Interesting though some of the material was, it just didn't hold my interest. (I also hadn't realized when I bought the book that it had originally been written in the 1930s. But when the author referred to the Australian Aborigines as "barbarians", that sentiment seemed so out of place with modern thinking that I checked the copyright information.) ( )

7/7/22

In Number : the language of science, Tobias Dantzig discusses numbers: real numbers, rational and irrational numbers, perfect numbers, primes, natural numbers, cardinal and ordinal numbers, finite numbers, imaginary numbers. Numbers, numbers, numbers. But there is much more. Dantzig discusses alphabet and its relationship to numbers, how we sense number when we don’t count, fractions and decimals, the theory of zero, infinity, and pi. The early mathematicians and their discoveries are recounted and in a section at the end, the author lists problems of math and their solutions.

My copy of Number is a 2007 edition edited by Joseph Mazur with the uncorrected text from Dantzig’s 4th edition. In addition to rearranging the text into two parts (number theory and mathematical problems), Mazur adds a forward by his brother Barry, endnotes to explain the text (although beware that the original text is not footnoted; Mazur only numbers the page to which he refers in his endnotes), a section on further reading which is up to date as of 2007, and an excellent index. Mazur also updates Dantzig’s work of 1954 to include solutions that had not been proven earlier.

The most interesting material for me was the early history of numbers, numbering and symbolism. I was also fascinated by the sections on geometry, especially referring to the Greeks. I learned of several mathematicians and their discoveries with which I was not familiar.

This is not a book for the faint of heart as Dantzig includes complex solutions. However, those that wish to skip these sections will still benefit from the rest of the text. Counting and arithmetic are second nature to us and learning that it wasn’t always so is a fascinating journey. ( )

A survey of "the number concept" as revealed by the evolving idea of the infinite, evolving because specific mathematical problems required a change to reach a tenable solution. Dantzig ends with a short table of "key dates" in this timeline, useful as an orientation map on second reading. The material is too unfamiliar after just one reading for me to re-construct the changes to the concept of the infinite, or for that matter to identify when innovations were not linked to the infinite. A nice yardstick for the next reading: outline in narrative form that evolution.

Dantzig adopts a philosophical approach, in an easy style and allowing much free play for personality and dramatic history-telling. There are equations and some material can be rough sledding, but it's the material not the telling.

My edition (the fourth, revised and augmented) includes Part I being the verbatim text of the first edition, and Part II "for all intents and purposes, a new book". In 2011-12, read just Part I; best to revisit Part I before embarking on Part II, which reviews similar territory but from perspective of other specific problems illustrating the mathematical ideas and innovations. ( )

Indeholder "Forord til tredje udgave", "Forord til fjerde udgave", "Første del. Talbegrebets udvikling", "Andel del. Nye og gamle problemer". "Navneregister".
Første del indeholder "I. Fingeraftryk.", "II. Den tomme søjle", "III. Tallære", "IV. Det sidste tal", "V. Symboler"; "VI. Det uudsigelige", "VII. Denne flydende verden", "VIII. Tilblivelsesakten", "IX. Hullerne fyldes ud", "X. Tallegemet", "XI. Det uendeliges anatomi", "XII. De to virkeligheder".
Anden del indeholder "A. Om at skrive tal", "B. Om hele tals egenskaber", "C. Om rødder og roduddragning", "D. Om principper og argumenter".

Der er lidt om venskabstal, perfekte tal, Eratostenes si, primtal, fermat-primtal. Wilsons teorem, Goldbach's postulat, Fermat's sidste sætning, osv.
Teorem af Fermat: n^p -n er et multiplum af p, hvis p er et primtal.
Teorem af Wilson: (n+1) går op i (n!+1) hvis og kun hvis n er et primtal.
Induktionsbevis, Reductio ad absurdum, bevis for at sqrt(2) er irrationel, algebraiske tal, Zenon og Zenons fire argumenter, Dedekind og Dedekind-snit, reelle tal, tallegemer, Cantor, kædebrøker, logaritmer, konvergente og divergente rækker, Algebraiske ligninger. Algebra og analyse. Descartes, analytisk geometri, Gauss, Argand, Sagredo, Simpicio, Salviati. Galilæi, transcendente tal, Cayley, Sylvester, matrix, matricer, komplekse tal, imaginære tal, Ramanujan, restklasser, binomialformlen, Blaise Pascal og Pascals trekant. Et sødt bevis for Wilsons teorem. Primtalsfordelingen. Euler, Legendre, Eulers sætning om at ethvert generisk polynomium må antage sammensatte værdier for i det mindste en værdi af argumentet. pythagoræiske taltripler, Sylvester, ulige perfekte tal, polynomiumsligninger, kædebrøker, uendelige kædebrøker, diofantiske ligninger, Dirichlet, Dirichlets distributionsprincip,

Denne bog var en kær ven, da jeg startede med at gå på opdagelse i matematikken. Der er en sjov blanding af nemme og svære ting i den. Den er dog skrevet for at være populærvidenskab, så der er mange afsnit, der bare er ord, ord og atter ord. ( )