Hello everyone, you will get some interesting results that might be not known to many of you. You will get some fun results at the end of this page. Many people are wondered about π but only a few of them are literally know about π and its origin. In this post, I will show you What is π, Its origin, Proof of the value of π.
Origin of π (Information taken from Wikipedia)
The number π is a mathematical constant. It is defined as the ratio of a circle's circumference to its diameter, and it also has various equivalent definitions. It appears in many formulas in all areas of mathematics and physics. The earliest known use of the Greek letter π to represent the ratio of a circle's circumference to its diameter was by Welsh mathematician William Jones in 1706. It is approximately equal to 3.14159. It has been represented by the Greek letter "π" since the mid-18th century, and is spelled out as "pi". It is also referred to as Archimedes' constant.
What is the exact value of π
Actually the value of π can't be calculated accurately since it is an irrational number. But there is some approximations for the value of π such as 22/7.
\[\frac{22}{7}\]
This is a close value to π. It is correct up-to 2 decimals. There is more accurate approximation for this value 355/113.
If you take a simple number 113355 then split it form middle (113-355) then flip it (355-113) now you got that number.
\[\frac{355}{113}\]
This value is correct up-to 6 decimals. Here I will show it.
22/7 = 3.14285714
355/113 = 3.1415929
π = 3.1415926......
Definition of π
π is commonly defined as the ratio of a circle's circumference C to its diameter d.
The ratio C/d is constant, regardless of the circle's size. For example, if a circle has twice the diameter of another circle, it will also have twice the circumference, preserving the ratio C/d. This definition of π implicitly makes use of flat (Euclidean) geometry; although the notion of a circle can be extended to any curve (non-Euclidean) geometry, these new circles will no longer satisfy the formula π = C/d.
Here, the circumference of a circle is the arc length around the perimeter of the circle, a quantity which can be formally defined independently of geometry using limits—a concept in calculus. For example, one may directly compute the arc length of the top half of the unit circle, given in Cartesian coordinates by the equation x2 + y2 = 1, as the integral.
How to calculate value of π on calculator.
Here I will show you two Technique with their Proofs by which you can calculate value of π on calculator.
1. By calculating area of circle
We know that area of circle is πr^2. If we consider r is 1 then area of circle will be π.
For calculating Area of circle, we draw an n-sided regular polygon inscribed in circle.
Then we divide that polygon in n number of triangles. Now we can easily calculate area of polygon since all triangles are similar as shown in figure:
Now the interior angel for all triangle is θ
\[θ=\frac{360}{n}\]
By using trigonometry, we can find value of h and x
\[h=r.cos\frac{θ}{2}=1.cos\frac{360}{2n}=cos\frac{180}{n}\]
\[\frac{x}{2}=r.sin\frac{θ}{2}=1.sin\frac{360}{2n}=sin\frac{180}{n}\]
\[x=2\times sin\frac{180}{n}\]
Area of polygon = No. of Triangles x Area of one triangle
Area of polygon = n x T
First we calculate value of T:
\[T=n\times x\times h\]
\[T=n\times cos\frac{180}{n}\times 2\times sin\frac{180}{n}\]
\[T=2\times n\times cos\frac{180}{n}\times sin\frac{180}{n}\]
\[T=\frac{n}{2}\times sin\frac{360}{n}\]
Now area of polygon
\[A=\frac{n}{2}\times sin\frac{360}{n}\]
In this if we increase the value of n then we got value close to π. Best thing in this that we can calculate it simply on calculator.
If n=100, we get 3.1395....
If n=1000, we get 3.14157....
If n=10000, we get 3.1415924....
If n=100000, we get 3.141592652....
True value of π = 3.14159265358....
It's formula can be written as:
\[\pi=\lim_{n \rightarrow \infty}\frac{n}{2}sin\frac{360}{n}\]
2. By calculating Perimeter of Circle
We know that perimeter of circle is 2πr. If we consider r is 1/2 then perimeter of circle will be π.
For calculating Perimeter of circle, we draw an n-sided regular polygon inscribed in circle.
And then we divide that polygon in n number of triangles. Now we can easily calculate Perimeter of polygon since all triangles are similar as shown in figure:
Now the interior angel for all triangle is θ
\[θ=\frac{360}{n}\]
By using trigonometry, we can find value of L
\[\frac{L}{2}=r.sin\frac{θ}{2}=\frac{1}{2}.sin\frac{360}{2n}=\frac{1}{2}.sin\frac{180}{n}\]
\[L=sin\frac{180}{n}\]
Perimeter of polygon = No. of Triangles x Length of outer side
Perimeter of polygon = n x L
\[P=n\times sin\frac{180}{n}\]
In this if we increase the value of n then we got value close to π. Best thing in this that we can calculate it simply on calculator.
If n=100, we get 3.1410....
If n=1000, we get 3.141587....
If n=10000, we get 3.141592602....
If n=100000, we get 3.14159265307....
True value of π = 3.14159265358....
As you can observe, this will give more accurate value of π
It's formula can be written as:
\[\pi=\lim_{n \rightarrow \infty}\ n.sin\frac{180}{n}\]
Some fun facts about π
1. (π^4 + π^5)^1/6 = 2.718281809 and e = 2.718281828
∴ (π^4 + π^5)^1/6 ≈ e
2. It's actually euler's identity\[e^{i\pi}+1=0\]
3. Infinite sequence for π:\[\frac{\pi}{6}=\frac{1}{2}\frac{3}{2}\frac{3}{4}\frac{5}{4}\frac{5}{6}\frac{7}{6}\frac{7}{8}\frac{9}{8}.....\]or\[\frac{\pi}{6}=\sum_1^\infty\frac{n}{n+1}.\frac{n+2}{n+1}\]Or simply write:\[\frac{1}{2}\frac{3}{2}\]And increase each value by 2 and you will get exactly same result.
4. Relation with value of g:\[\pi\approx\sqrt{g}\]
5. If you learn first 38 digits of π then you have a 38 digit prime number.31415926535897932384626433832795028841
6. By using 200 digits of π you can calculate volume of observable universe with precision of an atom but the fact is you have to calculate radius of observable universe with minimum 100 decimal of precision.
7. Some of you will think 'Total life was a lie, now I can die peacefully' by knowing this result.\[\pi<\frac{22}{7}\]
31415926535897932384626433832795028841
6. By using 200 digits of π you can calculate volume of observable universe with precision of an atom but the fact is you have to calculate radius of observable universe with minimum 100 decimal of precision.
7. Some of you will think 'Total life was a lie, now I can die peacefully' by knowing this result.
\[\pi<\frac{22}{7}\]
Value of π up-to 200 decimals
3.
14159 26535 89793 23846 26433 83279 50288 41971 69399 37510 58209 74944 59230 78164 06286 20899 86280 34825 34211 70679 82148 08651 32823 06647 09384 46095 50582 23172 53594 08128 48111 74502 84102 70193 85211 05559 64462 29489 54930 38196
Click here to get value of π up-to 1 million digits.
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