''Playing with Infinity: Mathematical Explorations and Excursions'' is a book in popular mathematics by Hungarian mathematician Rózsa Péter, published in German in 1955 and in English in 1961.

Publication history and translations

''Playing with Infinity'' was originally written in 1943 by mathematician Rózsa Péter, based on a series of letters Péter had written to a non-mathematical friend, . Because of World War II, it was not published until 1955, in German, under the title ''Das Spiel mit dem Unendlichen'', by Teubner. An English translation by

Zoltán Pál Dienes Zoltán Pál Dienes (anglicized as Zoltan Paul Dienes) (June 21, 1916 – January 11, 2014) was a Hungarian mathematician whose ideas on education (especially of small children) have been popular in some countries. He was a world-famous theorist ...

was published in 1961 by G. Bell & Sons in England, and by Simon & Schuster in the US. The English version was reprinted in 1976 by Dover Books, The German version was also reprinted, in 1984, by Verlag Harri Deutsch; the book has also been translated into Polish in 1962, and into Russian in 1967.; The Basic Library List Committee of the

Mathematical Association of America The Mathematical Association of America (MAA) is a professional society that focuses on mathematics accessible at the undergraduate level. Members include university, college, and high school teachers; graduate and undergraduate students; pure a ...

has suggested its inclusion in undergraduate mathematics libraries.

Topics

''Playing with Infinity'' presents a broad panorama of mathematics for a popular audience. It is divided into three parts, the first of which concerns counting, arithmetic, and connections from numbers to geometry both through visual proofs of results in arithmetic like the sum of finite arithmetic series, and in the other direction through counting problems for geometric objects like the diagonals of polygons. These ideas lead to more advanced topics including Pascal's triangle, the Seven Bridges of Königsberg, the

prime number theorem In mathematics, the prime number theorem (PNT) describes the asymptotic distribution of the prime numbers among the positive integers. It formalizes the intuitive idea that primes become less common as they become larger by precisely quantifying ...

and the

sieve of Eratosthenes In mathematics, the sieve of Eratosthenes is an ancient algorithm for finding all prime numbers up to any given limit. It does so by iteratively marking as composite (i.e., not prime) the multiples of each prime, starting with the first prime n ...

, and the beginnings of algebra and its use in proving the impossibility of certain straightedge and compass constructions. The second part begins with the power of inverse operations to construct more powerful systems of numbers: negative numbers from subtraction, and rational numbers from division. Later topics in this part include the countability of the rationals, the irrationality of the

square root of 2 The square root of 2 (approximately 1.4142) is a positive real number that, when multiplied by itself, equals the number 2. It may be written in mathematics as \sqrt or 2^, and is an algebraic number. Technically, it should be called the princip ...

, exponentiation and logarithms, graphs of functions, slopes and areas of curves, and complex numbers. Topics in the third part include non-Euclidean geometry, higher dimensions, mathematical logic, the failings of

naive set theory Naive set theory is any of several theories of sets used in the discussion of the foundations of mathematics. Unlike Set theory#Axiomatic set theory, axiomatic set theories, which are defined using Mathematical_logic#Formal_logical_systems, forma ...

, and

Gödel's incompleteness theorems Gödel's incompleteness theorems are two theorems of mathematical logic Mathematical logic is the study of logic, formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research i ...

. In keeping with its title, these topics allow ''Playing with Infinity'' to introduce many different ways in which ideas of

infinity Infinity is that which is boundless, endless, or larger than any natural number. It is often denoted by the infinity symbol . Since the time of the ancient Greeks, the philosophical nature of infinity was the subject of many discussions amo ...

have entered mathematics, in the notions of infinite series and limits in the first part, countability and transcendental numbers in the second, and the introduction of infinite points in projective geometry, higher dimensions,

metamathematics Metamathematics is the study of mathematics itself using mathematical methods. This study produces metatheories, which are mathematical theories about other mathematical theories. Emphasis on metamathematics (and perhaps the creation of the ter ...

, and undecidability in the third.

Audience and reception

Reviewer Philip Peak writes that the book succeeds in showing readers the joy of mathematics without getting them bogged down in calculations and formulas. On a similar note, Michael Holt recommends the book to mathematics teachers, as a sample of the more conceptual style of mathematics taught in Hungary at the time in contrast to the orientation towards practical calculation of English pedagogy. Reuben Goodstein summarizes it more succinctly as "the best book on mathematics for everyman that I have ever seen". By the time of Leon Harkleroad's review in 2011, the book had become "an acknowledged classic of mathematical popularization". However, Harkleroad also notes that some idiosyncrasies of the translation, such as its use of pre-decimal British currency, have since become quaint and old-fashioned. And similarly, although W. W. Sawyer in reviewing the original 1955 publication calls its inclusion of topics from graph theory and topology "truly modern", Harkleroad points out that more recent works in this genre have included other topics in their own quest for modernity such as "fractals, public-key cryptography, and internet search engines", which for obvious reasons Péter omits.

References

{{reflist, refs= {{citation , last = Goodstein , first = R. L. , author-link = Reuben Goodstein , date = May 1962 , doi = 10.2307/3611665 , issue = 356 , journal = The Mathematical Gazette , jstor = 3611665 , page = 157 , title = Review of ''Playing with Infinity'' , volume = 46, s2cid = 118074515 {{citation , last = Harkleroad , first = Leon , date = October 2011 , journal = MAA Reviews , publisher =

, title = Review of ''Playing with Infinity'' , url = https://www.maa.org/press/maa-reviews/playing-with-infinity {{citation , last = Holt , first = Michael , date = May 1977 , issue = 3 , journal = Mathematics in School , jstor = 30212436 , page = 35 , title = Review of ''Playing with Infinity'' , volume = 6 {{citation , last = Newman , first = James R. , author-link = James R. Newman , date = August 1962 , issue = 2 , journal = Scientific American , jstor = 24936655 , page = 146 , title = Review of ''Playing with Infinity'' , volume = 207 {{citation , last = Peak , first = Philip , date = March 1977 , issue = 3 , journal = The Mathematics Teacher , jstor = 27960825 , page = 282 , title = Review of ''Playing with Infinity'' , volume = 70 {{citation , last = Sawyer , first = W. W. , author-link = Walter Warwick Sawyer , journal = zbMATH , title = Review of ''Das Spiel mit dem Unendlichen'' , zbl = 0066.00201 Popular mathematics books 1943 non-fiction books 1955 non-fiction books 1961 non-fiction books Infinity