##
**The book of numbers.**
*(English)*
Zbl 0866.00001

Berlin: Springer-Verlag. ix, 310 p. (1996).

In ‘The book of numbers’, Conway and Guy explore the many ways in which the word “number” is used. Their intended audience is the inquisitive reader without any particular mathematical background. However, there is much in this book of value to the professional mathematician as well – new things, old things in unusual contexts, and the fascinating collision of old and new.

This is a book that rewards browsing. Teachers who read or browse here will find ideas, information, examples and even some pedagogical tools that will enrich their courses. The authors are playful, punning, and humorous, and their enthusiasm is contagious.

There are three main threads to the story. One is the development and enlargement of the idea of number from the counting numbers to cardinal, ordinal, and surreal numbers. Another is the special study of the integers and special sets or sequences of numbers such as prime numbers, Fibonacci numbers, and Bernoulli numbers. The third thread is the study of special numbers such as \(e,\pi\), Euler’s number \(\gamma\), Feigenbaum’s constant, and the like.

The first chapter – on number words – is worth the price of the book on its own. Browse and enjoy!

This is a book that rewards browsing. Teachers who read or browse here will find ideas, information, examples and even some pedagogical tools that will enrich their courses. The authors are playful, punning, and humorous, and their enthusiasm is contagious.

There are three main threads to the story. One is the development and enlargement of the idea of number from the counting numbers to cardinal, ordinal, and surreal numbers. Another is the special study of the integers and special sets or sequences of numbers such as prime numbers, Fibonacci numbers, and Bernoulli numbers. The third thread is the study of special numbers such as \(e,\pi\), Euler’s number \(\gamma\), Feigenbaum’s constant, and the like.

The first chapter – on number words – is worth the price of the book on its own. Browse and enjoy!

Reviewer: William J. Satzer Jr. (St. Paul)

### MSC:

00A05 | Mathematics in general |

00-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to mathematics in general |

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\textit{J. H. Conway} and \textit{R. K. Guy}, The book of numbers. Berlin: Springer-Verlag (1996; Zbl 0866.00001)

### Online Encyclopedia of Integer Sequences:

Catalan numbers: C(n) = binomial(2n,n)/(n+1) = (2n)!/(n!(n+1)!).Number of ways of writing n as a sum of 4 squares; also theta series of four-dimensional cubic lattice Z^4.

a(n) = 2^n - C(n,0)- ... - C(n,4).

Triangle of Euler-Bernoulli or Entringer numbers read by rows.

Characteristic function of factorial numbers; also decimal expansion of Liouville’s number or Liouville’s constant.

Numerator of sum( i+j+k = n, (i*j)/k) i,j,k >=1.

Triangle T(n,k) = abs( k *( (2*n+1)*(-1)^(n+k)+2*k-1) /4 ) read by rows, 1<=k<=n.

Floor of 4th root of pentatope numbers.

Decimal expansion of 3/Pi^2.

Products of three terms from A003627.

Number of prime divisors of n-th Conway and Guy second-order harmonic number (counted with multiplicity).

Numerators of third-order harmonic numbers (defined by Conway and Guy, 1996).

Denominators of third-order harmonic numbers (defined by Conway and Guy, 1996).

Erroneous version of A001333

Decimal expansion of square root of 221/25

Decimal expansion of the Pythagorean comma.

Fifth differences of 7th powers (A001015).

Triangle read by rows: coefficients in the sum of odd powers as expressed by Faulhaber’s theorem, T(n, k) for n >= 1, 1 <= k <= n.

a(n) is the numerator of the n-th hyperharmonic number of order n.

a(n) is the denominator of the n-th hyperharmonic number of order n.