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The Search For The Value Of Pi

The number represented by pi () is used in calculations whenever something round (or nearly so) is involved, such as for circles, spheres, cylinders, cones and ellipses. Its value is necessary to compute many important quantities about these shapes, such as understanding the relationship between a circles radius and its circumference and area (circumference=2r; area=r2).

Pi also appears in the calculations to determine the area of an ellipse and in finding the radius, surface area and volume of a sphere.

Our world contains many round and near-round objects; finding the exact value of pi helps us build, manufacture and work with them more accurately.

Historically, people had only very coarse estimations of pi (such as 3, or 3.12, or 3.16), and while they knew these were estimates, they had no idea how far off they might be.

The search for the accurate value of pi led not only to more accuracy, but also to the development of new concepts and techniques, such as limits and iterative algorithms, which then became fundamental to new areas of mathematics.

Finding the actual value of pi

Between 3,000 and 4,000 years ago, people used trial-and-error approximations of pi, without doing any math or considering potential errors. The earliest written approximations of pi are 3.125 in Babylon (1900-1600 B.C.) and 3.1605 in ancient Egypt (1650 B.C.). Both approximations start with 3.1 pretty close to the actual value, but still relatively far off.

Archimedes’ method of calculating pi involved polygons with more and more sides. Leszek Krupinski, CC BY-SA

The first rigorous approach to finding the true value of pi was based on geometrical approximations. Around 250 B.C., the Greek mathematician Archimedes drew polygons both around the outside and within the interior of circles. Measuring the perimeters of those gave upper and lower bounds of the range containing pi. He started with hexagons; by using polygons with more and more sides, he ultimately calculated three accurate digits of pi: 3.14. Around A.D. 150, Greek-Roman scientist Ptolemy used this method to calculate a value of 3.1416.

Liu Huis method of calculating pi also used polygons, but in a slightly different way. Gisling and Pbroks13, CC BY-SA

Independently, around A.D. 265, Chinese mathematician Liu Hui created another simple polygon-based iterative algorithm. He proposed a very fast and efficient approximation method, which gave four accurate digits. Later, around A.D. 480, Zu Chongzhi adopted Liu Huis method and achieved seven digits of accuracy. This record held for another 800 years.

In 1630, Austrian astronomer Christoph Grienberger arrived at 38 digits, which is the most accurate approximation manually achieved using polygonal algorithms.

Moving beyond polygons

The development of infinite series techniques in the 16th and 17th centuries greatly enhanced peoples ability to approximate pi more efficiently. An infinite series is the sum (or much less commonly, product) of the terms of an infinite sequence, such as , , 1/8, 1/16, 1/(2n). The first written description of an infinite series that could be used to compute pi was laid out in Sanskrit verse by Indian astronomer Nilakantha Somayaji around 1500 A.D., the proof of which was presented around 1530 A.D.

In 1665, English mathematician and physicist Isaac Newton used infinite series to compute pi to 15 digits using calculus he and German mathematician Gottfried Wilhelm Leibniz discovered. After that, the record kept being broken. It reached 71 digits in 1699, 100 digits in 1706, and 620 digits in 1956 the best approximation achieved without the aid of a calculator or computer.

In tandem with these calculations, mathematicians were researching other characteristics of pi. Swiss mathematician Johann Heinrich Lambert (1728-1777) first proved that pi is an irrational number it has an infinite number of digits that never enter a repeating pattern. In 1882, German mathematician Ferdinand von Lindemann proved that pi cannot be expressed in a rational algebraic equation (such as pi=10 or 9pi4 – 240pi2 + 1492 = 0).

Toward even more digits of pi

Bursts of calculations of even more digits of pi followed the adoption of iterative algorithms, which repeatedly build an updated value by using a calculation performed on the previous value. A simple example of an iterative algorithm allows you to approximate the square root of 2 as follows, using the formula (x+2/x)/2:

  • (2+2/2)/2 = 1.5
  • (1.5+2/1.5)/2 = 1.4167
  • (1.4167+2/1.4167)/2 = 1.4142, which is a very close approximation already.

Advances toward more digits of pi came with the use of a Machin-like algorithm (a generalization of English mathematician John Machins formula developed in 1706) and the Gauss-Legendre algorithm (late 18th century) in electronic computers (invented mid-20th century). In 1946, ENIAC, the first electronic general-purpose computer, calculated 2,037 digits of pi in 70 hours. The most recent calculation found more than 13 trillion digits of pi in 208 days!

It has been widely accepted that for most numerical calculations involving pi, a dozen digits provides sufficient precision. According to mathematicians Jrg Arndt and Christoph Haenel, 39 digits are sufficient to perform most cosmological calculations, because thats the accuracy necessary to calculate the circumference of the observable universe to within one atoms diameter. Thereafter, more digits of pi are not of practical use in calculations; rather, todays pursuit of more digits of pi is about testing supercomputers and numerical analysis algorithms.

Calculating pi by yourself

There are also fun and simple methods for estimating the value of pi. One of the best-known is a method called Monte Carlo.

A square with inscribed circle. Deweirdifier

The method is fairly simple. To try it at home, draw a circle and a square around it (as at left) on a piece of paper. Imagine the squares sides are of length 2, so its area is 4; the circles diameter is therefore 2, and its area is pi. The ratio between their areas is pi/4, or about 0.7854.

Now pick up a pen, close your eyes and put dots on the square at random. If you do this enough times, and your efforts are truly random, eventually the percentage of times your dot landed inside the circle will approach 78.54% or 0.7854.

Now youve joined the ranks of mathematicians who have calculated pi through the ages.

Xiaojing Ye, Assistant Professor of Mathematics and Statistics, Georgia State University

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1. Always dramatically announce absolutely every entrance.

2. And reserve a special announcement for your best assets.

3. Don’t let other people define who you are.

5. …Always feel her presence.

6. And pay homage to her when appropriate.


7. Ignore the people who are coming on too strong.

8. Even though maybe sometimes you come on a little too strong.

9. Never put limits on yourself.

10. Unless something’s too difficult… then, whatever.

11. Never let other people tell you what to do.

12. Unless they’re complimenting you. Then they can say whatever they want.

13. Stay strong in your convictions.

14. And question things you’re not sure about.

15. Never give up because you will eventually find your way.

16. Feel good about the fact that you’re better than everybody around you.

18. Pretty much everybody is falling in love with you.

19. Go on literally thousands of dates every year.

20. Never forget that you are, and will always be, the best thing that ever happened to everybody around you.

21. Dream about what makes you happy.

22. You can never, EVER, be too dramatic.

24. Everything is always, without fail, about you.

25. Do not tolerate children.

26. Learn as many skills as you can. You never know when they’ll come in handy.

27. Also, learn as many languages as possible.

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China is finally getting Apple’s latest toys. Apple said today it would begin selling both the iPhone 5, the fourth-generation iPad and the iPad mini in the country in December.

The two iPads officially go on sale Dec. 7, although only Wi-Fi versions will be available. There is no information on when or if the 4G LTE versions of the iPads will come to China. The iPhone 5 arrives on Dec. 14.

Tellingly, Apple’s press release announcing the news only highlights the iPhone 5 and iPad mini in the title — the fourth-generation iPad is mentioned only in the text. Apple is clearly downplaying its new flagship iPad, most likely to avoid potentially stealing the spotlight from the product it really wants to push this holiday: the iPad mini.

China has been a focal point for Apple’s business in the last few years. The potential market is obviously large, and Apple has opened flagship stores in major urban centers such as Shanghai and Beijing. Although the threat from knock-off gadgets (and knock-off stores) is much greater in China than other countries, it hasn’t hurt the appeal of Apple’s products in the country, and sales there have grown immensely.

Apple first introduced the iPhone 5 in September, and it’s now available in 47 countries. The iPad mini arrived in October alongside the blink-and-you-missed-it iPad Fourth Generation. All three products — as well as Apple’s new iPods — use the company’s new Lightning connector, a much smaller data and power port that allowed Apple to achieve the ultra-slim designs in both the iPad mini and iPhone 5.

Do you think Apple’s new products will be a hit in China? Let us know in the comments.

BONUS: iPad Mini Review

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