In mathematics, Pi is the symbol denoting the ratio of the circumference of a circle to its diameter. The ratio is approximately 3. 14159265, pi being an irrational number (one that cannot be expressed as a simple fraction or as a decimal with a finite number of decimal places) and a transcendental number (one without continuously recurrent digits).
Electronic computers in the late 20 th century have carried pi to more than 100, 000, 000 decimal places. Using a computer program, I calculated pi into 1000 decimals: 3. 14159265 358979323846 2643383279 5028841971 6939937510 5820974944 5923078164 0628620899 8628034825 3421170679 8214808651 3282306647 0938446095 5058223172 5359408128 4811174502 8410270193 8521105559 6446229489 5493038196 4428810975 6659334461 2847564823 3786783165 2712019091 4564856692 3460348610 4543266482 1339360726 0249141273 7245870066 0631558817 4881520920 9628292540 9171536436 7892590360 0113305305 4882046652 1384146951 9415116094 3305727036 5759591953 0921861173 8193261179 3105118548 0744623799 6274956735 1885752724 8912279381 8301194912 9833673362 4406566430 8602139494 6395224737 1907021798 6094370277 0539217176 2931767523 8467481846 7669405132 0005681271 4526356082 7785771342 7577896091 7363717872 1468440901 2249534301 4654958537 1050792279 6892589235 4201995611 2129021960 8640344181 5981362977 4771309960 5187072113 4999999837 2978049951 0597317328 1609631859 5024459455 3469083026 4252230825 3344685035 2619311881 7101000313 7838752886 5875332083 8142061717 7669147303 5982534904 2875546873 1159562863 8823537875 9375195778 1857780532 1712268066 1300192787 6611195909 2164201989.
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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 = r 2. 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 (3 rd 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 6 th century Chinese and Indian mathematicians had independently confirmed or improved the number of decimal places.
By the end of the 17 th century in Europe, new methods of mathematical analysis provided various ways of calculating pi. Early in the 20 th 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. ) Arc tangent 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.
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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.