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Irrational Numbers

Mathematical Association of America
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The Carus Monographs of the Mathematical Association of America are rich in memorable associations for a great many of us. I have a fond and vivid memory of scoring a copy of number sixteen in the series, Dedekind Sums, by Hans Rademacher and Emil Grosswald, at a Southern California MAA Section Meeting over ten years ago: even at that time, when I was perhaps not as inclined to reminiscences as I am now, the discovery of such a classic in its familiar deep-blue hard-cover format with the spare printing on the cover done in gold, surrounding the MAA Seal, triggered an irresistible feeling of nostalgia. I seized the book immediately.

The Introduction to Dedekind Sums was written by Ivan Niven, Chairman of the Committee on Publications of the MAA in 1972, the date of the appearance of no. 16, and himself the author of no. 11 in the Carus Monograph Series, Irrational Numbers, the book under present review. Niven first launched this wonderful book in 1956, and now, almost fifty years later, Irrational Numbers reappears in its fifth printing. Recent reprints in the Carus Series no longer sport the deep-blue hard cover, but come out in soft-cover format and varying color schemes: the fifth edition of Irrational Numbers has a cover the color of, well, mud. But all is well: once you crack the spine, “it’s déjà vu all over again.”

We encounter the original material verbatim, starting off with Niven’s elegant and succinct preface (all in the original type-face), carrying the author’s grateful acknowledgment of personal debts owed to Professors Olds and Zuckerman, and tantalizing allusions to Hardy and Wright (i.e. their legendary Introduction to the Theory of Numbers), Koksma, Perron, and Siegel. Accordingly it is crystal-clear that we are in for some very classical stuff, reminiscent (for those of us whose degrees are over twenty years old) of small, informal senior and graduate seminars run by mathematicians who themselves had made contributions to the subjects under consideration. In my own case, my undergraduate school counted Steinberg and Redheffer on its faculty, both of whom figure in Niven’s discussion of Theorem 9.1 (p. 117) on the linear independence of exponentials of distinct algebraic numbers (over the field of algebraic numbers), i.e. the generalized Lindemann Theorem. It is a spring-board for Niven’s proof, in a single paragraph (!) fourteen pages hence, that e and π are transcendental (and he proves a lot more besides!).

To be sure the book is fantastic and remains valuable even fifty years after its first appearance. It certainly qualifies (still) as a wonderful choice for a topics-in-number theory seminar or a tutorial or reading course. Individual chapters of Irrational Numbers already go a long way in this regard all by themselves. For example, the chapter on continued fractions is an outstanding succinct rendition of the subject’s foundations culminating in the theorem that a simple c.f. is periodic iff it represents a quadratic surd. Another example is had in the book’s culminating chapter, giving a complete treatment of the 1934 – 1935 resolution of Hilbert’s Seventh Problem by Gelfond and Schneider (independently). Here Niven notes that the proof he presents is based on Carl Ludwig Siegel’s presentation of the topic in his book (with Richard Bellman), Transcendental Numbers, which I recall very well indeed from a seminar given in the late 1970s by Basil Gordon, David G. Cantor, and E. G. Straus (requiescat in pace): “the more things change, the more they stay the same…”

All the foregoing having been said, and asking the reader’s indulgence for my nostalgic excesses, I recommend Irrational Numbers in the strongest possible terms. It is the perfect introduction to the indicated subject and serves as a fine precursor to such texts as the aforementioned book by Siegel and Bellman, Alan Baker’s gorgeous but everywhere dense Transcendental Number Theory, and, for instance, the compendium, New Advances in Transcendence Theory (ca. 1988), edited by Baker. Not being up on the latest stuff in this beautiful part of number theory I do not have more recent sources to recommend, but there can be no doubt that, whatever they are, their reader will greatly benefit from a preliminary scrutiny of Niven’s monograph. Beyond this Irrational Numbers is simply a terrific book to read for its own sake.

Michael Berg is professor of mathematics at Loyola Marymount University in Los Angeles, CA.

Date Received: 
Tuesday, August 16, 2005
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Ivan Niven
The Carus Mathematical Monographs 11
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Michael Berg
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Tuesday, June 5, 2012