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The number represented by π is used in calculations whenever something round is involved, such as for circles, spheres, cylinders... Its value is necessary to compute many important quantities about these shapes.

The first rigorous approach to finding the true value of pi was based on geometrical approximations. Around 250 B.C., 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. In 1630, Austrian astronomer Christoph Grienberger arrived at 38 digits, which is the most accurate approximation manually achieved using polygonal algorithms.

In 1665, English mathematician and physicist Isaac Newton used infinite series (which are the sum of the terms of an infinite sequence) to compute pi to 15 digits using calculus he and German mathematician Leibniz discovered. After that, the record kept being broken. It reached 620 digits in 1956, the best approximation achieved without the aid of a calculator or computer.

In 1946, 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!

There are also fun and simple methods for estimating the value of pi. One of the best-known is a method called “Monte Carlo.”: draw a circle and a square around it on a piece of paper. Imagine the square’s sides are of length 2. 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, eventually the percentage of times your dot landed inside the circle will approach 78.54% — or 0.7854. Now you’ve joined the ranks of mathematicians who have calculated pi through the ages.

Adapted from the article “The long search for the value of pi“, by Xiaojing Ye, The Conversation on March 14, 2016

Questions

1. Sum up the main arguments of the article.
2. Why is it important to know the value of pi?
3. Which areas of mathematics have been used in the approximations of pi?
4. Explain the proposition in the last paragraph “The ratio between their areas is pi/4, or about 0.7854.”
5. What do you think about the method called “Monte Carlo”?

## Vocabulary

• bounds: bornes
• accurate: précis
• eventually: tôt ou tard (faux ami !)