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{{Short description|Number theory book}}
{{Use list-defined references|date=November 2022}}
{{italic title}}
'''''From Zero to Infinity: What Makes Numbers Interesting''''' is a book in popular mathematics and [[Number theory|number theory]] by Constance Reid. It was originally published in 1955 by the Thomas Y. Crowell Company.{{r|gibb}} The fourth edition was published in 1992 by the Mathematical Association of America in their MAA Spectrum series.{{r|leamy|mor-mor|singmaster}} A K Peters published a fifth "Fiftieth anniversary edition" in 2006.{{r|belle|godshall|hoagland|lozano-robledo|papp|siegel}}

==Background==
Reid was not herself a professional mathematician, but came from a mathematical family that included her sister [[Biography:Julia Robinson|Julia Robinson]] and brother-in-law [[Biography:Raphael M. Robinson|Raphael M. Robinson]].{{r|reid-obit}} 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".{{r|gibb}} She became known for her many books about mathematics and mathematicians, aimed at a popular audience, of which this was the first.{{r|reid-obit}}

Reid's interest in number theory was sparked by her sister's use of computers to discover [[Mersenne prime]]s. She published an article on a closely related topic, [[Perfect number|perfect number]]s, in ''Scientific American'' in 1953, and wrote this book soon afterward.{{r|singmaster}} Her intended title was ''What Makes Numbers Interesting''; the title ''From Zero to Infinity'' was a change made by the publisher.{{r|lozano-robledo}}

==Topics==
The twelve chapters of ''From Zero to Infinity'' are numbered by the ten decimal digits, <math>e</math> ([[E (mathematical constant)|Euler's number]], approximately 2.71828), and <math>\aleph_0</math>, the [[Aleph number|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:{{r|singmaster|belle|siegel}}
*Chapter 0 discusses the history of number systems, the development of [[Positional notation|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|indeterminate form]]s arising from [[Division by zero|division by zero]].{{r|singmaster|belle|siegel}}
*Chapter 1 concerns the use of numbers to count things, arithmetic, and the concepts of [[Prime number|prime number]]s and [[Integer factorization|integer factorization]].{{r|singmaster|belle}}
*The topics of Chapter 2 include binary representation, its ancient use in peasant multiplication and in modern computer arithmetic, and its formalization as a number system by Gottfried Leibniz. More generally, it discusses the idea of number systems with different bases, and specific bases including [[Hexadecimal|hexadecimal]].{{r|singmaster|belle}}
*Chapter 3 returns to prime numbers, including the [[Sieve of Eratosthenes|sieve of Eratosthenes]] for generating them as well as more modern [[Primality test|primality test]]s.{{r|singmaster}}
*Chapter 4 concerns [[Square number|square number]]s, the observation by Galileo that squares are equinumerous with the counting numbers, the [[Pythagorean theorem]], [[Fermat's Last Theorem]], and [[Diophantine equation]]s more generally.{{r|singmaster|belle}}
*Chapter 5 discusses [[Figurate number|figurate number]]s, [[Partition (number theory)|integer partition]]s, and the [[Generating function|generating function]]s and [[Pentagonal number theorem|pentagonal number theorem]] that connect these two concepts.{{r|singmaster|belle}}
*In chapter 6, Reid brings in the material from her earlier article on [[Perfect number|perfect number]]s (of which 6 is the smallest nontrivial example), their connection to [[Mersenne prime]]s, the search for [[Largest known prime number|large prime numbers]], and Reid's relatives' discovery of new Mersenne primes.{{r|singmaster|belle}}
*Mersenne primes are the primes one unit less than a [[Power of two|power of two]]. Chapter 7 instead concerns the primes that are one more than a power of two, the Fermat primes, and their close connection to [[Constructible polygon|constructible polygon]]s. The [[Heptagon|heptagon]], with seven sides, is the smallest polygon that is not constructible, because it is not a product of Fermat primes.{{r|singmaster}}
*Chapter 8 concerns the [[Cube (algebra)|cubes]] and [[Waring's problem]] on representing integers as sums of cubes or other powers.{{r|singmaster|belle}}
*The topic of Chapter 9 is [[Modular arithmetic|modular arithmetic]], divisibility, and their connections to positional notation, including the use of [[Casting out nines|casting out nines]] to determine divisibility by nine.{{r|singmaster|belle|siegel}}
*In Chapter <math>e</math>, ''From Zero to Infinity'' shifts from the integers to [[Irrational number|irrational number]]s, [[Complex number|complex number]]s, [[Logarithm|logarithm]]s, and [[Euler's formula]] <math>e^{i\pi}=1</math>. It connects these topics back to the integers through the theory of [[Continued fraction|continued fraction]]s and the [[Prime number theorem|prime number theorem]].{{r|singmaster}}
*The final chapter, Chapter <math>\aleph_0</math>, provides a basic introduction to [[Aleph number]]s and the theory of infinite sets, including [[Cantor's diagonal argument]] for the existence of uncountable infinite sets.{{r|singmaster|belle}}

The first edition included only chapters 0 through 9.{{r|gibb}} The chapter on infinite sets was added in the second edition, replacing a section on the [[Interesting number paradox|interesting number paradox]].{{r|hamilton}} Later editions of the book were "thoroughly updated" by Reid;{{r|singmaster}} in particular, the fifth edition includes updates on the search for [[Mersenne prime]]s and the proof of [[Fermat's Last Theorem]], and restores an index that had been dropped from earlier editions.{{r|papp}}

==Audience and reception==
''From Zero to Infinity'' has been written to be accessible both to students and non-mathematical adults,{{r|singmaster}} requiring only high-school level mathematics as background.{{r|hoagland}} 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.{{r|godshall|siegel}}

In reviewing the fourth edition, mathematician [[Biography:David Singmaster|David Singmaster]] describes it as "one of the classic works of mathematical popularisation since its initial appearance", and "a delightful introduction to what mathematics is about".{{r|singmaster}} Reviewer Lynn Godshall calls it "a highly-readable history of numbers", "easily understood by both educators and their students alike".{{r|godshall}} Murray Siegel describes it as a must have for "the library of every mathematics teacher, and university faculty who prepare students to teach mathematics".{{r|siegel}}

Singmaster complains only about two pieces of mathematics 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 [[Constructible polygon|constructible polygon]]s can be reduced to the case of prime numbers of sides.{{r|singmaster}} Siegel points out another small error, on algebraic factorization, but suggests that finding it could make another useful exercise for students.{{r|siegel}}

==References==
{{reflist|refs=

<ref name=reid-obit>{{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}}</ref>

<ref name=gibb>{{citation
 | last = Gibb | first = E. Glenadine
 | date = February 1957
 | issue = 2
 | journal = The Mathematics Teacher
 | jstor = 27955358
 | page = 178
 | title = Review of ''From Zero to Infinity'', 1st ed.
 | volume = 50}}</ref>

<ref name=hamilton>{{citation
 | last = Hamilton | first = J. M. C.
 | doi = 10.2307/2687853
 | issue = 1
 | journal = Mathematics Magazine
 | jstor = 2687853?
 | mr = 1571022
 | pages = 43–44
 | title = Review of ''From Zero to Infinity'', 2nd ed.
 | volume = 34
 | year = 1960}}</ref>

<ref name=leamy>{{citation
 | last = Leamy | first = John
 | date = March 1993
 | issue = 3
 | journal = The Mathematics Teacher
 | jstor = 27968284
 | page = 265
 | title = Review of ''From Zero to Infinity'', 4th ed.
 | volume = 86}}</ref>

<ref name=mor-mor>{{citation
 | last2 = Morrison | first2 = Phylis
 | date = December 1992
 | department = Science books for young people
 | issue = 6
 | journal = Scientific American
 | jstor = 24939341
 | title = Review of ''From Zero to Infinity'', 4th ed.
 | volume = 267}}</ref>

<ref name=singmaster>{{citation
 | mr = 1154796
 | title = Review of ''From Zero to Infinity'', 4th ed.
 | work = [[MathSciNet]]
 | year = 1993
 | zbl = 0803.00002}}</ref>

<ref name=belle>{{citation
 | last = Belle | first = Vaishak
 | date = June 2011
 | doi = 10.1145/1998037.1998040
 | issue = 2
 | journal = ACM SIGACT News
 | pages = 10–11
 | title = Review of ''From Zero to Infinity'', 5th ed.
 | url = https://www.cs.umd.edu/~gasarch/bookrev/42-2.pdf
 | volume = 42}}</ref>

<ref name=godshall>{{citation
 | last = Godshall | first = Lynn
 | date = July 2007
 | journal = Convergence
 | title = Review of ''From Zero to Infinity'', 5th ed.
 | url = https://www.maa.org/press/periodicals/convergence/from-zero-to-infinity}}</ref>

<ref name=hoagland>{{citation
 | last = Hoagland | first = Kayana
 | date = April 2008
 | issue = 8
 | journal = The Mathematics Teacher
 | jstor = 20876226
 | pages = 622–623
 | title = Review of ''From Zero to Infinity'', 5th ed.
 | volume = 101}}</ref>

<ref name=lozano-robledo>{{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}}</ref>

<ref name=papp>{{citation
 | last = Papp | first = F.-J.
 | mr = 2198198
 | title = Review of ''From Zero to Infinity'', 5th ed.
 | work = [[MathSciNet]]
 | year = 2006}}</ref>

<ref name=siegel>{{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}}</ref>

}}

[[Category:Elementary number theory]]

{{Sourceattribution|From Zero to Infinity}}