Lindemann showed that pi is transcendental by showing that if the complex number z is algebraic (the root of an integral polynomial) then e^{z} +1 cannot be zero.

Lindemann also claimed, and Weierstrass proved rigorously, that this result can be generalised to the following: if each of z_{1}, … , z_{n} is algebraic, and N_{1}, … , N_{n} are algebraic and non-zero, then

cannot be zero.

It follows immediately that if z is algebraic and non-zero, e^{z}, sin z and cos z are transcendental and corresponding results can be obtained for the inverse functions.

Although proofs of the transcendence of pi are found in many texts, for example the proof in Alan Baker’s book ‘Transcendental Number Theory’ discussed in the previous post, full proofs of the Lindemann-Weierstrass Theorem are harder to find. Baker provides a two-page proof, which is widely cited, but it is extremely condensed and key steps are omitted. I have been forced to go back to Weierstrass’ original 1885 paper to find a full proof.

Weierstrass’ paper was published in the rather fulsomely titled ‘Sitzungsberichte der Königlich Preußischen Akademie der Wissenschaften zu Berlin’, pages 1067 to 1085. A link to the journal is provided on the ‘Journals’ page.

Weierstrass’ paper is in three parts, and in the next three posts I will discuss each of these parts in turn.

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This entry was posted on February 6, 2010 at 7:50 pm and is filed under Rational and algebraic numbers. You can follow any responses to this entry through the RSS 2.0 feed.
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August 17, 2010 at 8:59 pm |

I came across your blog while looking for simplified proofs on transcendence. Just thought I’d leave a comment to convey my appreciation. Perhaps you could shed some light on Mahler’s classification of transcendental numbers in the next few posts.

August 18, 2010 at 10:26 am |

Thank you for your comment Sandeep. I hope the posts were of use to you. At the moment I am looking at Roger Apéry’s proof of the irrationality of zeta(3) and have become rather bogged down in it. But I think I can see the light at the end of the tunnel. After that I want to examine the connections between irrationality proofs using continued fractions, and those using repeated integration by parts. This will keep me busy for some time to come, so I’m afraid it will be some time, if ever, that I come to Mahler’s classification.