Pi Day Countdown #8 – Pi and Viete

In the 16th century a French lawyer named Francois Viete discovers a simple yet profound formula for computing Pi.

Francois_Viete
Francois Viete, 1540 – 1603

Viete starts by using similar method to Archimedes. He starts by drawing a circle and then inscribe into it a square. By assuming that the circumference of the circle is equal to the perimeter of the square he estimates \pi to 4/ \sqrt{2}, approximately $latex 2.828… $. Viete then doubled the number of sides, the approximation then becomes:

2 * 2/ \sqrt{2} * 2/ \sqrt{2 + \sqrt{2}} \approx 3.061..

Each time Viete doubles the number of sides, he not only improves the approximation but he is multiplying into his approximation by a factor that is growing closer and closer to zero. For example the approximation of \pi for the octagon, is the approximation of \pi for the square times 2/ \sqrt{2 + \sqrt{2}}. From there Viete creates an infinite product that will give the exact solution for \pi:

\pi = 2 * 2/ \sqrt{2} * 2/ \sqrt{2 + \sqrt{2}} * 2/ \sqrt{2 + \sqrt{2 + \sqrt{2}}}* 2/ \sqrt{2 + \sqrt{2 + \sqrt{2 + \sqrt{2}}}} * ...

Notice that each term as a certain pattern to it. Each term will will have more nested square roots then the one previous to it. Unfortunately in order to produce an exact value of \pi one will need an infinite number of terms.

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