This wonderful book is crammed with fascinating facts and properties
about every kind of number. Many of the properties are given a geometric
interpretation, so there are plenty of marvelous diagrams. Even if you've
come across the material before, this geometric and diagrammatic approach
may give you a new perspective on things --- for example, I was
particularly taken with the 'telegraph pole' representation of infinite
ordinals in the last chapter. Many more weighty presentations tend to be
short on diagrams. I think this is a pity: although 'proof by diagram' is
rarely sound, and nearly always needs to be backed up by a symbolic proof,
diagrams themselves can do much to help prime the intuition.
The sign of a good book: I was fascinated by what was there, and was
left wanting more. Here's a brief overview of the contents, to give you a
flavour of the breadth covered.
- The Romance of Numbers. Names of numbers, and how number
names influence the rest of language. Written number systems: Babylonian
cuneiform, Greek, Roman, Egyptian, Hindu-Arabic. Other bases.
- Doing Arithmetic and Algebra by Geometry. Square
and triangular numbers, and other 2D shapes. Cube and tetrahedral
numbers, and other 3D shapes (rhombic dodecahedral numbers, anyone?) 4D,
and higher, shaped numbers. Wonderful geometric proofs of various
Tacked on the end comes a short section
very large (finite) numbers.
Fascinating, but I would have like a whole chapter on its own!
- What Comes Next? Factorials, Pascal's Triangle,
binomial coefficients. Sequences, differencing, Jackson's Difference
- Famous Families of Numbers. Bell numbers, Stirling
numbers, Ramanujan's numbers, Catalan numbers, Faulhaber's formula,
Bernoulli numbers, Euler numbers. Fibonacci
numbers, why sunflower heads and pineapple rinds grow in a
- The Primacy of Primes.
Prime numbers, sieve of
Eratosthenes. Arithmetic modulo p, unique factorisation,
perfect numbers, Fermat's
numbers and test for primes. Distribution of primes.
- Further Fruitfulness of Fractions. Farey fractions,
Ford circles, Euler's totient numbers, decimal expansions, shuffling,
Long primes, pythogorean fractions.
- Algebraic Numbers. Irrationality of root-2.
Continued fractions of irrationals. Lagrange numbers, Markov numbers,
Freiman's number. Ruler and compass constructions, construction of
regular 17-gon. Calbi's triangle, Graham's hexagon.
- Imaginary Numbers. Complex plane, adding and
multiplying. Gaussian integers and primes. The rational triangle.
Hypercomplex numbers: Hamilton's quaternions, Cayley numbers.
- Transcendental numbers. Pi, Liouville's number,
Gregory's numbers, Størmer's numbers. Logarithms, e,
harmonic numbers, Euler's number, Stirling's formula for n!
- Infinite and infinitesimal numbers. Cantor's ordinal
numbers, cardinal numbers. Surreal numbers.