Carl Louis Ferdinand von Lindemann on PI as a transcendental number

He and https://en.wikipedia.org/wiki/Charles_Hermite (Hermite) have an interesting list of doctoral students. -Cr6

Theorem: https://en.wikipedia.org/wiki/Lindemann%E2%80%93Weierstrass_theorem
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Carl Louis Ferdinand von Lindemann

Carl Louis Ferdinand von Lindemann (April 12, 1852 – March 6, 1939) was a German mathematician, noted for his proof, published in 1882, that π (pi) is a transcendental number, meaning it is not a root of any polynomial with rational coefficients.

Life and education

Lindemann was born in Hanover, the capital of the Kingdom of Hanover. His father, Ferdinand Lindemann, taught modern languages at a Gymnasium in Hanover. His mother, Emilie Crusius, was the daughter of the Gymnasium's headmaster. The family later moved to Schwerin, where young Ferdinand attended school.

He studied mathematics at Göttingen, Erlangen, and Munich. At Erlangen he received a doctorate, supervised by Felix Klein,[1] on non-Euclidean geometry. Lindemann subsequently taught in Würzburg and at the University of Freiburg. During his time in Freiburg, Lindemann devised his proof that π is a transcendental number (see Lindemann–Weierstrass theorem). After his time in Freiburg, Lindemann transferred to the University of Königsberg. While a professor in Königsberg, Lindemann acted as supervisor for the doctoral theses of the mathematicians David Hilbert, Hermann Minkowski, and Arnold Sommerfeld.

Transcendence proof

In 1882, Lindemann published the result for which he is best known, the transcendence of π. His methods were similar to those used nine years earlier by Charles Hermite to show that e, the base of natural logarithms, is transcendental. Before the publication of Lindemann's proof, it was known that if π was transcendental, then it would be impossible to square the circle by compass and straightedge.

https://en.wikipedia.org/wiki/Ferdinand_von_Lindemann
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Institutions
University of Munich

Doctoral advisor C. Felix Klein[1]

Doctoral students
Emil Hilb
David Hilbert
Martin Kutta
Alfred Loewy
Hermann Minkowski
Oskar Perron
Arthur Rosenthal
Arnold Sommerfeld


Hermite's list:

Institutions
   École Polytechnique
   Sorbonne

Doctoral advisor
Eugène Charles Catalan

Doctoral students
Léon Charve
Henri Padé
Mihailo Petrović
Henri Poincaré
Thomas Stieltjes
Jules Tannery

A brief look at his life

Carl Louis Ferdinand Lindemann was born in 12 avril 1852 in Hanovre. From 1870 to 1873, this great traveller did his studies at Göttingen, Erlangen, Munich, London and Paris. Then he teached at Wurzbourg (1877), Fribourg (1877-1883), Königsberg (1883-1893) and finnally Munich.

In 1882, Lindemann publish Die Zahl Pi which brings an end to the problem of squaring a circle and 25 centuries of questionning !

But Lindemann also brought his first efforts towards geometry and after his great success on Pi,he will look at Fermat's big theorem for the rest of his life, without finding a solution...

Around - A few words on its transcendental

Remember that we say that a complex number (hence also reals) is said to be algebraic if it is a root of a non nul polynomial with integers coefficients and is said to be transcendental otherwise.
i is algebraic for example (well it will be useful later on...) since i is a root of the polynomial x4-1=0.
It was only in 1844 that the existance of transcendental terms was proved by Liouville.
In1874, the great Georges Cantor proved thanks to his passion for set theory, that most of the reals are transcendental, or in fact that the set of algebraic real is countable (hence of size N !)
The transcendental of Pi is not as a meaningul result as we could think in the sense that it does not give us any practical interesting information on the decimals of Pi. Even more, as Cantor showed, the set of trancsendental is a lot more bigger than the one of algebraic, Pi, like any number taken at random, had it's chance to be with those! But since Lindemann's discover brought an end to one of the oldest problem in the world of mathematics, to know how to sqare a circle...
To draw a square (or a circle for the matter!) of the same area as a circle with a ruler and a compas lead to building a segment of length Pi with those same equipment. But I-can't-remember-who showed that only the additions, multiplication, roots and quotion could be build with the ruler and compas. Which is equivalent to the fact that Pi be the root of any polynomial with integers coefficients.
This dream (too beautiful!) died with Lindemann. His proved was heavely based on the method which Hermite used to prove the trancendance of e in 1873. He, himself predicted, after his exploit, that the method could be applied to Pi but even more complicate and did not have the courage to try...
Which will not be our case since the proof for the transcendance of e and then of Pi simplyfied by Weierstrass, Hilbert, Hurwitz and Gordan follow this paragraph. They are taken from Transcendental Number Theory by A. Baker and from le Fascinant Nombre Pi by J.P. Delahaye (The fascinating number Pi) (see Bibliography)

Note that if you can prove that e+ is rational or transcendental, you will have won the big prize since that problem is still not solved!

(note site is not secure: http://www.pi314.net/eng/lindemann.php )

PI Day is March 14:

https://en.wikisource.org/wiki/H._RES._224_Supporting_the_designation_of_Pi_Day,_and_for_other_purposes