 = 3.125 and 10 = 3.162 have been traced to much earlier dates:  though in defense of solomon's craftsmen it should be noted that the item being described seems to have been a very large brass casting, where a high degree of geometrical precision is neither possible nor necessary. 

there are some interpretations of this which lead to a much better value.

the fact that the ratio of the circumference to the diameter of a circle is constant has been known for so long that it is quite untraceable.  the earliest values of including the 'biblical' value of 3, were almost certainly found by measurement. in the egyptian rhind papyrus, which is dated about 1650 bc, there is good evidence for 4(8/9)² = 3.16 as a value for .

the first theoretical calculation seems to have been carried out by archimedes of syracuse (287-212 bc). he obtained the approximation

²²³/₇₁ < < ²²/₇.

archimedes knew, what so many people to this day do not, that does not equal 22/7, and made no claim to have discovered the exact value. if we take his best estimate as the average of his two bounds we obtain 3.1418, an error of about 0.0002.

here is archimedes' argument.

consider a circle of radius 1, in which we inscribe a regular polygon of 3 2^n-1 sides, with semiperimeter b_n, and ascribe a regular polygon of 3 2^n-1 sides, with semiperimeter a_n.

the diagram for the case n = 2 is on the right.

the effect of this procedure is to define an increasing sequence

b₁, b₂, b₃, ...

a₁, a₂, a₃, ...

using trigonometrical notation, we see that the two semiperimeters are given by

a_n = k tan(/k), b_n = k sin(/k),

where k = 3 2^n-1. equally, we have

a_n+1 = 2k tan(/2k), b_n+1 = 2k sin(/2k),

(1) . . . (1/a_n + 1/b_n) = 2/a_n+1

(2) . . . a_n+1b_n = (b_n+1)².

/3) = 33 and b₁ = 3 sin(

/3) = 33/2, calculated a₂ using (1), then b₂ using (2), then a₃ using (1), then b₃ using (2), and so on until he had calculated a₆ and b₆. his conclusion was that

b₆ < < a₆.

archimedes' constant

much more scholarly expositions concerning are available, for example, beckmann's book and borwein & borwein's book. our treatment of this, the most famous of the transcendental constants, is necessarily incomplete.

the area enclosed by a circle of radius 1 is =3.1415926535... and its circumference is 2. how is it that the same mysterious appears in both formulas? we give the answer here along with a little story. of course, appears in higher dimensional analogs (spherical volume and surface area) as well.

in the 3rd century b.c., archimedes considered inscribed and circumscribed regular polygons of 96 sides and deduced that

the following recursion

(often called the borchardt-pfaff algorithm) essentially gives archimedes' estimate on the fourth iteration. the mathcad plus 6.0 file wayman.mcd discusses this procedure further.  click here if you have 6.0 and don't know how to view web-based mathcad files.

another connection between geometry and arises in buffon's needle problem. suppose a needle of length 1 is thrown at random on a plane marked by parallel lines of distance 1 apart. what is the probability that the needle will land in a position which crosses a line? the answer is .

here is a completely different probabilistic interpretation of . suppose two integers are chosen at random. what is the probability that they are coprime, i.e., have no common factor exceeding 1? the answer is .

archimedes' constant was proved to be irrational by lambert in 1761 and transcendental by lindemann in 1882. the first truly attractive formula for computing decimal digits of was found by machin

the advantage of machin's formula is that the second term converges very rapidly and the first is nice for decimal arithmetic. using this, machin became the first individual to correctly compute 100 digits of .

we skip over many years of history and mention only one recent algorithm. the borwein quartically convergent algorithm is related to ramanujan's work on elliptic integrals:

then decreases monotonically to 1/ and

i believe that this is the basis for kanada's record-breaking evaluation of to over 200 billion digits.  quintically convergent algorithms and, more generally, pth-order iterative algorithms for p>4, were discussed by borwein & borwein.