Irrational Numbers and Their Representation by Sequences and Series

by Henry Parker Manning

Dr. Manning's book on irrational numbers contains a presentation in a simple form of another field of mathematical inquiry, such as is also eminently_ suited for placing in the hands of the ordinary schoolmaster. We have decided that the geometry of proportion shall be taught to schoolboys without reference to irrational quantities, but we have not yet eliminated a spirit of reckless extravagance in the quite unnecessary use of infinite series, often with total disregard for their convergency. In Dr. Manning's treatment an irrational number is defined as forming a point of separation between rational numbers of two classes, the numbers of one class being less than those of the other. This definition appears to involve the assumption (pp. 7, 10, &c.) that the point of separation is unique, in other words, that there cannot be two irrational numbers which have not some rational number separating them. Perhaps this assumption may be regarded as a definition of equality of irrational numbers; in any case, the inquiring reader would find it necessary to examine more fully the references to Dedekind's and Cantor's writings given on p. 56. Once the assumption or definition is made, the representation of numbers by sequences readily follows. The theory of limits is discussed on p. 57, and in the following chapter the notion of a sequence is shown to give rise to that of a series. The remaining portion of the book is mainly devoted to the study of convergence, and includes the well-known multiplication theorem and applications to the still better-known binomial and exponential series.
-Nature, Vol. 75

130 pages, Paperback

First published August 6, 2015

About the author

Henry Parker Manning was an American professor of mathematics. In 1889, he entered Johns Hopkins University to study mathematics, astronomy and physics. When he received his Ph. D. degree in 1891, his first printed paper had already appeared in the American Journal of Mathematics. When he was nearly seventy, Manning learned early Egyptian hieroglypics, and collaborated with Arnold Buffum Chace in his publication of the Rhind Mathematical Papyrus. He retired in 1930 and spent several years as associate editor of the American Mathematical Monthly. Amongst his other works are Non-Euclidean Geometry (1901), Irrational Numbers and Their Representation by Sequences and Series (1906), The Fourth Dimension Simply Explained (1910) and Geometry of Four Dimensions (1914)