by rynde

In mathematics number of math symbols and Constants are used. The word pi is one of the major constant in mathematics. The symbol pi is the Greek alphabet and pi symbol is denoted as (Pi, pi). The symbol pi is the 17th letter in the Greek system. The word pi is used in so many fields like mathematics and science operation. Pi is the irrational number that specifies value cannot be expressed exactly as fraction that is ‘a/b’ in these case a and b is the two integers.
More about Pi Derivation:

Pi equation can be calculated by using various algorithms. Some of the pi equations are given below,

Geometric Derivation for pi:

Pi is empirically estimated by sketching a large circle, and then determine its diameter and circumference and then dividing the circumference by the diameter.

Consider Euclidean plane geometry pi is specified as the ratio of a circle circumference ‘C’ to its diameter ‘d’

‘Pi=C/d’

In above formula c/d is constant. A specifies the area of the square and derivation of pi is

‘Pi=A/r^2’
Brent-Salamin algorithm:

In 1975 when Richard Brent and Eugene Salamin independently find the Brent-Salamin algorithm which uses only arithmetic to double the number of correct digits at The algorithm consists of setting

‘a_0=1’ , ‘b_0=1/sqrt(2)’ ,’ t_0=1/4′ , ‘p_0=1’

Use iterating process

‘a_(n+1)=(a_n+b_n)/2’

‘b_(n+1)=sqrt (a_n,b_n)’

‘t_(n+1)=t_n-p_n(a_n-a_(n+1))^2’

‘P_(n+1)=2p_n’

Using the 25 iterating process to find the pi equation

‘Pi=(a_n+b_n)^2/4t_n’

Using BBB formula to find the Derivation for pi is

‘Pi=sum_(k=0)^(oo)p(k)/(b^(ck)q(k))’

In the above equation b and c are positive integer and p, q are polynomial

Substitute ‘b=2 ‘ and ‘c=4 ‘ find the constant pi

‘Pi^k=sum_(n=1)^(oo)(a/(q^n-1))+(b/(q^(2n)-1))+c/(q^(4n)-1)’

Using the above equation to find the pi value.
Value for Derivation of Pi:

Pi does not have any physical unit. Pi value is already derived. The derived pi value is given below

Rational value of Pi (pi) =3,’22/7′ ,’355/133′ ,’103993/33102′ ,…

Binary value of Pi (pi) = 11.00100100001111110110…

Decimal value of pi (pi) =3.14159265358979323846264338327950288…

Hexadecimal value of pi (pi) = 3.243F6A8885A308D313….

Continued fraction of pi (Pi) =3,7,15,1,292,1,1,1,2,1,3,1,14,2,1,1,…

The number π (/paɪ/) is a mathematical constant that is the ratio of a circle’s circumference to its diameter. The constant, sometimes written pi, is approximately equal to 3.14159. It has been represented by the Greek letter “π” since the mid-18th century. π is an irrational number, which means that it cannot be expressed exactly as a ratio of two integers (such as 22/7 or other fractions that are commonly used to approximate π); consequently, its decimal representation never ends and never repeats. Moreover, π is a transcendental number – a number that is not the root of any nonzero polynomial having rational coefficients. The transcendence of π implies that it is impossible to solve the ancient challenge of squaring the circle with a compass and straight-edge. The digits in the decimal representation of π appear to be random, although no proof of this supposed randomness has yet been discovered.

For thousands of years, mathematicians have attempted to extend their understanding of π, sometimes by computing its value to a high degree of accuracy. Before the 15th century, mathematicians such as Archimedes and Liu Hui used geometrical techniques, based on polygons, to estimate the value of π. Starting around the 15th century, new algorithms based on infinite series revolutionized the computation of π, and were used by mathematicians including Madhava of Sangamagrama, Isaac Newton, Leonhard Euler, Carl Friedrich Gauss, and Srinivasa Ramanujan.

In the 20th century, mathematicians and computer scientists discovered new approaches that – when combined with increasing computational power – extended the decimal representation of π to over 10 trillion (1013) digits. Scientific applications generally require no more than 40 digits of π, so the primary motivation for these computations is the human desire to break records, but the extensive calculations involved have been used to test supercomputers and high-precision multiplication algorithms.

Because its definition relates to the circle, π is found in many formulae in trigonometry and geometry, especially those concerning circles, ellipses, or spheres. It is also found in formulae from other branches of science, such as cosmology, number theory, statistics, fractals, thermodynamics, mechanics, and electromagnetism. The ubiquitous nature of π makes it one of the most widely known mathematical constants, both inside and outside the scientific community: Several books devoted to it have been published; the number is celebrated on Pi Day; and news headlines often contain reports about record-setting calculations of the digits of π. Several people have endeavored to memorize the value of π with increasing precision, leading to records of over 67,000 digits.

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