A Brief History of Numbers

At undergraduate level, now students deal fluently with numbers, negative, rational, real, or complex and they solve linear and quadratic equations without a problem. What is nearly obvious today, is however relatively recent in the history of mathematics. A formal definition of the reals and even of the natural numbers was only given in the twentieth century. In this book, you can read how the number concept of the ancient Greek has evolved during the centuries till we arrived at our modern concepts. The book is written in such a way that it can be enjoyed by students at an undergraduate level. A proper level of readability is maintained throughout. The text is not overloaded with citations and references which sometimes happens in professional historical studies. Neither is it a popularizing account of anecdotes. For those who want to know more, very useful suggestions of texts for general reading as well as for the separate chapters, are added in an appendix

A short introductory chapter explains the number systems as we know it now, including the notion of algebraic, transcendental, and complex numbers and the cardinals and (transfinite) ordinals. This is followed by a brief survey of the notation for numbers in ancient Egypt, Babylon, and Greece, and of course the usual positional system that we are using now. The latter are only notational issues. The true development of the number concept is started when analyzing the Greek vision, who thought of numbers as magnitudes and they had proportions, which were not considered to be numbers. This makes quite a difference. For example, their geometric interpretation of the product of two numbers is the area of a rectangle. So different powers of a variable could not be added since for example adding a length and an area made no sense. What we would call a ratio of two natural number was a proportion for them and it corresponded to a geometric construction. When such a proportion could not be expressed by natural numbers they had a problem with incommensurability like 2‾√ and π since these are irrational. It is only with Diophantus (sometimes called the father of algebra) that we see some kind of algebraic manipulation to solve problems which were equations, given in the form of `similarity' of proportions. The constraint that they allowed only constructions using straight edge and compass is a myth. They were not really strict on that. This restriction was added later by copyists.

The Hindu-Arabic number system, including the zero was brought to Europe via Medieval Islamic authors. Al-Kwārizmī with his book of calculation the al-jabr (the origin of the word algebra) is the best known one. These Islamic authors solved quadratic and even cubic equations using computational rules that looked algebraic, although the underlying idea was often still geometric. For example $ax^2+bx=c$ and $ax^2+c=bx$ and other variants could be solved, but they required different techniques and they were not considered as special cases of our general quadratic equation. Moreover, the problems were still formulated rhetorically, such as `squares and roots equal numbers' or `squares and numbers equal roots' for the above examples. No symbols were in use yet. Since no quantities of dimension higher than three were conceivable, the cubic was the final frontier and of course only positive outcomes were considered to be solutions.

Mathematics revived in Europe in the 12th-16th century. The Hindu-Arabic number notation was popularized in Europe by Fibonacci's Liber Abbaci. Texts were copied and translated from the Arabic and Greek originals. Probably because of all this copying, abbreviations were introduced especially by the Cossists in Germany named after the influential Die Coss by Christoff Rudolf. This cos, represented by the letter c was a translation of the Italian cosa. Italians used it to name the unknown in an equation. The plus and minus signs and the square root sign were introduced, and as long as the final result was positive, also intermediate negative quantities could be used in a formal sense. However, soon negative numbers were accepted, be it reluctantly, as a result. Eventually Cardano, Tartaglia, and del Ferro, although fighting over precedence, solved a general cubic equation and Bombelli took roots of negative numbers, hence opening the gate to complex numbers.

With the scientific revolution at the end of the 16th century, Viète already used a notation that came close to ours, Simon Stevin finally identified proportions as proper rational numbers, Napier provided logarithms as a elegant computational tool. Descartes connected algebra and geometry and, using proportions and a geometric construction, he could define the product of two lengths to be another length, hence the product kept the same dimension as the factors. Wallis introduced the negative numbers as numbers and he also had the complex numbers and their representation in the plane. Numbers started to emerge as entities that lived independent of what they counted or measured. Newton already gave a loose definition. Euler and Gauss of course contributed to calculating, but it was Hamilton who first defined complex numbers as a couple of reals with particular computational rules. He extended it to quaternions and even further. It was however Dedekind at the late 19th century who really started to think about the definition of numbers and number systems and came up with the definition of a field. Later he also defined the reals with his cuts, filling up the real line to form a continuum. Finally the natural numbers, who were the basis of everything, were defined in an axiomatic way by Peano and Frege, based on set theory. This completes the number systems we are used to building them up in a nice hierarchical structure. With the continuum of the reals, a finite interval now contains infinitely many real numbers, as many as there are on the whole real line, seemingly a paradox. This has brought Cantor to study the transfinite ordinals.

It is clear that the last chapters of the book where Dedekind enters the stage, the mathematics become a bit more demanding, but Corry progresses at low pace and the development should be still understandable for undergraduates. Of course the formal proofs are not and should not be included in this book and some of the more technical or additional material that is not essential for the development of the history is moved to appendices added at the end of each chapter. For example the `casting out nines' test is explained in such an appendix, something that is probably unfamiliar to undergraduates in an era of computers and calculators.

It is a highly recommended and pleasant read, not pedantic, but not casual either. You learn for example that it is not because one is the first to use a concept, that one is recognized for it. It will only be accepted by a community if it is adopted by an authority. Also, every copy or translation will add something to the original so that after a while the difference can be considerable. The book is written with great care, but nevertheless I could spot a puzzling typo on page 23 where it says that a gigabyte is a hundred million bytes whereas it should be a thousand million.

Adhemar Bultheel
Book details

Written at an undergraduate level, this is a very accessible historical survey of how the concept of numbers and number systems has evolved in Western Europe since the ancient Greek till the definition of the real numbers by Dedekind and the transfinite numbers of Cantor.

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978-0-19-870259-7 (hbk)