The Number π

In our school, we had to choose a Science or Mathematics project for post-summer. Friends from my class selected Science projects. So I decided to go for a Math project and convinced a group of five friends. 

We planned to dedicate the project on the number Pi (my favourite number). The theme for the Maths project was ‘Maths in our Daily Life’, so we planned to show the importance of Pi.  

We agreed that for the project we would give a bit of information about Pi (even though almost everyone knows about it), then we would calculate Pi in 3 different ways (the fun part) and finally prove that Pi is very important. 

For the 1st part, we decided we would include – it’s geometrical definition, when the symbol ‘π’ was first used for it, it’s irrationality and transcendentality and some rational approximations ( $latex \frac{22}{7} &s=-2$, $latex \frac{355}{113} &s=-2$).

For the 2nd part, we decided to calculate Pi through – Ramanujan’s Pi Formula – $latex \begin{aligned} \frac{1}{\pi } \ =\ \frac{\sqrt{8}}{99^{2}} \ \sum ^{\infty }_{k\ =\ 0} \ \frac{(4k)!\ \times \ (1103\ +26390k)\ }{\left( 4^{k} k!\right)^{4} \ \times \ 99^{4k}} \  \end{aligned} $

and Gregory Series –

$latex \pi = \begin{aligned} \ 4\ \sum ^{\infty }_{k\ =\ 1}\frac{(-1)^{k+1}}{2k\ -\ 1} \end{aligned}$

and Pendulum (T = 2π $latex \begin{aligned} \sqrt{\frac{l}{g}} \end{aligned} &s=-3$ where T = Time taken to complete one oscillation, l = length of string and g = acceleration due to gravity). The Ramanujan’s Pi Formula converges rapidly but Gregory Series doesn’t converge rapidly. 

For the pendulum, we decided to make l = 0.6125 meters so that   $latex \begin{aligned} \sqrt{\frac{l}{g}} \end{aligned} &s=-3$ = $latex \begin{aligned} \frac{1}{16} \end{aligned} &s=-3$  , which means that π = T $latex \begin{aligned} \times \end{aligned} &s=-3$ 2. Then we would do 10 swings so that we get some time to get prepared to stop the stopwatch and dividing by 10 is easy.  So T = $latex \begin{aligned} \frac{t}{10} \end{aligned}&s=-3$ ( t = time it takes to complete 10 oscillations) which means π = $latex \begin{aligned} \frac{t\ \times \ 2}{10}  \end{aligned}&s=-3$  .

For the Ramanujan’s Pi Formula and the Gregory Series, I decided that I would write a Python program to run the formulas where you give the program the upper bound for the series and the formula and it would give the digits of Pi according to that. As the value of the upper bound increases, the value of Pi becomes more and more accurate.

And for the 3rd and the last part, we decided to include – What would happen to aircraft without Pi? (aircraft travels in an arc, so to calculate the fuel required as per distance it is important to know the length of the arc, and

$latex arc length = \begin{aligned} 2\pi r\left(\frac{\theta }{360}\right)  \end{aligned}$ 

which has π),  the volume and surface areas of a cylinder, a cone, a sphere, fracture of a cone, etc require π and the perimeter and area of a circle require π. 

NASA uses 15 digits of Pi to keep the International Space Station in its orbit and to launch rockets. And 40 digits are enough to calculate the volume of the entire observable Universe to the nearest atom. Then why do we need more digits of Pi? (a program running Ramanujan’s Pi Formula can calculate around 3000 digits of Pi in seconds and on the internet, you can find that people have calculated trillions of digits of π).

The reason for this is that sometimes computers are made to calculate the digits of Pi to test the speed of the computer and more and more digits of Pi may help to prove Pi is a normal number where digits are evenly distributed for a base of natural numbers. And it would be awesome if we could find a pattern that exists in Pi’s decimal expansion. But the main reason is that it is fun to get more and more digits of Pi.

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