Pi

The circle number π (Pi), also Ludolph's number, Ludolf's number, or Archimedes' constant, is a mathematical constant defined as the ratio of the circumference of a circle to its diameter. This ratio is independent of the size of the circle. is a transcendental and therefore also irrational number. The decimal fraction expansion of the circle number begins with π though in practical calculations many times only three significant digits of π used: .

The circle number does not only occur in geometry, but also has significance in other mathematical subfields and theories. For example, it can be used to link the solution of the classical Basel problem with the theory of Fourier series.

History of the designation

The circular number and some of its properties were already known in ancient times.

The designation with the Greek letter Pi () (after the initial letter of the Greek word περιφέρεια - to Latin peripheria, "peripheral area" or περίμετρος - perimetros, "circumference") was first used by William Oughtred in his 1647 paper Theorematum in libris Archimedis de Sphæra & Cylyndro Declaratio. In it he expressed by π the ratio of half the circumference (semiperipheria) to half the diameter (semidiameter), i.e. π

The same terms were used around 1664 by the English mathematician Isaac Barrow.

David Gregory took π(1697) for the ratio of circumference to radius.

59 years later than Oughtred, in 1706, the Welsh mathematician William Jones was the first to use the lowercase Greek letter π to express the ratio of circumference to diameter in his Synopsis Palmariorum Matheseos.

It was not until the 18th century that π popularized by Leonhard Euler. He first used π for the circle number in 1737, having previously used . Since then, due to Euler's importance, this designation has been in common use.

Definition

Several equivalent approaches exist to define the circle number π .

The first (classical!) definition in geometry is that according to which the circle number is a ratio numerically equal to the quotient π formed from the circumference a circle and the associated diameter second approach to geometry is related to this and consists in understanding by the circle number the quotient π formed by the area a circle and the area of a square constructed over a radius (of length ). (This radius is called the radius of the circle. ) This second definition is expressed by the proposition that the area of a circle is π of the surrounding square.

In analysis, one often proceeds (after Edmund Landau) to first define the real cosine function via its Taylor series and then to define the circle number as the double of the smallest positive zero of the cosine. Further analytical approaches go back to John Wallis and Leonhard Euler.