History of Calculating PiDate: 05/20/98 at 15:57:40 From: Josh Subject: What is the formula for pi? Hi: I was doing a report on pi and wondered what the formula for pi was. How did they figure out that it is: 3.14159265358979323846 ... Josh Date: 05/20/98 at 19:12:01 From: Doctor Wilkinson Subject: Re: What is the formula for pi? This is a very good question. The first mathematician to calculate pi with reasonable accuracy was Archimedes, around 250 B.C. Using the formula: A = pi r^2 for the area of a circle, he approximated pi by considering regular polygons with many sides inscribed in and circumscribed around a circle. Since the area of the circle is between the areas of the inscribed and circumscribed polygons, you can use the areas of the polygons (which can be computed just using the Pythagorean Theorem) to get upper and lower bounds for the area of the circle. This was the first general method for calculating approximations to pi, and at least theoretically it could be used to get any degree of accuracy if you could just do the computations. Archimedes showed in this way that pi is between 3 1/7 and 3 10/71. The same method was used by the early seventeenth century with polygons with more and more sides to compute pi to 35 decimal places (Van Ceulen did the biggest 674 calculations.) When Newton and Leibnitz developed calculus in the late seventeenth century, more formulas were discovered that could be used to compute pi. For example, there is a formula for the arctangent function: arctan(x) = x - x^3/3 + x^5/5 - x^7/7 + ... If you substitute x = 1 and notice that arctan(1) is pi/4 you get a formula for pi. This is not useful because it takes too many terms to get any accuracy, but there are some related formulas that are very useful. The most famous of this is Machin's formula: pi/4 = 4 arctan(1/5) - arctan(1/239) This formula and similar ones were used to push the accuracy of approximations to pi to over 500 decimal places by the early eighteenth century (this was all hand calculation!) In the twentieth century there have been two important developments: the invention of electronic computers and the discovery of much more powerful formulas for pi. For example, in 1910 the great Indian mathematician Ramanujan discovered a formula that in 1985 was used to compute pi to 17 million digits. Other even better methods have been developed since, and computers are getting ever more powerful. The current record is about 51 billion decimal places. -Doctor Wilkinson, The Math Forum Check out our web site! http://mathforum.org/dr.math/ Date: 04/05/2007 at 15:21:41 From: Shantea Subject: first computer calculation of pi? Who came up with the first computer calculation of pi? Date: 04/05/2007 at 22:07:15 From: Doctor Wilkinson Subject: Re: first computer calculation of pi? That's a very good question, Shantea! It seems to have been George Reitwiesner who first used an electronic computer to calculate pi, in 1949 on an ENIAC, a very early computer. He computed 2037 decimal places. See A Chronology of Pi http://www-history.mcs.st-and.ac.uk/history/HistTopics/Pi_chronology.html for more information. - Doctor Wilkinson, The Math Forum Check out our web site! http://mathforum.org/dr.math/ Date: 02/28/2003 at 12:58:36 From: Ahmed Subject: Proving Machin's formula Machin's formula says : pi/4= 4arctan(1/5)- arctan(1/239) could you please prove this formula ? What is the relation between the numbers (pi/4), (1/5) , (1/239) ? Thanks in advance. Ahmed Date: 02/28/2003 at 18:13:18 From: Doctor Wilkinson Subject: Re: Proving Machin's formula Hi, Ahmed. There's a simple proof at Trigonometric Equivalences - McRae http://mcraefamily.com/MathHelp/GeometryTrigEquiv.htm - Doctor Wilkinson, The Math Forum Check out our web site! http://mathforum.org/dr.math/ |
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