The beauty of mathematics is so vast that you can never imagine what next it will bring out to you!! “There are patterns everywhere in the nature and in Mathematics, these patterns appear in the most beautiful forms” Late Ramanujan said. That’s indeed true….. The shine of sunlight on water, the texture of soil in places, the movement of leaves with the breeze all show different patterns!! Ramanujan also said “An equation has no meaning to me unless it expresses the thought of god in it”. He claimed that his god “NAMAGIRI” gave him ideas of new theorems in his dreams during his sleep!! Now let’s talk about our topic today which is one of the wonders of mathematics- THE NEVER ENDING NUMBER ‘Π’. Π or pi (pronounced as p-a-i) is an irrational number… that’s all which defines pi!! An irrational number is a number which is not rational or which cannot be expressed p/q where q is not equal to 0 and p and q are co-primes. Pi is an non-terminating non-recurring decimal which means after the decimal point in this number there are infinitely many numbers and these numbers do not recur or the same numbers do not repeat in a pattern. Pi or Π is a transcendental irrational number which means it is not the solution of any polynomial equation. Other important transcendental irrational numbers are for example ‘e’ – Euler’s number, ‘φ’ – Golden ratio and etc. Now, you may not know what is a solution of a polynomial equation. So, let there be a polynomial equation f(x) with the variable x. The solution of f(x) is said to be a number ‘a’ which is substituted in the place of x such that f(a)=0. Therefore Π can never be a solution of a polynomial equation… This is how pi can be easily defined. Now let’s look on to the next section where we see how it was originated which most of you may be knowing!!

THE RELATION TO PI TO A CIRCLE:

Actually pi was defined in terms of the elements of a circle. It was observed 1000’s of years ago that the ratio of the circumference to the diameter of a circle is always a constant no matter how big the circle is!! Interesting right?? So let there be a circle(o,r) (this is how a circle can be represented) where o is the center and r is the radius of the circle then the diameter is 2r. Now Π=Circumference/2r. As before we didn’t knew how to calculate the circumference of the circle after the observation of the constant Π, Circumference=2r x Π which is 2Πr. Now the question arises that what is the value of this amazing constant Π.

APPROXIMATIONS OF PI:

As said before pi being irrational is a non-terminating and non-recurring decimal. Therefore the value of Π can extend indefinitely. One can think of one trillion digits of Π but still there is more. Since ancient times mathematicians have tried to approximate the  value of Π as they can never be accurate in the name of Π. You may be knowing some of the fractions that are used to easily approximate Π. There are rational number such as 22/7, 355/113, 223/71 and so on but remember that Π≈ 22/7 or 355/113 or any other fraction. In the 3rd century bce, one of the greatest mathematicians Archimedes proved that 223/71<Π<22/7 which means the value of Π lies somewhere between 3.1408450704 and 3.142857142857. This indeed was a great achievement which helped in calculating hundreds or may be millions of digits!! Now notice that if you divide 22/7 you get a value like 3.142857 where the ‘142857’ after the decimal point starts recurring and thus this fraction is used for many practical purposes and also for teaching. Many other people took different fractions such as 25/8=3.125, 256/81≈ 3.16, 377/120 = 3.1416, 3927/1250  = 3.1416 which were correct up to several decimal places!! The Chinese accurately calculated pi up to 7 seven decimal places. In the Gupta-era, Aryabhata calculated pi to be 62832/20000 = 3.1416 which he used  to calculate the circumference of Earth. In the 14th century, the indian mathematician Madhava discovered an infinite series to calculate pi up to 12 correct decimal places. The Madhava-leibniz infinite series was : \pi ={\sqrt {12}}\sum _{k=0}^{\infty }{\frac {(-3)^{-k}}{2k+1}}={\sqrt {12}}\sum _{k=0}^{\infty }{\frac {(-{\frac {1}{3}})^{k}}{2k+1}}={\sqrt {12}}\left(1-{1 \over 3\cdot 3}+{1 \over 5\cdot 3^{2}}-{1 \over 7\cdot 3^{3}}+\cdots \right) Infinite series are the continuous summation of some numbers which tend to infinity giving some beautiful values. A very aesthetic definition of infinite series right?? If you want to know more about the infinite series click here. It is not a ad it will redirect you to wikipedia!! So, Madhava used the first 21 terms of the series to obtain the value of Pi as 3.14159265359 which is correct to 11 decimal places. In the 20th century, Srinivasa Ramanujan discovered another series which could give futher 8 decimals of Pi with each term you use: {\frac {1}{\pi }}={\frac {2{\sqrt {2}}}{9801}}\sum _{k=0}^{\infty }{\frac {(4k)!(1103+26390k)}{(k!)^{4}396^{4k}}} His series are the basis for the fastest algorithms which are used nowadays.. And now this pi was making people more curious about it’s value and this curiosity lead to the calculation of Pi with 12 trillion decimal places. In the 21st century Pi was calculated on computers and calculators making it faster and easier to calculate.  Yasumasa Kanada used a 64-node supercomputer with 1 terabyte of main memory, to calculate π to roughly 1.24 trillion digits in around 600 hours. There were further sevral records that were made and broken and the latest record of calculating Pi is 22.4 trillion digits of Pi calculated by Peter Trueb in November 2016 using his y-cruncher!!! Crazy enough right??

USING CONTINUED FRACTIONS TO COMPUTE Π:

Continued fractions are special type of fractions which extend indefinitely toward infinite. Every irrational number can be represented in the form of  a continued fractions instead of a common fraction including Pi. The continued fraction for pi is: {\displaystyle \pi =3+\textstyle {\cfrac {1}{7+\textstyle {\cfrac {1}{15+\textstyle {\cfrac {1}{1+\textstyle {\cfrac {1}{292+\textstyle {\cfrac {1}{1+\textstyle {\cfrac {1}{1+\textstyle {\cfrac {1}{1+\ddots }}}}}}}}}}}}}}} As you can see this fraction continues to infinity. It is simply 3+1/7+1/15+1+…… Taking this continued fraction at any point gets you values discussed before like 22/7, 333/106, and 355/113.

USES OF PI:

Pi is generally used in calculations of the geometry of a circle, sphere, hemisphere, ring and etc. There are generalised formulas such as Πr2 , 4/3Πr3 . It is used in the mensuration of figures where a circle is used such as Cylinder, Cone. It is also used in Euler’s identity where most of important constants in mathematics is used. Euler’s identity  is eiΠ + 1=0 where e is the Euler’s number, i is an imaginary number and Π is the constant Pi.   In the meantime, I was also trying to approximate Pi and I corrected it up to 8 decimal places!! I used the fraction 357/113.63662939 which is equal to 3.1415926529. Did you ever come over to an approximation of Pi?? If so comment it below and always remeber that all of us are merely the wanderers as formulae are not created by us and they do exist but it is the work of one the greatest minds to find it (which may be you!!). Want to share interesting facts visit our “BE A WRITER YOURSELF” page and let the world know what you want them to know.

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