# 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.

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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