''From Zero to Infinity: What Makes Numbers Interesting'' is a book in

popular mathematics Popular mathematics is the presentation of mathematics to an aimed general audience. The difference between recreational mathematics and popular mathematics is that recreational mathematics intends to be fun for the mathematical community, and ...

and

number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Mat ...

Constance Reid Constance Bowman Reid (January 3, 1918 – October 14, 2010) was the author of several biographies of mathematicians and popular books about mathematics. She received several awards for mathematical exposition. She was not a mathematician but ...

. It was originally published in 1955 by the Thomas Y. Crowell Company. The fourth edition was published in 1992 by the Mathematical Association of America in their MAA Spectrum series.

A K Peters A K Peters, Ltd. was a publisher of scientific and technical books, specializing in mathematics and in computer graphics, robotics, and other fields of computer science. They published the journals ''Experimental Mathematics'' and the ''Journal ...

published a fifth "Fiftieth anniversary edition" in 2006.

Background

Reid was not herself a professional mathematician, but came from a mathematical family that included her sister

Julia Robinson Julia Hall Bowman Robinson (December 8, 1919July 30, 1985) was an American mathematician noted for her contributions to the fields of computability theory and computational complexity theory—most notably in decision problems. Her work on Hilber ...

and brother-in-law Raphael M. Robinson. She had worked as a schoolteacher, but by the time of the publication of ''From Zero to Infinity'' she was a "housewife and free-lance writer". She became known for her many books about mathematics and mathematicians, aimed at a popular audience, of which this was the first. Reid's interest in number theory was sparked by her sister's use of computers to discover

Mersenne prime In mathematics, a Mersenne prime is a prime number that is one less than a power of two. That is, it is a prime number of the form for some integer . They are named after Marin Mersenne, a French Minim friar, who studied them in the early 17th ...

s. She published an article on a closely related topic,

perfect number In number theory, a perfect number is a positive integer that is equal to the sum of its positive divisors, excluding the number itself. For instance, 6 has divisors 1, 2 and 3 (excluding itself), and 1 + 2 + 3 = 6, so 6 is a perfect number. ...

s, in ''

Scientific American ''Scientific American'', informally abbreviated ''SciAm'' or sometimes ''SA'', is an American popular science magazine. Many famous scientists, including Albert Einstein and Nikola Tesla, have contributed articles to it. In print since 1845, it ...

'' in 1953, and wrote this book soon afterward. Her intended title was ''What Makes Numbers Interesting''; the title ''From Zero to Infinity'' was a change made by the publisher.

Topics

The twelve chapters of ''From Zero to Infinity'' are numbered by the ten decimal digits,

e

(

Euler's number The number , also known as Euler's number, is a mathematical constant approximately equal to 2.71828 that can be characterized in many ways. It is the base of the natural logarithms. It is the limit of as approaches infinity, an expressi ...

, approximately 2.71828), and

\aleph_0

, the smallest infinite cardinal number. Each chapter's topic is in some way related to its chapter number, with a generally increasing level of sophistication as the book progresses: *Chapter 0 discusses the history of number systems, the development of positional notation and its need for a placeholder symbol for zero, and the much later understanding of zero as being a number itself. It discusses the special properties held by zero among all other numbers, and the concept of

indeterminate form In calculus and other branches of mathematical analysis, limits involving an algebraic combination of functions in an independent variable may often be evaluated by replacing these functions by their limits; if the expression obtained after this s ...

s arising from

division by zero In mathematics, division by zero is division where the divisor (denominator) is zero. Such a division can be formally expressed as \tfrac, where is the dividend (numerator). In ordinary arithmetic, the expression has no meaning, as there is ...

. *Chapter 1 concerns the use of numbers to count things, arithmetic, and the concepts of

prime number A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...

s and integer factorization. *The topics of Chapter 2 include

binary representation A binary number is a number expressed in the base-2 numeral system or binary numeral system, a method of mathematical expression which uses only two symbols: typically "0" (zero) and "1" (one). The base-2 numeral system is a positional notation ...

, its ancient use in peasant multiplication and in modern computer arithmetic, and its formalization as a number system by

Gottfried Leibniz Gottfried Wilhelm (von) Leibniz . ( – 14 November 1716) was a German polymath active as a mathematician, philosopher, scientist and diplomat. He is one of the most prominent figures in both the history of philosophy and the history of mathem ...

. More generally, it discusses the idea of number systems with different bases, and specific bases including hexadecimal. *Chapter 3 returns to prime numbers, including the sieve of Eratosthenes for generating them as well as more modern primality tests. *Chapter 4 concerns

square number In mathematics, a square number or perfect square is an integer that is the square of an integer; in other words, it is the product of some integer with itself. For example, 9 is a square number, since it equals and can be written as . The usu ...

s, the observation by Galileo that squares are

equinumerous In mathematics, two sets or classes ''A'' and ''B'' are equinumerous if there exists a one-to-one correspondence (or bijection) between them, that is, if there exists a function from ''A'' to ''B'' such that for every element ''y'' of ''B'', the ...

with the counting numbers, the Pythagorean theorem,

Fermat's Last Theorem In number theory, Fermat's Last Theorem (sometimes called Fermat's conjecture, especially in older texts) states that no three positive integers , , and satisfy the equation for any integer value of greater than 2. The cases and have been ...

, and Diophantine equations more generally. *Chapter 5 discusses

figurate number The term figurate number is used by different writers for members of different sets of numbers, generalizing from triangular numbers to different shapes (polygonal numbers) and different dimensions (polyhedral numbers). The term can mean * polyg ...

integer partition In number theory and combinatorics, a partition of a positive integer , also called an integer partition, is a way of writing as a sum of positive integers. Two sums that differ only in the order of their summands are considered the same part ...

s, and the generating functions and

pentagonal number theorem In mathematics, the pentagonal number theorem, originally due to Euler, relates the product and series representations of the Euler function. It states that :\prod_^\left(1-x^\right)=\sum_^\left(-1\right)^x^=1+\sum_^\infty(-1)^k\left(x^+x^\right ...

that connect these two concepts. *In chapter 6, Reid brings in the material from her earlier article on

s (of which 6 is the smallest nontrivial example), their connection to

s, the search for large prime numbers, and Reid's relatives' discovery of new Mersenne primes. *Mersenne primes are the primes one unit less than a

power of two A power of two is a number of the form where is an integer, that is, the result of exponentiation with number two as the base and integer as the exponent. In a context where only integers are considered, is restricted to non-negativ ...

. Chapter 7 instead concerns the primes that are one more than a power of two, the

Fermat prime In mathematics, a Fermat number, named after Pierre de Fermat, who first studied them, is a positive integer of the form :F_ = 2^ + 1, where ''n'' is a non-negative integer. The first few Fermat numbers are: : 3, 5, 17, 257, 65537, 429496 ...

s, and their close connection to

constructible polygon In mathematics, a constructible polygon is a regular polygon that can be constructed with compass and straightedge. For example, a regular pentagon is constructible with compass and straightedge while a regular heptagon is not. There are infinite ...

s. The

heptagon In geometry, a heptagon or septagon is a seven-sided polygon or 7-gon. The heptagon is sometimes referred to as the septagon, using "sept-" (an elision of ''septua-'', a Latin-derived numerical prefix, rather than '' hepta-'', a Greek-derived nu ...

, with seven sides, is the smallest polygon that is not constructible, because it is not a product of Fermat primes. *Chapter 8 concerns the

cubes In geometry, a cube is a three-dimensional space, three-dimensional solid object bounded by six square (geometry), square faces, Facet (geometry), facets or sides, with three meeting at each vertex (geometry), vertex. Viewed from a corner it i ...

and

Waring's problem In number theory, Waring's problem asks whether each natural number ''k'' has an associated positive integer ''s'' such that every natural number is the sum of at most ''s'' natural numbers raised to the power ''k''. For example, every natural numb ...

on representing integers as sums of cubes or other powers. *The topic of Chapter 9 is

modular arithmetic In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" when reaching a certain value, called the modulus. The modern approach to modular arithmetic was developed by Carl Friedrich Gauss in his boo ...

divisibility In mathematics, a divisor of an integer n, also called a factor of n, is an integer m that may be multiplied by some integer to produce n. In this case, one also says that n is a multiple of m. An integer n is divisible or evenly divisible by ...

, and their connections to positional notation, including the use of

casting out nines Casting out nines is any of three arithmetical procedures: *Adding the decimal digits of a positive whole number, while optionally ignoring any 9s or digits which sum to a multiple of 9. The result of this procedure is a number which is smaller th ...

to determine divisibility by nine. *In Chapter

e

, ''From Zero to Infinity'' shifts from the integers to

irrational number In mathematics, the irrational numbers (from in- prefix assimilated to ir- (negative prefix, privative) + rational) are all the real numbers that are not rational numbers. That is, irrational numbers cannot be expressed as the ratio of two inte ...

complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the fo ...

logarithm In mathematics, the logarithm is the inverse function to exponentiation. That means the logarithm of a number to the base is the exponent to which must be raised, to produce . For example, since , the ''logarithm base'' 10 of ...

s, and

Euler's formula Euler's formula, named after Leonhard Euler, is a mathematical formula in complex analysis that establishes the fundamental relationship between the trigonometric functions and the complex exponential function. Euler's formula states that fo ...

e^=1

. It connects these topics back to the integers through the theory of

continued fraction In mathematics, a continued fraction is an expression obtained through an iterative process of representing a number as the sum of its integer part and the reciprocal of another number, then writing this other number as the sum of its integer ...

s and the prime number theorem. *The final chapter, Chapter

\aleph_0

, provides a basic introduction to

Aleph number In mathematics, particularly in set theory, the aleph numbers are a sequence of numbers used to represent the cardinality (or size) of infinite sets that can be well-ordered. They were introduced by the mathematician Georg Cantor and are named a ...

s and the theory of infinite sets, including

Cantor's diagonal argument In set theory, Cantor's diagonal argument, also called the diagonalisation argument, the diagonal slash argument, the anti-diagonal argument, the diagonal method, and Cantor's diagonalization proof, was published in 1891 by Georg Cantor as a m ...

for the existence of uncountable infinite sets. The first edition included only chapters 0 through 9. The chapter on infinite sets was added in the second edition, replacing a section on the interesting number paradox. Later editions of the book were "thoroughly updated" by Reid; in particular, the fifth edition includes updates on the search for

s and the proof of

, and restores an index that had been dropped from earlier editions.

Audience and reception

''From Zero to Infinity'' has been written to be accessible both to students and non-mathematical adults, requiring only high-school level mathematics as background. Short sets of "quiz questions" at the end chapter could be helpful in sparking classroom discussions, making this useful as supplementary material for secondary-school mathematics courses. In reviewing the fourth edition, mathematician

David Singmaster David Breyer Singmaster (born 1938) is an emeritus professor of mathematics at London South Bank University, England. A self-described metagrobologist, he has a huge personal collection of mechanical puzzles and books of brain teasers. He is mo ...

describes it as "one of the classic works of mathematical popularisation since its initial appearance", and "a delightful introduction to what mathematics is about". Reviewer Lynn Godshall calls it "a highly-readable history of numbers", "easily understood by both educators and their students alike". Murray Siegel describes it as a must have for "the library of every mathematics teacher, and university faculty who prepare students to teach mathematics". Singmaster complains only about two pieces of mathamatics in the book: the assertion in chapter 4 that the Egyptians were familiar with the 3-4-5 right triangle (still the subject of considerable scholarly debate) and the omission from chapter 7 of any discussion of why classifying

s can be reduced to the case of prime numbers of sides. Siegel points out another small error, on algebraic factorization, but suggests that finding it could make another useful exercise for students.

References

{{reflist, refs= {{citation , date = October 20, 2010 , publisher = Mathematical Association of America , title = Author and longtime MAA member Constance Reid dies at 92 , url = https://www.maa.org/news/math-news/author-and-longtime-maa-member-constance-reid-dies-at-92 , work = MAA News {{citation , last = Gibb , first = E. Glenadine , date = February 1957 , issue = 2 , journal =

The Mathematics Teacher Founded in 1920, The National Council of Teachers of Mathematics (NCTM) is a professional organization for schoolteachers of mathematics in the United States. One of its goals is to improve the standards of mathematics in education. NCTM holds an ...

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Mathematics Magazine ''Mathematics Magazine'' is a refereed bimonthly publication of the Mathematical Association of America. Its intended audience is teachers of collegiate mathematics, especially at the junior/senior level, and their students. It is explicitly a j ...

, jstor = 2687853? , mr = 1571022 , pages = 43–44 , title = Review of ''From Zero to Infinity'', 2nd ed. , volume = 34 , year = 1960 {{citation , last = Leamy , first = John , date = March 1993 , issue = 3 , journal =

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MathSciNet MathSciNet is a searchable online bibliographic database created by the American Mathematical Society in 1996. It contains all of the contents of the journal ''Mathematical Reviews'' (MR) since 1940 along with an extensive author database, links ...

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, jstor = 20876226 , pages = 622–623 , title = Review of ''From Zero to Infinity'', 5th ed. , volume = 101 {{citation , last = Lozano-Robledo , first = Álvaro , date = May 2006 , journal = MAA Reviews , publisher = Mathematical Association of America , title = Review of ''From Zero to Infinity'', 5th ed. , url = https://www.maa.org/press/maa-reviews/from-zero-to-infinity-what-makes-numbers-interesting {{citation , last = Papp , first = F.-J. , mr = 2198198 , title = Review of ''From Zero to Infinity'', 5th ed. , work =

, year = 2006 {{citation , last = Siegel , first = Murray H. , date = February 2007 , issue = 6 , journal = Mathematics Teaching in the Middle School , jstor = 41182422 , page = 350 , title = Review of ''From Zero to Infinity'', 5th ed. , volume = 12 Popular mathematics books 1955 non-fiction books Elementary number theory