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Pi: mysteries, history and utility

In honor of #piday (because 3/14 in American date format) here is a new article paying tribute to the mysterious and essential number Pi! From college to higher education, this symbol follows us to the trace in math and science. It is also the subject of intrigues in cinema, literature or music. It even exists in the Guinness Book of Records and that's classy!

Fun facts and other mysteries around Pi

First, Pi, it is probably the most famous mathematical number of all. A priori, e (from exp, if you no longer know, booster shot here) it probably speaks to you less, phi, the golden ratio, even less, and i squared probably not or more. Well, if you know all that, it's because you're probably one of those who do or have done a lot of math. But that is not the subject here.

Fact#1 What is the length of Pi?

An approximation of Pi, using an online calculator (here are some free ones here), it's 3,14159265359. It is often abbreviated orally to 3,14. Do you relocate why Pi Day is March 14 in case you didn't get it at first glance 😉?
So, there are already 11 decimals (digits after the decimal point) of Pi. How many are there exactly?

Hint: the latest record has just exceeded 100 decimals – 000 trillion – 000 billion! This record is obtained by calculating on Compute Engine, Google's Cloud solution, thanks to Emma Haruka Iwao, a Japanese engineer employed by Google.

Answer: it cannot be defined. Its length seems to be infinite. Indeed, Pi is a number irrational ! This means that it is not possible to write pi algebraically as a fraction of whole numbers. And for the more seasoned, see here to see how to demonstrate it
Reminder, definition of rationality:

\begin{array}{l}\text{Let n be a rational number, and p and q two integers (without comma) then:}\\ n = \dfrac{p}{q}\end{array}

Therefore, there is no fraction of whole numbers such that

 \pi = \frac{p}{q}

Moreover, Pi does not seem to contain any repeating patterns in its decimals. A pattern being a sequence of recurring decimals, id is which recur regularly in the total sequence.

Fact#2 Reciting Pi, a Guinness World Record in its own right

For many people, Pi is fascinating, to the point that some have had fun learning its decimals by heart. A good first way is to have a Pi cup at home, like you can find at home. City of Sciences in Paris – La Villette :

mug pi
Mug Pi, ideal gift for a budding mathematician / mathematician (we have more ideas like this here)

There are books and many websites where you can read and learn the decimals of pi, like on gecif.net for example. So people had fun memorizing them and reciting them!

A great French stand-uper and youtuber in this area is Fabien Olicard who would know several thousand decimals of pi! You can see footage from one of his comedic shows below. He promotes memorization and lends himself to the exercise of reciting Pi, in a surprising and unexpected way.

Official Guinness Pi recitation records are held in 2005 by Chinese Lu Chao, who recites 67 890 decimals in 24 hours and 4 minutes. Then in March 2015, the record goes to 70 decimal places recited in 9 hours 27 minutes by the Indian Rajveer Meena. Then in October 2015 to another Indian Suresh Kumar Sharma who recited 70 030 decimals in 17 hours and 14 minutes.

Fact#3 A lake named Pi

There is a Lake in Canada by the simple name of " Lake 3.1416 where 3.1416 is exactly the first 4 decimal places of pi rounded to the nearest unit. According to the Toponymy Commission of Canada (those who distribute the names of streets, forests, cities, etc.), the name refers to three vacationers who owned a camp whose dimensions are 14 feet (4,26 m) wide by 16 feet (4,9 m) long.

ft lake
Lake 3.1416 coordinates 49° 45′ 07″ nord, 68° 06′ 48″ ouest

Fact#4 Pi in culture

The movie Pi

Pi is a psychological thriller film released in 1998. This film tells the story of a young mathematician who thinks that "Nature is a book written in mathematical language", this being a direct reference to GalileoExtract, the Assayer, published in 1623. In the film, religion, money, the stock market and mathematical obsessions are intertwined. The protagonist would thus discover in the decimals the true name of God or how to become rich with the stock markets. We don't recommend doing the same, but to get rich with high school math, you can watch this way.

Pi in a novel

In the pure fiction novel Contact by Carl Sagan published in 1985, we discover Ellie Arroway, a young astronomer convinced of the existence of intelligent extraterrestrial life. In his laboratory, one day, the computers pick up a rational message emitted not from Earth but from Vega, a star located about 25 light years. Ellie then throws herself wholeheartedly into her decryption, where Pi is led to play a decisive role.

Pi: music!

The singer-songwriter Kate Bush, rock, progressive rock, alternative rock influenced music dedicated to Pi on his 2005 album Aerial. The title is simply Pi. the number ¨Pi. He would have been inspired by Daniel tammet, an English poet and hyperpolyglot who recited 22 digits of Pi in 5 hours, 9 minutes and 24 seconds in March 2004.

Fact#5 Pi the logo of Progresser-en-maths.com

And yes ! Pi is also in our logo. We decided to take pi as an emblem, because it is known to the general public and we aim to popularize math. We also chose Pi because, as you will see later, it is an object that remains a great mystery for mathematicians. In particular because it has been known for a very long time and its properties pose a definitively insoluble problem, even a shame: squaring the circle.

progress in math
Progress in Maths

History and usefulness of Pi

Pi was almost born at the same time as mathematics. Indeed, the beginning of geometry involving the drawing of circles quickly involved the object Pi. This in order to understand the proportions and the laws governing the figures.

Let's start with a small animation to understand the relationship between Pi and the radius or diameter of a circle.

File:Pi-unrolled-720.gif - Wikipedia

Babylon and Pi

The first to look for a relationship between the perimeter of the circle and its diameter were the Babylonians. As well as the search for the ratio between the square of the radius and the area of ​​the disc is also associated with them. We already have in 2000 BC, the following approximation :

\pi \ =\ 3\ +\ \frac{1}{8}\ =3,125

Not bad ! Although the method a bit complex if you click on the reference link.

Pi in ancient Egypt, for the pyramids

Also around 2000 BC, the rhind papyrus from the Egyptian scribe Ahmes offers another method to evaluate Pi. His idea is simple, calculate the area of ​​a disc of radius 1, by inscribing it in a square of side 2:

upper frame pi

Of course the area of ​​the square is:

Area_{square}\=\c\ \cdot \c\=2\ \cdot 2\=\4

Ahmes therefore says that the area of ​​the circle is smaller and is worth something (Pi) multiplied by the radius squared. He then attempts to square the circle. This consists of obtaining a square with an area equal to a disc. It is therefore still an unresolved problem today. So Ahmes seems to find iteratively that the area of ​​a disc with a diameter of 9 units is substantially equal to the area of ​​a square with a side of 8 units. This approximation results in this equality: 

\begin{array}{c}Area_{square}\=\ c\ \cdot \ c\=8\ \cdot 8\=\ 8^{2}\\ Area_{circle}\=\ \pi \ \cdot \left(\frac{diameter}{2}\right)^{2}\ =\ \pi \ \cdot \left(\frac{9}{2}\right)^{2}\\ Or\ Area_{ square}\ =\ Area_{circle}\ \\ So\ \pi \ \cdot \left(\frac{9}{2}\right)^{2}\ =\ 8^{2\ }\\ Let\ \pi \ =\ \frac{8^{2}}{\left(\frac{9}{2}\right)^{2}}\ =\frac{256}{81}\\ So\ \pi \ \approx 3,16\end{array}

To revise the formulas, on the areas, it is here

Pi in India, for astronomy

The study of the stars led the Indians to find a remarkable approximation. the indian Aryabhata gives, at the beginning of the VIe century AD. J.-C., a more precise approximation with the following fraction: 62/832 ≈ 20. The result is exact to 000−5 near. On Wikipedia, there seems to be no justification for this result. Moreover, Aryabhata would favor the use of roots of 10 (approximately equal to 3,1623) in his astronomical calculations.

Pi in Greek antiquity

Archimedes (287 to 212 BC), a mathematician and physicist of Greek Antiquity, to whom we owe Archimedes' Push, is a major player in the understanding of Pi. In his treatise Of the measure of the circle, Archimedes uses several geometry tricks in order to provide a framework for the value of Pi and it's pretty cool. For this, he inscribes a circle in a square, and a regular polygon in the circle.

To make it simple, we will take a hexagon. It is a regular polygon with 6 sides. Archimedes will use a regular polygon of 96 sides for more precision. This idea comes from the fact that we can try to cut a circle into a triangle, like this:

Cutting a circle into 8 portions, r is the radius, p is the number of portions.

Frame by perimeter

Let's take the first figure used and write a hexagon in it.

pi frame
Circle inscribed in square of side 2 and hexagon inscribed in circle of radius 1. Approximation of Pi by Archimedes' method

Note that the perimeter of the square is 2 * 4 = 8. The perimeter being the line which delimits the contour of a figure. So for the square the length of the side (2) times the number of sides (4). Note that that of the hexagon is 6. Indeed, if we cut the hexagon into triangles as in the figure, these are isosceles of length 1. There are 6 sides, therefore 6 * 1 = 6.

We therefore have a first framework given by:

\begin{array}{l}Perimeter\ square\ >\ Perimeter\ circle\ >\ Perimeter\ hexagon\ \\ 8\ >\ \pi \ \cdot 2\cdot radius\ >\ 6\ \\ \Leftrightarrow \ \ frac{8}{2}\ >\ \pi \cdot radius\ >\ \frac{6}{2}\\ or\ radius\ =\ 1\\ \Leftrightarrow \ 4\ >\ \pi \ >\ 3 \end{array}

Frame by area calculation

We take up the remarks of the scribe Ahmes. By relying again on the diagram just before, we find that the area of ​​the square side x side = 2 x 2 = 4. For the calculation of the area of ​​the hexagon it is also simple. We calculate the area of ​​one of the triangles of the hexagon which we then multiply by the number of triangles, i.e. 6.

\begin{array}{l}Area\ d^{\prime}un\ \triangle \ :\ \frac{Base\ \cdot \ Height}{2}\end{array}

So let's cut out one of our triangles to find the base, like this.

Pi estimate

We have the hypotenuse = 1 by definition and the short side = 1/2 = 0,5. To find the length of the long side which will be our base we use the theorem of Pythagoras (here is a reminder here).

\begin{array}{l}&1^{2}=0,5^{^2}+x^{2}\\ \Leftrightarrow &1^{2}-0,5^{^2}\ =x^ {2}\\ \Leftrightarrow &x^{2}=\left(1-0,5\right)\cdot \left(1+0,5\right)\\ \Leftrightarrow &x^{2}\ =0,5, 1,5\cdot 3\\ \Leftrightarrow &x\=\ \sqrt{\frac{4}{XNUMX}}\\ &\end{array}

So our basis is root three-quarters. And the area of ​​our isosceles equilateral triangle is:

\begin{array}{l}Area_{triangle}\ =\ \frac{\left(1\cdot \sqrt{\frac{3}{4}}\right)}{2}\\ \\ Area_{triangle }\ =\ \frac{\frac{\sqrt{3}}{\sqrt{4}}}{2}\\ \\ Area_{triangle}\ =\ \frac{\sqrt{3}}{\sqrt {4}}\cdot \frac{1}{2}\\ \\ Area_{triangle}\ =\ \frac{\sqrt{3}}{2}\cdot \frac{1}{2}\\ \ \Area_{triangle}\=\\frac{\sqrt{3}}{4}\end{array}

So finally, here is the area of ​​our hexagon.

\begin{array}{l}Area_{hexagon}\=\6\cdot \ \frac{\sqrt{3}}{4}\\ \\ Area_{hexagon}\=\ \frac{3\sqrt{3 }}{2}\\ \\ Area_{hexagon}\ \ \approx \ 2,5981\end{array}

The bounding by the area of ​​pi becomes:

\begin{array}{l}Area_{square}\ >\ Area_{circle}\ >Area_{hexagon}\\ c\cdot c\ >\ \pi \ \cdot \ radius^{2}\ >6\cdot c\cdot \frac{\left(base\cdot height\right)}{2}\\ 4\ >\pi \ \cdot \ radius^{2}\ >\ \frac{3\sqrt{3)}{ 2}\\ Or\ radius\ =\ 1\ and\ 1^{2}=1\\ So\ 4\ >\pi \ >\ \frac{3\sqrt{3}}{2}\end{array }

Archimedes gives this framing of Pi:

\begin{array}{l}3+\frac{10}{71}\ <\ \pi \ <\ 3+\frac{10}{70}\\ \\ With\ 3+\frac{10}{ 71}\ \approx 3,1408\\ \\ And\ \ \ \ \ 3+\frac{10}{70}\ \approx 3,1429\end{array}

Why is Pi called Pi exactly?

According to Wikipedia : the greek letter "π" is the first of the Greek word περιφέρεια (periphery, that is to say circumference). It therefore symbolizes the ratio of the circle to its diameter from the XNUMXth century.

"The first to simply use π is William Jones in his book Synopsis palmariorum mathesios published in 1706, about the clever calculation of this number by the series of his friend John thing. Mathematicians, however, continue to use other notations. Among these Euler takes up the notation of Jones in his correspondence from 1736. He adopts it in his book Introduction in analysin infinitorum published in 1748, which certainly had a great influence. The notation ends up imposing itself towards the end of the XNUMXth century”.

Indeed, Pi was also used in reference to the diameter and therefore its value was doubled. Mathematicians sometimes wrote π/δ where δ designates the diameter.

Pi since the 1900s and computing

The focus was on calculating as many decimal places of Pi as possible. From a math problem, Pi has become a computer problem. How, using the available resources, calculate the most decimals of Pi?
3 solutions:

  • Increase available resources. But this is not always possible. And if we want to double the resources, we will have to double the price.
  • Use formulas that converge faster.
  • Time is a doubly important factor: The longer we run the algorithm, the more decimals of Pi we can calculate. And also if we wait years, without running the algorithm, then Moore's law comes into play. And so for the same price, we will have access to many more resources.

For most purposes, 355/113 gives an approximation of Pi with 7 decimal places, is more than enough. But the fact of wanting to calculate the most possible decimals of Pi has a double utility. On the one hand, it is necessary to have greater precision in the calculations. On the other hand, it is also an indicator of technological progress. The more decimals of Pi we are able to calculate, the more technologically advanced we are.

Bonus: The elegance of Pi in math formulas

For the beauty of the math, here is a list of formulas that result in π.

With the arctangent formula

\pi = 4 \sum_{n=0}^{+\infty} \frac{(-1)^n}{2n+1} 

From Fourier series

\begin{array}{l}\pi\=\ \sqrt{6\ \sum_{n=1}^{+\ \infty}\frac{1}{n^2}}\end{array}
\begin{array}{l}\pi \=\ \sqrt[4]{90\ \sum _{n=1}^{+\ \infty }\frac{1}{n^4}}\end{ array}

Thanks to Ramanujan's formulas

\pi\ =\frac{9801}{2\sqrt{2}\sum_{k=0}^{+\infty}\frac{\left(4k\right)!\left(1103+26390k\right)}{\left(k!\right)^4369^{4k}}}\ 

One of Machin's formulas

\pi\ =\ 16\ \arctan\left(\frac{1}{5}\right)\ -\ 4\ \arctan\left(\frac{1}{239}\right)

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