Pi and the prime numbers. A journey of discovery into mathematics. (Pi und die Primzahlen. Eine Entdeckungsreise in die Mathematik.) (German) Zbl 1478.00005

Weitz, Edmund

Berlin: Springer (ISBN 978-3-662-62879-9/hbk; 978-3-662-62880-5/ebook). iv, 273 p. (2021).

The book is aimed at readers with small mathematical background. It takes the reader on a journey through the landscape of mathematics and wants to show him or her the peculiarity of mathematical thinking and the beauty of mathematics. The goal of this journey is a proof of Leibniz’s series for Pi. The reader should be able to follow this proof on 230 pages. The basic idea of the proof is to count the points with integer coordinates inside a circle with radius n and divide this number by \(n^2\). If n goes to infinity, you get a series for Pi. But the author adheres to Confucius’ motto: The way is the goal.
The journey takes you through geometry, algebra, and number theory, touching on the following topics: natural, integer, rational, and irrational numbers; the Cartesian coordinate system; prime numbers; modular arithmetic; complex numbers and Gaussian numbers; Fermat’s little theorem; Spivak’s proof of Fermat’s two squares theorem.
The book contains anecdotes about famous mathematicians such as Archimedes, Fermat, Leibniz, Newton, and Gauss.
The book is a distinct reading pleasure even for readers with a solid mathematical background.

Reviewer: Ralf Socher (Brandenburg an der Havel)

MSC:

00A09

Popularization of mathematics

Keywords:

Leibniz series for pi; history of mathematics; prime numbers; modular arithmetic; Gaussian numbers