CALCULUS Understanding Its Concepts and Methods

Carl Louis Ferdinand von Lindemann (1852--1939) --- Historical Sketch

The German mathematician Carl Louis Ferdinand von Lindemann is celebrated for his proof that is transcendental, that is, is not a root of any polynomial with rational coefficients. In 1873, the year Lindemann was awarded his doctorate, Charles Hermite proved that is transcendental. Using methods similar to those of Hermite, Lindemann showed in 1882 that is transcendental. His proof was based on Hermite's proof that is transcendental together with the fact, proved by Euler, that .

The problem of squaring the circle, namely constructing a square with the same area as a given circle using only a straightedge and compass, is one of the classical problems of Greek geometry. Lambert had proved in 1761 that is irrational, but this was not enough to prove the impossibility of squaring the circle with straightedge and compass because some irrational numbers can be so constructed. Lindemann's proof that is transcendental finally established that squaring the circle with straightedge and compass is impossible.

Lindemann studied first at Göttingen, then at Erlangen (where he earned his doctorate in mathematics), and at Munich. He also visited universities in France and England. He became interested in physics as it concerned the idea of the electron. He was a professor at the Universities of Freiburg, Königsburg, and Munich.

Lindemann was one of the founders of the modern German educational system. He emphasized the development of the seminar and in his lectures communicated the latest research results. He also supervised forty-seven doctoral students, including the famous mathematicians David Hilbert and Hermann Minkowski.

A number is transcendental if it is not the root of any polynomial equation with rational coefficients. Can the word rational be replaced by the word integral in this definition?

Copyright © 2006 Darel Hardy, Fred Richman, Carol Walker, Robert Wisner. All rights reserved. Except upon the express prior permission in writing, from the authors, no part of this work may be reproduced, transcribed, stored electronically, or transmitted in any form by any method.