transcendental numbers



An Introduction to the History of Transcendental Numbers

(dedicated to Fred Piper, my teacher, with affection and admiration)

A note of late March 2017. You may encounter a problem opening my Maple worksheets (as indeed I do myself) here at my web site - it would appear to depend on the internet browser being used. Thus, if I attempt to open one of my worksheets using Interner Explorer there is never a problem, whereas if I use Firefox then all that one sees - this is just an example - is something like this: {VERSION 3 0 "IBM INTEL NT" "3.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 0 128 0 128 1 0 1 0 0 1 0 0 0 0 } {CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 } {CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "2D Output" 2 20 "" 0 0 0 128 0 1 0 1 0 0 0 0 0 0 0 } {CSTYLE " " -1 256 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 257 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 258 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 259 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" and so on (almost at infinitum)...

I asked Maplesoft for advice on this and they recommended doing the following (and I found it worked):

  1. Don't attempt directly to open the worksheet (by clicking on my link), instead right click and save the worksheet to (say) the download folder.

  2. Now that the worksheet is in the download folder (cut it out if you wish and put into whatever folder you wish) you may open the worksheet in the usual way (I should add that while I see this 'works', I have absolutely no idea as to why it does... ) [END OF NOTE].

is the title of a Maple-based talk I gave on May 6th 2004 to the Dublin Branch of the Irish Mathematics Teachers Association.

In my talk I covered only a small part of the very substantial Maple worksheet (along the lines of my Austria 2001 Fermat's little theorem talk) which I prepared for the occasion. The active mws file, with all outputs removed, is available here, and the text only html version (including all the Maple outputs) is available here. A hard copy runs to 77 A4 printed pages.

I highly recommend this wonderful book, which celebrates the 60th birthday of the remarkable English mathematician Alan Baker:

A Panorama of Number Theory or The View from Baker's Garden

Edited by Gisbert Wustholz, Published by the Cambridge University Press, September 2002, 372 pages 1 line diagram 3 tables, Hardback ISBN: 0521807999

On 24th February 2018 I sent the following email to the Irish Mathematics Departments listserver:

The remarkable mathematician (and, incidentally, a Fields Medalist) Alan Baker died (from a stroke) earlier this month, on Sunday 4th. His passing should not go unnoticed. When Serre was awarded a Fields Medal in 1954, Hermann Weyl remarked at the ceremony that Serre was the brightest star in the mathematical firmament (I am relying on my memory here, having read this some 50 years ago; it was in the Proceedings of the Zurich Congress). The same could be truly said of Baker when he burst onto the scene in the mid 60's with regard to his work in Transcendental Numbers, Diophantine approximations and equations. He truly revolutionized these subjects. In particular he proved a vast extension of the (true claim) of Hilbert's seventh problem (which had been independently settled by Gelfond and Schneider in 1934).

As reported by the great C.L. Siegel, Hilbert himself had declared (in a lecture attended by C.L.S.) that he expected the Riemann hypothesis would be settled in his lifetime, perhaps Fermat's Last Theorem in the lifetime of those listening, but his seventh problem? "No one present would live to see that!" (source: Constance Reid's Hilbert biography). No newspaper - not even in England (A.B. was born in London, and - a happy coincidence for me - shared my mother's birthday, 19th August) - seems to have published an obituary, a sad reflection on their values.

I am offering to give a talk on Baker's work in any department in the country (from Tralee to Derry), a talk that even first year students could follow.

John Cosgrave

P.S. Some fourteen years ago I prepared a document (in Maple, and also html converted) 'A History of transcendental numbers'. It began as a small document as a basis for a talk to the Irish Mathematics Teachers Association, but it took on a life of its own, and the final (77 A4 pages to print) document is on this page:

The html document is here:

and the Alan Baker part is here:

Not one Mathematics dept. took up my offer.

Resources in Transcendental Numbers

Michel Waldschmidt generously provides a wealth of valuable papers and transparencies

(Note added February 2011. I had the great pleasure of meeting Michel Waldschmidt in Vancouver, in July 2007.)

Tanguy Rivoal (zeta function at 3, 5, 7, ... )

Yuri Bilu's homepage

Paula B. Cohen (now Tretkoff)'s homepage

Dorian Goldfeld's homepage

T. N. Shorey's homepage

Robert Tijdeman's homepage

In April 2004 I sent an email to the Number Theory Mailing List in which I wrote "... I would very much appreciate hearing from anyone who offers courses - with web related material - on transcendental numbers (I'd like to make links to them). Does anyone teach any transcendence proofs at undergraduate level? I'd be very interested to know."

    An Historia Mathematica email from Roger Cooke on the subject Leibnitz on Pi. I thank Paula Cohen for informing me with the following excellent site (which, so far, is the only one of which I've heard):

David Angell's University of New South Wales course on Irrationality and Transcendence


Contact details. jbcosgrave at