# Numbers: Their Tales, Types, and Treasures

Numbers and (elementary) number theory is often used in books about popular mathematics since many problems there are easy to formulate at a generally understandable level. Moreover the history of numbers and their notation goes along with the history and the evolution of mathematics. So we see that numbers feature in books like Single digits (M. Chamberlain, 2015), Professor Stewart's incredible numbers (I. Stewart, 2015), Those Fascinating Numbers (JM De Koninck, 2009), The Magic Numbers of the Professor (O. O'Shea, 2007), Numbers at Work (R. Taschner, 2007), and this list goes on and on, and then there are the many books on the history of numbers like *The Universal History of Numbers* by G. Ifrah (Wiley, 2000).

And here is yet another popular book about numbers. We do find some elements that can also be found elsewhere, but there are still some differences. The book starts in its first chapter with the obligatory association of numbers with counting and what counting really means, which is conceptually not as trivial as it seems since to do computations, you have to detach numbers from the objects you are counting. Only then the abstract notation of numbers is initiated. A more detailed history of the notation of numbers is given in chapter 3. The interlude of chapter 2 elaborates on the psychological aspects of numbers. We instinctively grasp small amounts up to three or four, which is reflected in the separate, non-systematic names that are given to these numbers in practically all languages, even the most primitive ones. If there are more than 5 objects, we start counting or catch their number by grasping couples or triples and calculate. Counting is not intuitive. We have to learn it by practicing. A child that can name the numbers from one to ten is reciting a poem, which is quite different from counting.

Once the abstraction from counting objects to numbers is properly made, one can start detecting patterns like triangular, square, or rectangular numbers, meaning that we can arrange that amount of stones or objects or dots in the form of a triangle, square, or rectangle. These arrangements can lead to well known summation formulas. Pentagonal and tetrahedral numbers are less common and certainly less popular.

The fifth chapter is called *Counting for Poets* and may come a bit as a surprise in a book on numbers. However certain patterns of metrical rhythm are imposed by the type of the poem. It may require stressing syllables by duration or loudness. Verse meters are as old as the Vedic literature. The duration or weight of a verse is expressed in units of moras. For example a meter is a sequence of short and long syllables, of weight respectively one and two moras. One may see here an analogy with a music score. So, given a the total length of the verse, how many different meters are then possible? And so, triggered by Pingala's historical work on Sanskrit prosody, this is how the authors turn this into a chapter on numbers and combinatorial problems. Even the Fibonacci sequence shows up. The Fibonacci's sequence, the golden section, and Pascal's triangle are further discussed in the next chapter.

With several patterns that can be detected in Pascal's triangle, the kickoff is given for other arrangements of numbers like magic squares and how to construct them. Napier's rods or bones are sticks with numbers written on it, and when arranged in an appropriate way, this can be used to multiply (moderately) large numbers by recognizing certain patters. It is basically an Arabic invention, but it was improved by Napier who is better known for his invention of the logarithms. You may read more about this in *John Napier: Life, Logarithms, and Legacy* (John Havil, 2014).

Chapter 8 finally arrives at the unavoidable prime numbers, but this is kept rather short, culminating in a list of unsolved problems. Other special numbers are perfect numbers, Kaprekar numbers, Armstrong numbers and some isolated numbers with surprising properties.

Relationships between numbers are the subject of the next chapter like amicable numbers of all sorts. Most attention goes however to Pythagorean triples, their construction and some curious properties and patterns they reveal. Also checks for divisibility by 2,3,...,17 are explored.

The keyword for chapter 10 is proportions. Continued fractions are introduced for example when subdividing an interval with the ratio of Fibonacci numbers, which of course results in the golden ratio (again), but also pi as the ratio of the circumference of the circle to its diameter and its long history are the subject here.

The last chapter is about numbers and philosophy as in the 20th century numbers were defined in a formal way by logicists, formalists as well as constructivist. You will find here also some thoughts about the modeling of nature, the so called *Unreasonable effectiveness of mathematics in the natural sciences* as Wigner once stated it.

Some tables of numbers (Fibonacci, primes, perfect numbers, Kaprekar numbers, Armstrong numbers, amical numbers, and palindromic numbers) conclude the book.

This survey illustrates that there are some well known topics discussed, but also some unexpected ones, that makes this book certainly advisable. No special mathematical training is needed. Especially the chapters on the development of the concept of numbers in a child and the link with prosody are refreshing. Some of the intriguing patterns presented are unexpected and new to me. So I can full-heartedly recommend to read this book both for the professional mathematician and the generally interested reader.

**Submitted by Adhemar Bultheel |

**17 / Aug / 2015