The circle number π

The experiment on the circle number \pi (“Pi”) in the Erlebnisland Mathematik shows that and at which position of the sequence of digits of \pi the date of birth of any visitor can be found.

For example, if March 14, 1941 is the date of birth of that visitor, he should enter the sequence of digits 140341. The result can be read on the screen “in seconds”: This sequence of digits appears in the decimal expansion of \pi for the first time at the 976,229th position.

Figure 1: The exhibit “What is Pi?”

And now … the mathematics of it:

The so-called circle number \pi (also called Archimedes constant or Ludolph’s number) is defined as the ratio p/d of the circumference p and the diameter d of any circle in the plane, i.e. a circle with a diameter of 1 has a circumference of exactly \pi. It is a mathematical constant.

The designation of the circle number with the small Greek letter \pi (“Pi”) can be justified by the fact that the two Greek words περιφερεια (periphereia — english “edge area”) and περιμετρως (perimetros — english “circumference”) begin with this letter.

First use of the notation \pi is found in the Welsh mathematician William Jones“Synopsis palmariorum matheseos” (survey of the principal works of mathematical science), published in 1706. After his Swiss colleague Leonhard Euler adopted this notation in 1737, the designation of the circle number with the lowercase Greek letter \pi became common.

However, the fascination with \pi has lasted for millennia: For example, as early as 250 B.C., the Greek mathematician Archimedes recognized that the quotient of the circumference and diameter of a circle is a constant number which, according to his calculations, must lie between 3.1408450 and 3.1428450.

In the Old Testament (1. Book of Kings 7,23) we find the measurements of a round water basin, which the Israelite king Solomon had built for the temple in Jerusalem: “Then he made the sea. It was cast of bronze and measured 10 cubits from one edge to the other; it was completely round and 5 cubits high. A cord of 30 cubits could encircle it all around.” For the ratio of circumference to diameter thus the value 3 results. More exact were the data of Egyptian scholars: The oldest known arithmetic book of the world, the arithmetic book of Ahmes from the 17th century B.C., calls the value (16/9)^2\approx 3,1604.

In Babylon (in today’s Iraq) one used a little later as approximation for \pi the value 3+1/8=3,125.

The Indian “cord rules” for the construction of altars from the middle of the first millennium B.C. give the value (26/15)^2\approx 3,0044 for \pi for the circle calculation. In the 6th century A.D. the Indian mathematician Aryabhata already determined the value very exactly to 3.1416.

For a long time the question whether \pi was a rational or an irrational number could not be answered. Only in the second half of the 18th century the mathematician Johann Heinrich Lambert could prove the irrationality of \pi. Before that, the English mathematician John Wallis had discovered Wallis’ product, named after him, in 1655:

    \[\frac{\pi}{2}=\frac{2}{1}\cdot\frac{2}{3}\cdot\frac{4}{3}\cdot\frac{4}{5}\cdot\frac{6}{5}\cdot\frac{6}{7}\cdots.\]

Gottfried Wilhelm Leibniz found the following series representation in 1682:

    \[\frac{\pi}{4}=\sum_{n=0}^\infty{\frac{(-1)^n}{2n+1}}=1-\frac{1}{3}+\frac{1}{5}-\frac{1}{7}\pm\cdots.\]

An amazing discovery was made by the Indian mathematician Srinivasa Ramanujan in 1914:

    \[\frac{1}{\pi}=\frac{2\sqrt{2}}{9801}\sum_{n=0}^\infty{\frac{(4n)!(1103+26390n)}{(n!)^4\cdot 396^{4n}}.\]

This series is characterized by its comparatively fast convergence.

In 1996, David Harold Bailey, Peter Borwein, and Simon Plouffe discovered a novel series representation (soon to be called the BBP series) for \pi:

    \[\pi=\sum_{n=0}^\infty{\frac{1}{16^n}\left(\frac{4}{8n+1}-\frac{2}{8n+4}-\frac{1}{8n+5}-\frac{1}{8n+6}\right)}.\]

Later, other BBP series were found. These formulas, due to their favorable shape for the hexadecimal system and their good convergence (which is, however, worse than the convergence of Ramanujan’s formula), allow a very efficient algorithm for calculating the decimal places of \pi, which has become the standard in many applications nowadays (the so-called BBP algorithm). Für

    \[\pi=3,1415926535897932384626433832795028841971693993751\ldots\]

result, for example, if the first fifty digits of the partial sums

    \[S_N=\sum_{n=0}^N{\frac{1}{16^n}\left(\frac{4}{8n+1}-\frac{2}{8n+4}-\frac{1}{8n+5}-\frac{1}{8n+6}\right)}$</em> <!-- /wp:paragraph --> <!-- wp:paragraph --> determined for $N=76,77,78$, the values\]

S_{76}=3,1415926535897932384626433832795028841971693993757,

    \[\]

S_{77}=3,1415926535897932384626433832795028841971693993750,

    \[\]

S_{78}=3. 1415926535897932384626433832795028841971693993751.

The approximate values of \pi and the procedures for their determination have long been very valuable, especially for practical applications, e.g., in civil engineering. The approximate values determined in the last decades, however, already have so many digits that a practical use is hardly given any more. This is shown for example by the question, how many digits of \pi are necessary to calculate the largest real circle imaginable in the universe with the best accuracy. According to the latest cosmological considerations it results that the light of the big bang in the form of the background radiation reaches us from a distance which is the product of the assumed age of the world (about 1.3\cdot 10^{10} years) with the speed of light (about 300,000\mathrm{km}/\mathrm s), i.e. about 1.3\cdot 10^{26}\mathrm m. The circle with this radius would have a circumference of \pi\cdot 1.3\cdot 10^{26}\mathrm m, so about 8.17\cdot 10^{26}\mathrm m. The smallest physically meaningful length unit is the so-called Planck length of about 10^{-35}\mathrm m. The imaginary circumference would thus consist of 8.17\cdot 10^{61} Planck lengths. In order to calculate its circumference from the given and to one Planck length exactly known radius with the accuracy of again one Planck length, already 62 decimal places of \pi would be sufficient.


But what are the number theoretic properties of the number \pi? We want to illuminate this in the following: The value of \pi has an infinite, non-periodic decimal fraction expansion 3.14159265359…. In other words, \pi is (as mentioned above) non-rational, so it cannot be written as a fraction m/n of two integers m and n (where n\neq 0). Therefore one says \pi is irrational. But even more is true: the number \pi does not even satisfy a polynomial equation a_n x^n+a_{n-1}x^{n-1}+\cdots+a_0=0 with integers a_0,\ldots,a_n\in\mathbb Z, a_n\neq 0, n>0. Thus, in particular, it cannot be rational, because any rational number m/n (m,n\in\mathbb Z, n\neq 0) satisfies the equation nx-m=0. This was first proved by the German mathematician Carl Louis Ferdinand Lindemann. Such numbers (and thus also \pi) are called transcendent. From this property it now follows that it is impossible to represent the number \pi as an expression containing only integers, fractions and roots.

With this observation, Lindemann’s theorem has the following famous consequence: it is impossible to construct, using only compass and ruler, a square having exactly the area of a circle of given radius r (say r=1). The side length of such a square would have to be exactly \sqrt{\pi} and could then be represented as an expression containing only integers, fractions and square roots (because these are exactly the numbers that can be constructed with compass and ruler). The problem just mentioned (proven to be unsolvable) is also called squaring the circle.

In the number sequence of \pi behind the decimal point no regularity is recognizable; also it satisfies statistical tests for coincidence. These observations justify a conjecture (at present still unanswered): Namely, that \pi is a so-called normal number. These are real numbers in whose decimal places every given sequence of digits of a certain length \ell occurs with the same asymptotic probability p (namely with p=10^{-\ell}). For example, this meant that in the decimal places of \pi the digit sequences 23 and 45 occur in approximately the same number, considering only enough digits. Normal numbers still contain any sequence of digits of finite length in their decimal places. So if the assumption “\pi is a normal number” is true, the content of every book written so far and also to be written in the future is contained in binary coding in the binary representation of \pi! On the other hand, the task of the exhibit in Erlebnisland Mathematik is much simpler: The assumption that every six-digit number sequence occurs in the decimal representation of \pi has always been confirmed so far. It should be noted at this point at last that almost every (in a strict mathematical sense) randomly chosen real number is normal. In this sense, normal numbers behave as if they were randomly chosen.


Finally, some news worth mentioning about the circular number π

  • At the 1,142,905,318,634th decimal place of \pi one finds the digit sequence 314159265358 for the first time again according to the Japanese mathematician and chair of computer science Yasumasa Kanada (*1948). Until 2009 this held the “world record” with the determination of the number of decimal places of \pi.
  • Friends of the number \pi, on the one hand, commemorate the circular number on March 14 with \pi Day because of the American date notation 3-14. On the other hand, a \pi Approximation Day is celebrated on July 22 to honor Archimedes’ approximation 22/7 for \pi.
  • In “Sternstunden der modernen Mathematik” by Keith Devlin (cf. bibliography) there is another example in which \pi surprisingly plays a role: If one throws a match on a board divided by parallel lines, each one match length apart, then the probability that the match falls so that it intersects a line is exactly 2/\pi. This is a variant of the famous paradox in Buffon’s needle experiment.
  • The unofficial world record for memorizing \pi is held by Akira Haraguchi of Japan, who is said to have recited 100,000 decimal places of \pi “off the top of his head” in 16 hours on October 4, 2006.

Literature

[1] Behrends, E.: \pi und Co. Kaleidoskop der Mathematik, Berlin / Heidelberg, 2008.

[2] Beutelspacher, A.: Mathematik zum Anfassen, Gießen, 2005.

[3] Delahaye, J.–P.: \pi — Die Story, Basel, 1999.

[4] Devlin, K.–J.: Sternstunden der modernen Mathematik. Berühmte Probleme und neue Lösungen, 2. Auflage, München, 1992.

[5] Jones, W.: Synopsis palmariorum matheseos: or, A new introduction to mathematics containing the principles of arithmetic & geometry demonstrated, in a short and easy method; with their application to the most useful parts thereof … Design’d for the benefit, and adapted to the capacities of beginners, London, 1706.

[6] Schmidt, K.–H.: \pi. Geschichte und Algorithmen einer Zahl, Norderstedt, 2001.

[7] Tietze, H.: Mathematische Probleme. Gelöste und ungelöste mathematische Probleme. Vierzehn Vorlesungen für Laien und Freunde der Mathematik, München, 1990.

[8] Zschiegner, M.A.: Die Zahl \pi — faszinierend normal! in: Mathematik lehren 98, S. 43–48, 2000.