# Is pi greater than 5

## Cheers to the number pi

In honor of the number π, the so-called Pi Day is celebrated on March 14th. This date was chosen because the American spelling of the date (3/14) already hides the first digits of the famous circle number. Π is an irrational number, i.e. it cannot be represented as a fraction and has an infinite number of decimal places.

The constant π describes the ratio of the circumference and diameter of a circle. The value of π is needed, for example, to calculate the circumference (U = 2πr) or the area (A = πr^{2}) of a circle. The best way to understand the relationship between the number π and the circumference and area of a circle is to derive the approximate value of π yourself, either through a rough visual estimate or through a mathematical approximation method.

An illustrative approximation of

For example, π can be obtained by stretching a cord around a circle and then checking how often the diameter of the circle can be measured with the cord, namely more than three times. Thus the circumference is more than three times as long as the diameter of the circle.

In order to estimate the area of a circle with radius r at a glance, one could cover three quarters of the circle with three squares of edge length r. Such a square has the area r^{2} and is far larger than the covered section of the circle. The protruding surface sections of the three squares fit into the still empty segment of the circle, whereby the circle is still not completely covered. This estimate is also very imprecise, but it shows why the area of a circle is more than three times r^{2} amounts to.

The determination of the number π has occupied mankind for thousands of years. Particularly noteworthy is the process developed by Archimedes to approximate the circular area by inscribing and circumscribing a regular polygon. The more corners the polygon has, the more precise the area of a circle and thus the number

π can be determined. By using a 96-point, Archimedes was able to calculate the first two decimal places of π:

Over time, the Archimedes algorithm has been steadily improved. This is how Ludolph van Ceulen succeeded at the beginning of the 17th century with the help of a 262 corner

Calculating π to 35 decimal places, which gave the circle number the name Ludolph's number.

The geometric approximation method was finally replaced by the development of infinite series, such as the formula developed by Gottfried Wilhelm Leibniz:

However, this series converges so slowly that only for the calculation of the first two decimal places one has to calculate up to the point k = 49.

The Indian mathematician Srinivasa Ramanujan discovered a much faster converging series at the beginning of the 20th century. With its formula, eight decimal places of π can be calculated per iteration:

In the age of computers, the calculation of the number π has advanced rapidly and can now be determined with an accuracy of several trillion digits. Not that this degree of precision is of any practical use, but it gives us an even deeper insight into the fascinating number π that we want to celebrate today, preferably in a picturesque setting with pizza, a beer and the obligatory piece of pie.

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