This booklet is the English translation of the German original from 2013: *Zahlen, Geschichte, Gesetze, Geheimnisse* The book starts and ends with the question: What is a number? Beutelspacher gives answers by sketching the historical evolution of the concept of numbers from counting one-two-three-many up to complex numbers.

The author is a popularizer of mathematics, well known in Germany through his columns, and his many books. He is also the founder of the *Mathematikum*, a math museum in Giessen. The present book, his most recent, tells the evolution of the concept of numbers, and their representation through the centuries. The idea is that the text should be accessible for anyone with no or just a minimal knowledge of mathematics. This subject has been covered by many other authors before. For example Havil's *The irrationals* or Stewart's *Professor Stewart's incredible numbers* or Ifrah's monumental *The Universal History of Numbers*, and there are of course many more. So what is new here? Well, it is short, and yet nothing essential has been left out and it is truly explaining all that is needed for the layman to understand what is going on. However, it certainly is not a flat executive summary because it still has details and anecdotes to keep the attention of the reader.

The content is organized in five chapters. That it should start with the integers is obvious. In fact the first chapter starts from counting in a primitive society. Gradually the concept of a natural number emerges and the early mathematicians investigated number patterns like even and odd numbers, square and triangular numbers, magic squares, and Pythagorean triples, but also prime numbers. A brief excursion is made to Fermat's last theorem and the ultimate proof by Wiles. Some cryptography and the basic idea of RSA coding are explained. This shows how important natural numbers still are in our modern society.

The second chapter deals with the representation of numbers. The origin is of course tallying, and different notations and number systems. There were the Egyptian and Roman systems, which were not so useful for computing. The Babylonians had a place value sexagesimal system, which was much more useful. We inherited our 60 minutes in an hour and 60 seconds in a minute. For example their number 234 could for example denote 2 hours, 3 minutes and 4 seconds. It is however via the Arabic mathematicians that the Indian decimal system as we know it, including the zero, was introduced in Europe. The chapter also gives a good explanation of how the abacus was used for computing. Also some divisibility rules are explained. Division by 2, 5 and 10 are trivial of course, but still, the check digit at the end of our EAN barcodes is based on the remainder modulo 10 of the weighted sum of the digits in the code. Finally, there is obviously the binary system. It was described by Shannon in 1948 as the basis for communication, although Leibniz envisioned already a binary computer, but he did not elaborate on it.

The story of the rationals and irrationals is told in the next chapter. The step to be made from the geometric concept of proportion to the ratio of two integers and then to the rational number that this ratio represents is not so obvious. The Egyptians had an ingenious system of unit fractions to compute with, but the true concept comes again from the Indians and it can be connected with decimal representation of numbers (containing a finite number of digits). But rationals had their limitations and gradually, starting with the incommensurability problem of the Greek, the irrationals conquer their way into the minds of mathematicians. The golden ratio which appears in the pentagram, was already scrutinized by the Greek. The marvelous proof of the irrationality of the square root of 2 is included. It is also pointed out that there are algebraic irrationals and transcendental irrationals.

Chapter 4 prolongateso this idea and continues the dissection of the transcendentals. That requires the introduction of limits of number sequences. The rational numbers are now extended with the limits of these sequences. This gives for example the result that 0.999... represents a limit that is actually the same as 1, a fact not so easily accepted by a general reader. We are further instructed about the approximations to pi by the Greek, the introduction of Euler's number e and we are introduced to Cantor's theory of the infinite and how his diagonal technique could prove that there are infinitely many transcendental numbers

The imaginary and complex numbers became necessary when one wanted to solve polynomial equations. We learn how al-Khwārizmī solved quadratic equations with geometric constructions, while Cardano had a formula which made him believe that a solution to such an equation can also be a negative number. In a geometric context, numbers are lengths, and then a negative solution is not acceptable. The algebra made possible what geometry could not deliver. For the cubic equation, we get the story of the Tartaglia-Fior duel and how Cardano pilfered Tartaglia's formula so that he could publish it and that is how today it gets Cardano's name attached to it. The quintic equation bears the dramatic story of Abel and Galois who both died at a young age. Abel from pulmonary tuberculosis and Galois from the consequences of a physical duel over a love affair. This stopped the race to find algebraic formulas for the solution a polynomial equation of higher degree. This doesn't mean that there are no solutions to the equation. Already quadratic equations required complex numbers, but they were not recognized. A quadratic with complex roots was considered to have none. It was Cardano who first used complex numbers implicitly, but without recognizing them in his computations for the cubic. After complex numbers were accepted, the fundamental theorem of algebra stating that every polynomial equation of degree *n* has *n* real or complex roots was soon formulated. It was however Gauss who finally proved it almost two centuries after Roth had given a first hint.

The conciseness of the text and the objective of readability for laymen, necessitates some loose formulations that are strictly speaking not a hundred percent correct when isolated from the context. For example `Every equation can be solved!' (p.85) means actually `Every polynomial equation has a complex solution', or '[The binary system] is the representation of numbers used by modern computers' (p.37) which is only partially true since they work mostly with the hexadecimal number system with a lot of bells and whistles attached. And there are more such examples. But these are of course nitpicking and clearly, when seen in context, the lay reader will certainly have no problem with such formulations. To make every sentence unambiguous would only harm readability. It is only when they are used as a section title (like the one on page 85) that makes a mathematician frown.

The text is pleasant to read and is well illustrated. The reader is gently guided through number wonderland, avoiding any abstraction or complexity that could deter the innocent reader. And yet, in all its simplicity such a reader is still challenged to follow the author in some of his scratching the surface of algebraic equations or some other true mathematical issues that are a bit more technical. Let me end by noting that the translators (A. Bruder, A. Easterday, J.J. Watkins) did an excellent job. In fact they have added an appendix with additional notes that contain sometimes more details (e.g. they give a proof of the nine test), often they refer to other (mostly recent) books or even websites where more information can be found. A useful addition because it is quite acceptable that this guided tour gets some readers hooked who want to read more about this fascinating world of numbers.