Circle π-PI (The Mysterious number)
The circle we are all familiar with from school, but what makes it intriguing? The world you see is not always linear; a waterfall, a ball being thrown, etc. are all in a curve. Pi always has a curve-like structure. Four thousand years ago, the Egyptians used a method to determine the area of a circle; the result offered an approximation of 3.1605 for.
The area of a circle was estimated by the ancient Babylonians by multiplying the radius by three, which produced a pi value of three. Approximately 1900–1680 BC Babylonian tablets show a value of 3.125 for, which is a more accurate estimate.
A supercomputer calculated pi to a record-breaking 62.8 trillion digits, breaking the previous record. Pi value was concealed in the universal nature, and we only dug it up to get those values.
What brings PI here?
If it is simply useful for calculating circumference, area, and a few other things, that’s OK, but when it appears in so many other fields, I was unable to go asleep for a night.
Buffon’s Needle
Buffon’s Needle is one of the oldest problems in the field of geometrical probability, It involves dropping a needle on a lined sheet of paper and determining the probability of the needle crossing one of the lines on the page. The remarkable result is that the probability is directly related to the value of pi.
Online stimulation
https://mste.illinois.edu/activity/buffon/Collisions give pi value
The collision of two box with two different weights , we get the collision of pi value why ? no one knows , why pi is here ? its just blew my mind paper
To find the orbital speed of the satellite
v = (2 • pi • R)/T ,general equation for circular motion.
where R is the radius of the orbit.
The aforementioned equation, which also includes the value of pi, can be used to compute the orbit’s radius.
For Einstein’s field equation of general relativity:
It is a gravity theory.
The fundamental assumption is that gravity is a curving or warping of space rather than an invisible force that draws items together.
An item warps the space around it more dramatically the more large it is.
Searching for the value of PI
circumference =2Π r ,diameter=2r .
Every circle on this planet, from the smallest to the largest , has a circumference or diameter of 3.14, but why 3.141? Why not use other numbers?
Let’s think about it…
this letter Π on the left side is named from the Greek letter.
PI green word meaning circumference in English, π is pronounced as “pie” (/paɪ/ PY).
Let’s go for a circle first…
we know Area=𝜋 r², consider that we do not know the pi value for now, so then how to find an area of a circle?
We can divide it into equal parts and combine them to create a rectangle because we are aware that a rectangle’s area can be calculated using the formula l*w.
Therefore, we cut that into n number of slices, and merging it will answer with some minimal error since the more slices we take, the more precise the area value we can obtain.
Since they have been divided equally and combined, the length of this rectangle is 1/2 its circumference. Next, w is the radius, and since we also require pi for the circumference, we must get the value that would give us the area directly, pi * r2.
So how do you calculate pi?
We were generally aware that the formula for pi is circumference divided by diameter for any circle, small or large.
We’ll draw a circle, a square above it, and a six-sided hexagon inside of it.
then taking into account that the circle’s area falls within those two shapes.
We learned that pi is between three and four by using the method described above. but how to estimate it with some degree of accuracy.
Therefore, by enlarging the hexagon’s sides, we can obtain a higher pi value.
now by increasing the hexagon into 96 sides we will get
𝜋 >3.1408 and 𝜋 < 3.1429
The last words attributed to Archimedes are "Do not disturb my circles" (Latin, "Noli turbare circulos meos"; Katharevousa Greek, "μὴ μου τοὺς κύκλους τάραττε"), a reference to the circles in the mathematical drawing that he was supposedly studying when disturbed by the Roman soldier.
The majority of Ludolph van Ceulen’s life was devoted to determining the precise value of the mathematical constant, effectively following the same procedures as Archimedes some 1 700 years earlier.
Ludolph van Ceulen discovered 35 digits in pi by calculating the Billions and Billions of side of hexagon
2⁶² =4quadrillion sides of hexagon===> for 35 point in pi
Then, many mathematicians attempted a billion different sides, but is this approach truly practical?
Isaac Newton appears and finds a more effective way to do it.
we know the binomial series
above it counts from 0 but what if? -1 or m/n form that was tried of Isaac.
if n==-1
then
(1+x)^-1 = 1 − 1x + 1x2 − 1x3 + 1x4 − 1x5 +..series continuous forever.
else if n==1/2
then
(1+x)^1/2 = 1+1/2∙x +(1/2)(1-(1/2))x^2/2!+....series continuous forever.if n<1:series continuous forever.
for now solving above we will get an series like
Newton’s Series
(1-x²)¹/2= 1–1/2x² +1/8(x)⁴+1/16(x)⁶+….
Check out brilliant for integrating area searching.
Newton tried the above integration from 0 to 1/2 instead of 0 to 1, which will speed up the calculating process and yield a far better result of pi.
Pi is equal to 3.1416 after the integration (0,1/2). The more terms you utilize in Newton’s series, the more digits you will get.
2⁶² =4quadrillion sides of hexagon===> for 35 point in pi (Ludolph van Ceulen almost spent is 25 years in his life)
equals to
newton’s series 50 term calculation ===> for 35 point in pi
there are many formulas that came after that Chudonovsky algorithm by Ramanujam, Machin-like formulae …
For us, the scale of one that we derived and decided it, but perhaps nature itself has scale of Pi as beginning one or starting point of something, it has opened many fields in math, quantum mechanics, etc., and it will open many new as well. Pi value is always a mystery. It is not just talking with the circle, but with all geometric space and time related things and many of them.
