The circle number π

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

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

And now … the mathematics:

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 — “border area“) and περιμετρως (perimetros — “circumference“) begin with this letter.

The first use of the notation $\pi$ is found in the work of the Welsh mathematician William Jones in his “Synopsis palmariorum matheseos” ( overview of the main 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 Greek lower-case letter $\pi$ became common.

However, the fascination with $\pi$ has lasted for millennia: As early as 250 BC, the Greek mathematician Archimedes recognised that the quotient of the circumference and diameter of a circle is a constant number that, according to his calculations, must lie between 3.1408450 and 3.1428450.

In the Old Testament (1 Kings 7:23) we find the measurements of a round water basin that 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 stretch around it.” The ratio of circumference to diameter is therefore 3. The figures of Egyptian scholars were more precise: the oldest known arithmetic book in the world, the arithmetic book of Ahmes from the 17th century BC, gives the value $(16/9)^2\approx 3.1604$ .

In Babylon (in present-day Iraq), the value 3+1/8=3.125 was used a little later as an approximation for $\pi$ .

The Indian “string rules” for the construction of altars from the middle of the first millennium BC give the value $(26/15)^2\approx 3.0044$ for $\pi$ for the circle calculation. In the 6th century AD, the Indian mathematician Aryabhata already determined the value very precisely to be 3.1416.

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

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

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 astonishing 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 characterised by its comparatively fast convergence.

In 1996, David Harold Bailey, Peter Borwein and Simon Plouffe discovered a novel row 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, more BBP series were found. These formulas, because of their favourable 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 for many applications nowadays (the so-called BBP algorithm). For

$\pi=3,1415926535897932384626433832795028841971693993751\ldots$

for example, determining the first fifty digits of the partial sums

$S_N=\sum_{n=0}^N{\frac{1}{16^n}\left(\frac{4}{8n+1}-\frac{2}{8n+4}-\frac{1}{8n+5}-\frac{1}{8n+6}\right)}$

determined for $N=76,77,78$ , the values

$\begin{align*}S_ {76} &=3.1415926535897932384626433832795028841971693993757, \\ S_{77} &=3,1415926535897932384626433832795028841971693993750, \\ S_{78} &= 3. 1415926535897932384626433832795028841971693993751.\end{align*}$

The approximate values of $\pi$ and the procedures for determining them were very valuable for a long time, especially for practical applications, e.g. in the building industry. The approximate values determined in the last decades, on the other hand, already have so many digits that there is hardly any practical use left. This can be seen, for example, in the question of how many digits of $\pi$ are required to calculate the largest real circle imaginable in the universe with the best accuracy. According to the latest cosmological observations, the light from the Big Bang in the form of background radiation reaches us from a distance that 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 unit of length is the so-called Planck length of about $10^{-35} \mathrm{m}$ . The imaginary circumference would therefore consist of $8.17\cdot 10^{61}$ Planck lengths. In order to calculate its circumference from the given radius, which is known to an accuracy of one Planck length, 62 decimal places of $\pi$ would be sufficient.

But what are the number-theoretical properties of the number $\pi$ ? We will shed light on this in the following: The value of $\pi$ has an infinite, non-periodic decimal fractional expansion 3.14159265359…. In other words, $\pi$ is (as mentioned above) not rational, so it cannot be written as a fraction $m/n$ of two integers $m$ and $n$ (where $n\neq 0$ ). They therefore say $\pi$ 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 every rational number $m/n$ ( $m,n\in\mathbb Z$ , $n\neq 0$ ) satisfies the equation $nx-m=0$ . This was first demonstrated by the German mathematician Carl Louis Ferdinand Lindemann. Such numbers (and thus also $\pi$ ) are called transcendent. It now follows from this property 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 a square with exactly the area of a circle of given radius $r$ (say $r=1$ ) using only a compass and ruler. 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 a compass and ruler). The problem just mentioned (proven to be unsolvable) is also called squaring the circle.

There is no recognisable regularity in the sequence of $\pi$ after the decimal point; nor does it satisfy statistical tests for randomness. These observations justify a (currently still unanswered) assumption: 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 means that in the decimal places of $\pi$ the digit sequences 23 and 45 occur in approximately the same number, if only enough digits are considered. 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 $\pi$ ! The task at the exhibit in Maths Adventure Land, on the other hand, 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 that almost any randomly chosen real number (in a strict mathematical sense) is normal. In this sense, normal numbers behave as if they were chosen at random.

Finally, some news worth mentioning about the circular number π.

At the 1,142,905,318,634th decimal place of $\pi$ , the sequence of digits 314159265358 is found again for the first time, according to the Japanese mathematician and chair of computer science Yasumasa Kanada (*1948). Until 2009, this held the “world record” for determining the number of decimal places of $\pi$ .
Friends of the number $\pi$ , on the one hand, commemorate the circular number on 14 March with $\pi$ Day because of the American date notation 3-14. On the other hand, a $\pi$ Approximation Day is celebrated on 22 July to honour Archimedes’ approximation 22/7 for $\pi$ .
In the “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 memorising $\pi$ is held by the Japanese Akira Haraguchi, who is said to have recited 100,000 decimal places of $\pi$ “off the top of his head” in 16 hours on 4 October 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.