05. Pi – Michel Tombroff

Michel Tombroff (BE)

Pi

Texts by Luc Lemaire

Back to our 40,000 kilometre-long rope wrapped around the equator and to which a meter’s length has been added. Based on intuition, we suppose that it could be lifted less than a millimetre across the equator.

But the formula\(p=2\times \pi \times r\) states that if we increase the perimeter by a metre, the radius increases by \(r=\frac{1}{2\pi}\). The rope is lifted by by about 16 centimetres all around the equator.

The Greeks, in particular Archimedes (287-212 BCE), studied the properties of pi, based on precise definitions and rigorous demonstrations.

We shall begin with the formula \(p=2\times \pi \times r\). It is a definition, but also a theorem that states: for two circles with different radii, the number “\(\pi\)” is the same.

A result by Archimedes pertains to the area \(A\) of the disc in the circle: \(A=\pi\times r^2\). Here is a simple demonstration.

Cut the disc into quarters as shown in the first drawing. Then place them side by side as shown in the second drawing. As the number of quarters increases, the figure becomes a rectangle whose height equals the radius (\(R\)) and whose width is half its perimeter, in other words \(\pi\times R\).  The product of both is \(A=\pi\times R^2\).

Archimedes also rigorously calculated the value of \(\pi\).

By using a polygon inscribed within a circle and a polygon circumscribed outside a circle, he concluded that the length of the circle is between the lengths of both hexagons. Archimedes actually went as far as a 96-sided polygon to obtain an approximate value of \(3,14\).

For centuries, the computation of pi decimal places has progressed slowly, with a circle’s geometry as a starting point.

When initiating analysis and differential calculation, however, everything changes.

John Wallis (1616-1703), and subsequently James Gregory (1638-1675), obtained formulas for pi which were not directly related to geometry.  For example, pi is represented by an infinite sum of terms:

\[\pi=4\times (1-\frac{1}{3}+\frac{1}{5}-+\frac{1}{7}+…)\]

The sum of an infinite number of terms is called a series. In certain cases, a series does not result in a finite number (for example, sum 1+1+1+1…). In other cases, the sum of an infinite number of terms comes close to a finite number, or converges towards a finite number. It is the case in the series shown below, which converges towards pi.
The more terms we calculate, the greater the number of exact decimals.

All subsequent progress in calculating pi is based on increasingly ingenious series with no relation to geometry.
For example, Leonhard Euler (1707-1783) discovered the formula:

\[\frac{\pi^2}{6}=1+\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+…\]

Let’s talk about a completely different subject: prime numbers.

A prime number is an integer larger or equal to 2, that can only be divided by 1 and itself. For example, 6 is not a prime number since 6 = 2 . 3, but 7 and 11 are prime numbers.  There is an infinite number of prime numbers, beginning with: 2, 3, 5, 7, 11, 13, 17, 19, 23…

A rather complicated calculation shows that \(\frac{\pi^2}{6}\) is also the result of an “infinite product”:

\[\frac{\pi^2}{6}=\frac{1}{(1-\frac{1}{2^2})}\times \frac{1}{(1-\frac{1}{3^2})}\times \frac{1}{(1-\frac{1}{5^2})}\times \frac{1}{(1-\frac{1}{7^2})}\times \frac{1}{(1-\frac{1}{11^2})}\times … =\prod_{p\ prime}\frac{1}{(1-\frac{1}{p^2})}\]

where only prime numbers appear.

An incredible formula! The circle’s perimeter on the left; prime numbers on the right. Two completely different subjects are actually related. The uniqueness of mathematics is once again evident: it is not possible to study one branch of the field without the others.

Let’s go back to calculating the ever-growing number of decimals in pi.

In theory, all the decimals can be obtained with the following formula, but very slowly: we must calculate a large number of terms in the series to obtain relatively few exact decimals. In the case of

\[\pi=4\times (1-\frac{1}{3}+\frac{1}{5}-+\frac{1}{7}+…)\]

we must add 500 terms to obtain \(3,14\), and 5,000 to obtain \(3,141\).

Instead of calculating decimals, efforts turned to finding increasingly efficient formulas.

Among these numerous formulas, the following one was invented by one of history’s most enigmatic mathematicians: Srinivasa Ramanujan (1887-1920). Although he did not receive a proper education, he worked on his own in India and filled notebooks with formulas without formal demonstration. He wrote to British mathematician G.H. Hardy and sent him 120 formulas, once again, without demonstration. Hardy invited him to Cambridge where they worked together for five years. Ramanujan died from pneumonia. Ramanujan’s notes provided work for numerous mathematicians who then sought to generate demonstrations for his countless formulas.

Here is very strange formula that he wrote for pi. It is not known how he found it.

\[\pi=\frac{9801}{\sqrt{8}}(\sum_{n=0}^{\infty}\frac{(4n)!\times (1103+26390 n)}{(n!)^4\times 396^{4n}})^{-1}\]

Here, \(n!\) is called the « factorial of \(n\) » and it is the product of all integers between \(1\) and \(n\). For example, \(4!=1\times 2 \times 3 \times 4=24\).

In such a series, each added term to the sum adds eight exact decimals to the value obtained for pi.  We are far from series

\[\pi=4\times (1-\frac{1}{3}+\frac{1}{5}-+\frac{1}{7}+…\]

which gives only three decimals after 5,000 terms.

This 1914 formula – with the help from a computer – was used by William Gosper in 1985 to calculate 17 million de decimals of pi.

Pi has currently been calculated out to 31 trillion decimals. To do so, we use increasingly powerful computers as well as increasingly ingenious series formulas.  The goal is to find terms that give more and more decimals, but to make sure that the time it takes to calculate these terms is worth the gain.

Why is it useful to know all these decimals? They obviously do not have a practical value. The research itself, however, has led to the development of increasingly fast and ingenuous algorithms and has advanced our knowledge about mathematics and computer science.

Pi is undoubtedly the most essential number in mathematics, but it is joined on the podium by two other numbers: the complex number “\(i\)”, the square root of “\(-1\)”, and the Euler number “\(e\)”, the basis of the exponential function.

To learn more, we invite you to read the contribution Pi, e, et i : les trois nombres incontournables written by Luc Lemaire for the exhibition Order of Operations.

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