Ludolph van Ceulen in 1596 using the polygon method, first came to calculate 20 decimal digits, then 35. Van Ceulen wasn't the last to use the method: for example Willebrord Snellius in 1621 calculated 34 digits, while the Austrian astronomer Christoph Grienberger in 1630 reached a record 38 digits using a 1040-sided polygon: this result is the most accurate ever achieved using the polygon method.
The infinite series supplanted this method: the first to use them in Europe was the French mathematician François Viète in 1593
\[\frac2\pi = \frac{\sqrt2}2 \cdot \frac{\sqrt{2+\sqrt2}}2 \cdot \frac{\sqrt{2+\sqrt{2+\sqrt2}}}2 \cdots\]
And in 1655 John Wallis
\[\frac{\pi}{2} = \frac{2}{1} \cdot \frac{2}{3} \cdot \frac{4}{3} \cdot \frac{4}{5} \cdot \frac{6}{5} \cdot \frac{6}{7} \cdot \frac{8}{7} \cdot \frac{8}{9} \cdots\]
European mathematics, however, had come to this method only after Indian mathematics, albeit independently. In India, in fact, there is evidence of first approaches of this kind between 1400 and 1500. The first infinite series used to calculate \(\pi\) is found, in fact, on the pages of the Tantrasamgraha (literally "compilation of systems") of the Indian astronomer Nilakantha Somayaji, circa 1500-1501. The series, presented without any proof (later published in the Yuktibhāṣā, circa 1530), was attributed by Nilakantha to the mathematician Madhava of Sangamagrama, who lived between 1350 and 1425 circa. Apparently Madhava discovered several infinite series, including many that contain the sine, cosine, and tangent. The Indian mathematician used these series to reach up to 11 digits around 1400, a value that was improved around 1430 by the Persian mathematician Jamshīd al-Kāshī using the polygon method.
Showing posts with label pi day. Show all posts
Showing posts with label pi day. Show all posts
Pi stories: the Cyclometricus and other tales
It was 1621 when the Cyclometricus by Willebrord Snellius, a pupil of Ludolph van Ceulen, was published. Snellius proved that the perimeter of the inscribed polygon converges to the circumference twice than the circumscribed polygon. As a good pupil of van Ceulen, Snellius managed to get 7 decimal places for the $\pi$ by using a 96-sided polygon. His best result, however, was 35 decimal places, which improved his master's result, 32.
The next improvement is dated 1630 by Christoph Grienberger, the last mathematician to evaluate $\pi$ using the polygon method, while the first successful method change came out thanks to the british mathematician and astronomer Abraham Sharp who determined 72 decimal places of $\pi$, of which 71 correct, using a series of arctangents. A few years later, John Machin improved Sharp's result with the following formula and that allowed him to achieve the remarkable result of 100 decimal places! \[\frac{\pi}{4} = 4 \arctan \frac{1}{5} - \arctan \frac{1}{239}\] Machin's approach proved successful, so much so that the slovenian baron Jurij Vega improved on two occasions the above formula obtaining a greater number of decimal digits of $\pi$, the first time in 1789 with a formula similar to Euler's one \[\frac{\pi}{4} = 5 \arctan \frac{1}{7} + 2 \arctan \frac{3}{79}\] then in 1794 with a Hutton-like formula \[\frac{\pi}{4} = 2 \arctan \frac{1}{3} + \arctan \frac{1}{7}\] The arctangent era continued with William Rutherford \[\frac{\pi}{4} = 4 \arctan \frac{1}{5} - \arctan \frac{1}{70} + \arctan \frac{1}{99}\] and with Zacharias Dase \[\frac{\pi}{4} = \arctan \frac{1}{2} + \arctan \frac{1}{5} + \arctan \frac{1}{8}\] Finally comes the british William Shanks that pushing the full potential of the Machin's formula managed to get 707 decimal places, of which only 527 were correct after Ferguson's controls in 1946. Here, however, we are going in the era of mechanical calculation, prologue to computer era.
The next improvement is dated 1630 by Christoph Grienberger, the last mathematician to evaluate $\pi$ using the polygon method, while the first successful method change came out thanks to the british mathematician and astronomer Abraham Sharp who determined 72 decimal places of $\pi$, of which 71 correct, using a series of arctangents. A few years later, John Machin improved Sharp's result with the following formula and that allowed him to achieve the remarkable result of 100 decimal places! \[\frac{\pi}{4} = 4 \arctan \frac{1}{5} - \arctan \frac{1}{239}\] Machin's approach proved successful, so much so that the slovenian baron Jurij Vega improved on two occasions the above formula obtaining a greater number of decimal digits of $\pi$, the first time in 1789 with a formula similar to Euler's one \[\frac{\pi}{4} = 5 \arctan \frac{1}{7} + 2 \arctan \frac{3}{79}\] then in 1794 with a Hutton-like formula \[\frac{\pi}{4} = 2 \arctan \frac{1}{3} + \arctan \frac{1}{7}\] The arctangent era continued with William Rutherford \[\frac{\pi}{4} = 4 \arctan \frac{1}{5} - \arctan \frac{1}{70} + \arctan \frac{1}{99}\] and with Zacharias Dase \[\frac{\pi}{4} = \arctan \frac{1}{2} + \arctan \frac{1}{5} + \arctan \frac{1}{8}\] Finally comes the british William Shanks that pushing the full potential of the Machin's formula managed to get 707 decimal places, of which only 527 were correct after Ferguson's controls in 1946. Here, however, we are going in the era of mechanical calculation, prologue to computer era.
Pi and the Basel's problem
The Euler's demonstration, published in its final form in 1741, is particularly interesting: Euler supposed that it's possible to apply the rules of the finite polynomials even those endless.
We start with the development in Taylor series for the sine function in 0: \[\sin(x) = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \cdots\] Dividing by $x$ both terms, we obtain: \[\frac{\sin(x)}{x} = 1 - \frac{x^2}{3!} + \frac{x^4}{5!} - \frac{x^6}{7!} + \cdots\] whose roots are $\pi$, $-\pi$, $2\pi$, $-2\pi$, $3\pi$, $-3\pi$, $\ldots$ By changing the variable as $z = x^2$, the polynomial above becomes: \[\frac{\sin(\sqrt{z})}{\sqrt{z}} = 1 - \frac{z}{3!} + \frac{z^2}{5!} - \frac{z^3}{7!} + \cdots\] whose roots are $\pi^2$, $4\pi^2$, $9\pi^2$, $\ldots$
Now, given a polynomial $a_n x^n + \cdots + a_3 x^3 + a_2 x^2 + bx + 1$, for the formulas of Viète, we have that the sum of the reciprocals of its roots has as result $-b$. Applying this result for finished polynomials to infinite polynomial in $z$ above, we get: \[\frac{1}{3!} = \frac{1}{6} = \frac{1}{\pi^2} + \frac{1}{4\pi^2} + \frac{1}{9\pi^2} + \frac{1}{16\pi^2} + \cdots\] and so: \[\frac{\pi^2}{6} = 1 + \frac{1}{4} + \frac{1}{9} + \frac{1}{16} + \cdots = \sum_{n=1}^\infty \frac{1}{n^2}\] It's simple to observe the connection between Mengoli's series and Riemann's zeta \[\zeta(s) = \sum_{n=1}^\infty \frac{1}{n^s}\] with $s=2$.
Last observation: in 1982 on the magazine Eureka, it appeared a rigorous proof of Euler's result signed by John Scholes, although it seems that such a demonstration circulated already to late sixties between Cambridge corridors.
Ludolph van Ceulen: in searching of pi
Another fundamental challenge was the particular historical period in which Ludolph lived. At that time, in fact, life for Protestants was rather complicated: the Spanish Inquisition was, in fact, powerful enough to extend his long hands even in Germany. The Van Ceulen's, as Protestants, were forced, like many in the same conditions, to migrate to the most welcoming Netherlands of Prince William of Orange.
On the other hand Ludolph himself was a traveler: immediately after his father's death became a little travel first in the region of Livonia (Latvia and Estonia in our age), then to Antwerp to visit his brother Gert and then to Delft in the Netherlands where he settled for a time, since there was born one of his five daughter on May 4th, 1578.
His wife, Mariken Jansen, died in 1590, but Ludolph remarriage on June 17th of that year with Adriana Simondochter, widow of Bartholomew Cloot, accounting and math teacher, with whom he had generated eight children, for a total of 13 mouths to feed. The two families, Cloot and Van Ceulen, were in a close relations of friendship, so it's pretty obvious to imagine that the marriage between Ludolph and Adriana was the best solution to avoid losing a strong relationship.
Pi day: Zilienski, Wallis and the square
A good pi day to all readers! I hope that the following post, that I cannot review after the first writing, could be interesting to all!
The technique used by the ancient Greek for their geometric constructions was called "ruler and compass". In this way you can build a lot of regular polygons, for example, but there are three problems that are impossible unless you use different techniques: the angle trisection, doubling the cube, squaring the circle.
In particular for the squaring, it is easy to calculate the relation between the radius $r$ of the circke and the side $l$ of the square with the same area of the starting circle: \[L = \sqrt {\pi} r \] Now, since $\pi$ is a transcendental number, the formula above is the simplest representation of the impossibility of squaring the circle using ruler and compass: with these devices you can treat rational and irrational numbers, such as $\sqrt{2}$ (in this case simply draw a square of side 1).
So, using these two tools it is possible to obtain an approximate construction and, therefore, a corresponding approximate value for $\pi$: during the XX century there are produced a lot of approximations, for example by CD Olds (1963), Martin Gardner (1966), Benjamin Bold (1982). They are all variations of the geometric construction discovered by Srinivasa Ramanujan in 1913, that approached $\pi$ with the fraction \[\frac{355}{113} = 3.1415929203539823008 \dots\] right up to the sixth decimal place.
In 1914, Ramanujan discovered a more accurate approximation (to eight decimal places), always using ruler and compass: \[\left (9^2 + \frac{19}{22}^2 \right)^{1/4} = \sqrt[4]{\frac{2143}{22}} = 3.1415926525826461252 \dots\]
In particular for the squaring, it is easy to calculate the relation between the radius $r$ of the circke and the side $l$ of the square with the same area of the starting circle: \[L = \sqrt {\pi} r \] Now, since $\pi$ is a transcendental number, the formula above is the simplest representation of the impossibility of squaring the circle using ruler and compass: with these devices you can treat rational and irrational numbers, such as $\sqrt{2}$ (in this case simply draw a square of side 1).
So, using these two tools it is possible to obtain an approximate construction and, therefore, a corresponding approximate value for $\pi$: during the XX century there are produced a lot of approximations, for example by CD Olds (1963), Martin Gardner (1966), Benjamin Bold (1982). They are all variations of the geometric construction discovered by Srinivasa Ramanujan in 1913, that approached $\pi$ with the fraction \[\frac{355}{113} = 3.1415929203539823008 \dots\] right up to the sixth decimal place.
In 1914, Ramanujan discovered a more accurate approximation (to eight decimal places), always using ruler and compass: \[\left (9^2 + \frac{19}{22}^2 \right)^{1/4} = \sqrt[4]{\frac{2143}{22}} = 3.1415926525826461252 \dots\]
A brief history of pi: part 2
Today is the pi day, so I continue the brief history of $\pi$
Johann Heinrich Lambert was the first to show the irrational nature of $\pi$ in 1761 in Mémoire sur quelques propriétés remarquables des quantités transcendantes circulaires et logarithmiques: that could be written in this way: \[\tan(x) = \cfrac{x}{1 - \cfrac{x^2}{3 - \cfrac{x^2}{5 - \cfrac{x^2}{7 - {}\ddots}}}}\] Lamberd proved that if $x$ is not null and rational, then the previous expression must be irrational. So the irrationality of $\pi$ follows from $\tan (\pi /4) = 1$. A good synthesis of Lambert's proof is on The world of $\pi$.
In 1997 Laczkovich proposed a simplification of this demonstration, while another variation was proposed in 2009 by Li Zhou, using the integral calculus. In particular the second demonstration is inspired by the proof that Charles Hermite written in two letters to Paul Gordan and Carl Borchardt in 1873. Following Harold Jeffreys in Scinetific interference (1973), a simplification of this proof, that used a reductio ab adsurdum is proposed by Mary Cartwright.
Another proof of one page about the irrationality of $\pi$ is dued by Ivan Niven in 1946.
At the other hand, the transcendence of $\pi$ is a direct consequence of the Lindemann-Weierstrass theorem:
If $\alpha_1$, $\cdots$, $\alpha_n$ are algebraic numbers that are linearly independent over rationals, then $e^{\alpha_1}$, $\cdots$, $e^{\alpha_n}$ are algebraically independent over rationals.where an algebraic number is the solution of a polynomial equation with rational coefficients.
In 1882 Lindemann, using this theorem, showed that $e$ is transcendental, and, like a consequence of the Euler's identity, also $\pi$ is transcendental.
A very brief story of the Pi
As you know, the $\pi$ is defined as the ratio of the circumference to its diameter. This number, which is transcendental, was, apparently, known since ancient times. There are, in fact, some Egyptologists who believe that $\pi$, or perhaps $\tau = 2 \pi$, was known to them since the age og the Giza's pyramid, built between 2589 and 2566 BC, because the relationship between the perimeter and the height is 6.2857.
There are no explicit proof of the fact that, at the time, Egyptian mathematics became aware of a number such as $\pi$, however, between 600 and 1000 years later on a Babylonian tablet it is geometrically established the first value of $\pi$: $25/8 = 3.1250$. From documents written more or less in the same period it can be deduced that also the Egyptians calculated the value of $\pi$, obtaining $(16/9)^2 \simeq 3.1605$.
Indian mathematics, however, seems a little late: in 600 BC on Shulba Sutras, it is calculated for the $\pi$ value like $(9785/5568)^2 \simeq 3.088$, which will be updated later in 150 BC as $\sqrt{10} \simeq 3.1622$, which is a value much closer to the value calculated by the Egyptians.
A good approximation of $\pi$ value is in Mishnat ha-Middot, a geometric treatise by Rabbi Nehemiah: $3 + 1/7 \simeq 3.14286$.
The approximation, however, the most amazing not only for accuracy but also for the method is that proposed by Archimedes, the italo-greek mathematician who invented the method of polygons in order to calculate $\pi$, a costant that for a millennium became known simply as the Archimedes' constant.
He simply calculated the perimeter of polygons inscribed and circumscribed in a circle, thus obtaining a lower and an upper estimate of the value of the constant: \[223/71 < \pi < 22/7\] or \[3.1408 < \pi < 3.1429\] It's clear that his method of calculation is very modern and above suggests that Archimedes was well aware of the transcendental nature of the constant, which could be known only through approximations.
Today $\pi$ is known to 5 trillion digits and if you try to type the symbol $\pi$ on modern scientific calculators, the value they give you is, to the first decimal place, 3.14159265...
There are no explicit proof of the fact that, at the time, Egyptian mathematics became aware of a number such as $\pi$, however, between 600 and 1000 years later on a Babylonian tablet it is geometrically established the first value of $\pi$: $25/8 = 3.1250$. From documents written more or less in the same period it can be deduced that also the Egyptians calculated the value of $\pi$, obtaining $(16/9)^2 \simeq 3.1605$.
Indian mathematics, however, seems a little late: in 600 BC on Shulba Sutras, it is calculated for the $\pi$ value like $(9785/5568)^2 \simeq 3.088$, which will be updated later in 150 BC as $\sqrt{10} \simeq 3.1622$, which is a value much closer to the value calculated by the Egyptians.
A good approximation of $\pi$ value is in Mishnat ha-Middot, a geometric treatise by Rabbi Nehemiah: $3 + 1/7 \simeq 3.14286$.
The approximation, however, the most amazing not only for accuracy but also for the method is that proposed by Archimedes, the italo-greek mathematician who invented the method of polygons in order to calculate $\pi$, a costant that for a millennium became known simply as the Archimedes' constant.
He simply calculated the perimeter of polygons inscribed and circumscribed in a circle, thus obtaining a lower and an upper estimate of the value of the constant: \[223/71 < \pi < 22/7\] or \[3.1408 < \pi < 3.1429\] It's clear that his method of calculation is very modern and above suggests that Archimedes was well aware of the transcendental nature of the constant, which could be known only through approximations.
Today $\pi$ is known to 5 trillion digits and if you try to type the symbol $\pi$ on modern scientific calculators, the value they give you is, to the first decimal place, 3.14159265...
Pi Day and the BBP formula
In the last few days I must write the 47th italian Carnival of Mathematics (Google translation), so I cannot write something about Pi Day. So, in the end of the day, I try to write a little post about this day.
$\pi$ is one of the irrational numbers: we cannot write an irrational numbers like a ratio between twointeger numbers, so Pitagora's team called them irrational and hidden their existence. We can define $\pi$ like the ratio between circle and his diameter, and it's impossible that they are contemporary integer. This assertion was proofed in 1761 by Johann Heinrich Lambert, while in 1882 Ferdinand von Lindemann proofed that $\pi$ is a trascendental number, where a trascendental number is a number that we cannot calculate with usual algebric operations.
In ancient times, the first mathematician who evaluate $\pi$ was Archimedes, who used the method of exhaustion, that is a method to evaluate the area of a shape by inscribing inside it a series of polygons that converge to area's shape. In this way Archimedes find $\pi = \frac{211875}{67441} = 3.14163 \cdots$
Today we know 5 triolions of digits, thanks to the work of Shigeru Kondo, who modify a usual home pc in order to perform the calculations using the softwerdeveloped by Alexander Yee. They design an internet site to describe methods, software and results.
Following David Bailey and Jonathan Borwein(1), I find some others interesting site about the calculations of $\pi$'s digits (Inverse Symbolic Calculator, Electronic Geometry Models). Other links at Experimental mathematics. But... what is experimental mathematics?
In ancient times, the first mathematician who evaluate $\pi$ was Archimedes, who used the method of exhaustion, that is a method to evaluate the area of a shape by inscribing inside it a series of polygons that converge to area's shape. In this way Archimedes find $\pi = \frac{211875}{67441} = 3.14163 \cdots$
Today we know 5 triolions of digits, thanks to the work of Shigeru Kondo, who modify a usual home pc in order to perform the calculations using the softwerdeveloped by Alexander Yee. They design an internet site to describe methods, software and results.
Following David Bailey and Jonathan Borwein(1), I find some others interesting site about the calculations of $\pi$'s digits (Inverse Symbolic Calculator, Electronic Geometry Models). Other links at Experimental mathematics. But... what is experimental mathematics?
(...) namely the utilization of modern computer technology as an active tool in mathematical research.(1)For example using the PSLQ algorithm(2) in 1995 was found the BBP formula for $\pi$: \[\pi = \sum_{k=0}^\infty \frac{1}{16^k} \left ( \frac{4}{8k+1} - \frac{2}{8k+4} - \frac{1}{8k+5} - \frac{1}{8k+6} \right )\]
This formula permits one to directly calculate binary or hexadecimal digits beginning at the $n$-th digit, without needing to calculate any of the first $n − 1$ digits, using a simple scheme that requires very little memory and no multiple-precision arithmetic software.(1)You can find others BBP formulas in the David Bailey's compendium (pdf).
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