Determination of π value

Determination of π value

Author: Ashokkumar Gokuldas Heda

Corresponding address:

Ashokkumar Gokuldas Heda

Sahyog Medicals,

Kedia plots, Convent Roads,

Akola – 444 005 (M.S.)

INDIA

Phone: +91 724 2453081

Email: aaheda@gmail.com

INTRODUCTION

Pi is a mathematical constant whose value is the ratio of any circle’s circumference to its diameter; this is the same value as the ratio of a circle’s area to the square of its radius. The symbol π was first proposed by the Welsh mathematician William Jones in 1706. It is one of the most important mathematical and physical constant: many formulae from mathematics, science and engineering involve π. Throughout the history of mathematics; there has been much effort to determine π more accurately and to understand its nature. Reports on the latest, most precise calculation of π are common news items; but one question always remained – What is the exact numerical value of Pi?

The first theoretical calculation seems to have been carried out by Archimedes of Syracuse (287 – 212 BC). He obtained the approximation

223/71 < π < 22/7

Archimedes knew, what so many people to this day do not, that π does not equal to 22/7, and made no claim to have discovered the exact value. According to Archimedes, if π does not equal to 22/7 and rather is less than that; the question arises how less it is than 22/7?

Fascination with the number has even carried over into non-mathematicians, like me. So, I have made an attempt to determine this numerical value of π by performing a small experiment as detailed below.

EXPLANATION

I hope everyone is aware of one of the number 7’s greatest properties that any number (except multiple of 7 like 7, 14, 21, 28, 35, etc) if divided by 7, the numbers after the decimal will always contain the endlessly reoccurring six digit numerical pattern of 142857 142857 142857 142857 142....... and so on.

For a circle having diameter 7 cm, the circumference is measured to be 22 cm and the ratio of circumference to diameter on a calculator is displayed as 3.142857142857....; an infinite number. Hence, it is for sure that the value of π after the decimal will contain reoccurring six digit pattern of 142857 142857 142..... and so on.

However; in my opinion, for any circle, the ratio of circumference to its diameter must be constant and not infinity. So, where exactly the things might be going wrong? In my opinion; the length of circumference measured as 22 cm may be wrong and it might be little less than that. Let’s see how?

STEP I – Diagrammatic Experiment

The border of a circle at circumference is having somewhat thickness, so we can say that it has outer and inner edges (Fig. 1a). When we cut a circle at a point and stretch open to a straight line, we get crossed ends on either sides of a straight line (Fig. 1b). Even if the length of outer side is 22 cm, the length of inner side is lesser that 22 cm. Let the length of inner side be 21.98 cm. Cut the line exactly in the middle as shown in Fig. 1c and then join together at cross ends as shown in Fig. 1d. The length on outer as well as inner side now is equal to 21.99. This provides us the clear evidence that circumference measured definitely must be lesser than 22 cm. Now next question comes, how less is it than 22 cm?

STEP II – Approximation of decimal expression of circumference

The basic fundamental idea of this work is removal of infinity from π value by approaching the closest approximate decimal expression of circumference. In order to find out the correct decimal places for π, let us first find out the correct decimal expression of circumference of a circle having diameter 7 cm. This is demonstrated in the table below:

Circumference (C)	Diameter (d)	π = C/d	Verification* π x 7
21.9	7	3.12857142857...∞	21.8999999999...∞
21.99	7	3.14142857142...∞	21.9899999999...∞
21.999	7	3.14271428571...∞	21.9989999999...∞
21.9999	7	3.14284285714...∞	21.9998999999...∞
21.99999	7	3.14285571428...∞	21.9999899999...∞
21.999999	7	3.142857	21.999999
21.9999999	7	3.14285712857...∞	21.9999998999...∞
21.99999999	7	3.14285714142...∞	21.9999999899...∞
21.999999999	7	3.14285714271...∞	21.9999999989...∞
21.9999999999	7	3.14285714284...∞	21.9999999998...∞
21.99999999999	7	3.14285714285...∞	21.9999999999...∞

 * as calculated on 12 digit calculator

When the decimals places for circumference are increased up to 5 digits, the value of π is infinity. But, at six digit decimal places, π = 3.142857, which is a constant. Further, as the decimal places are increased up to 11 places, the value of π again is infinity.

Moreover, when these calculated values of π are re-multiplied by 7 for verification, circumference value changes from the original for all anticipated values except 21.999999.    

Therefore, for a circle having diameter 7 cm, the correct circumference is 21.999999 cm which is only 1 micro-millimetre less than 22 cm. Practically, this minute difference in circumference may not be measured by scale. In larger scale units, this difference is 1 mm in 22 km; i.e. 21km 999m 99cm 9mm.where the diameter of the circle is 7 Km. This is also in agreement with Archimedes’s opinion that value of π is less than 22/7.

CONCLUSION

The idea of this work is based on the fact; for any circle, the ratio of circumference and diameter must be constant and not infinity. Hence, for a circle having diameter 7 cm, the best approximation of circumference is equal to 21.999999 cm and correspondingly the constant value of π is equal to 3.142857. The number 142857 obtained after the decimal is nothing but a magic figure. All the numerals from 0 to 9 appear directly or indirectly in this specific value of π without repetition.

·      Numerals 1, 2, 3, 4, 5, 7 and 8 are directly present in the value of π.

·      There are six digits after the decimal in this value, thus 6 is indirectly involved.

·      1+4+2+8+5+7 = 27; 2+7 = 9. Thus, 9 is also indirectly involved in this value.

·      What about zero? A circle is after all a Zero.

 Thus π = 3 + (One Millionth of the magic figure)
             = 3.142857, which is hidden in the Indian old currency.

         

The Circles, Themselves are able to prove the value of π

The Circles, Themselves are able to prove the value of π

The Circles, Themselves are able to prove the value of π