Pi occurs in various mathematical calculations. The circumference (c) of a circle can be determined by multiplying the diameter (d) by : c = d. The area (A) of a circle is determined by the square of the radius (r): A = r2. Pi is applied to mathematical problems involving the lengths of arcs or other curves, the areas of ellipses, sectors, and other curved surfaces, and the volumes of solids.
It is also used in various formulas of physics and engineering to describe such periodic phenomena as the motion of pendulums, the vibration of strings, and alternating electric currents. In very ancient times, 3 was used as the approximate value of pi, and not until Archimedes (3rd century BC) does there seem to have been a scientific effort to compute it; he reached a figure equivalent to about 3. 14. A figure equivalent to 3. 1416 dates from before AD 200.
By the early 6th century Chinese and Indian mathematicians had independently confirmed or improved the number of decimal places. By the end of the 17th century in Europe, new methods of mathematical analysis provided various ways of calculating pi. Early in the 20th century the Indian mathematical genius Srinivasa Ramanujan developed ways of calculating pi that were so efficient that they have been incorporated into computer algorithms, permitting expressions of pi in millions of digits.
On the next couple of pages, I will outline problems and formulas to solve Pi. There are 4 ways to calculate the value of Pi: 1. ) Arctangent Formulas for Pi – This is one of the oldest methods to calculate PI, it uses the power series expansion of arctan (x). This method gets you about 1. 4 correct decimals for every term of the series you calculate. 2. ) Gauss-Legendre – This method doubles the number of correct decimals per iteration. 3. ) Ramanujan I This method comes from the theory of complex multiplication of elliptic curves, and was discovered by S. Ramanujan.
It is a linear formula, so it will get you 14 correct decimals for every term you calculate. 4. ) Ramanujan II – This formula is also based on the work of the eccentric Indian mathematician Srinivasa Ramanujan. It is a quadratic iteration based on an elliptic function called the Singular Value Function of the Second Kind. It converges exponentially to 1/PI as its argument increases. Each iteration of this formula gives approximately four times the number of decimal places as the previous one.
