Customer Reviews (1)

Distributions of blocks of digits
A gem of a book. Fun, easy to read. Starts with a number theoretic contribution of Viete (who made a step toward formulating algebraic questions with alphabetic symbols, a task later carried out by his countrymen Descartes and Fermat) and goes on to discuss the statistical distributions of blocks of digits appearing in decimal expansions of numbers. For a related discussion see Niven's "Irrational Numbers". Statistical distributions of blocks of digits are used implicitly and explicitly in symbollic dynamics in deterministic chaos. A word of warning: so-called 'normal numbers' occupy the continuum with measure one, but most irrational numbers (with measure one) are not computable. Champernowne's number (in any integer base of arithmetic) is the only known computable normal number. The convergence of its digit blocks to an even distribution is too slow for use in generating pseudorandom numbers. On the other hand, the digits of pi play good 5-handed poker for about 500000 base 10 digits, but we do not know if pi is normal to any base! ... Read more

Book Description

Customer Reviews (5)

Excellent overview by two outstanding mathematicians
The audience for ths book is people with background in mathematics. They teach many branches briefly with their examples in Chapter 1, but the presentation is at a fairly sophisticated level.

I found the historical and philosophical remarks very valuable, coming from authorities like these two men. Examples (slightly edited by me):

The great analysts of the 18th and 19th centuries (e.g., Newton, Leibniz, Bernoulli, Euler, Lagrange ...) had an almost unerring instinct for presenting valid results and plausible proofs without a firm basis in formal systems and without strict adherence to standards of logical rigor... Mathematical intuition in the hands of people of genius has such a clarity and unity that it anticipates special formalisms.

Mathematics is a science; it is also an art. The criteria of judgment in mathematics are always aesthetic, at least in part... One looks for `usefulness,' for `interest,' and also for `beauty.' Beauty is subjective, yet it is surprising that there is usually considerable agreement among mathematicians concerning aesthetic values.

It is a distinctive feature of mathematics that it can operate effectively and efficiently without defining its objects. Points, straight lines and planes are not defined... One need not know what things are so long as one knows what statements about them one is allowed to make [the axioms]... Other statements involving these undefined words can then be deduced by logic alone. This permits geometry to be taught to a blind man and even to a computer!

A Classic of Popular Mathematics
With new mathematics growing wild in the trees in the new century, one should review those things that have givenmodern mathematics direction and flavor. This classic little book sits along side of Sawyer's " Prelude to Mathematics" as the go to books for basic understanding. To these books I have added an unlikely candidate " Elliptical Curves" by McKean and Moll.If one has these three books, he will have a crack in the damof mathematics...

Important and Useful but not Engaging
The book shows what the power of mathematics is, how it changes, and how it expands to new areas. Unlike books that aim to popularize math, the book does not pontificate a mystic view of the meaning of mathematics; rather, it gives a sober perspective of what has happened in mathematics and what can be expected of the field.

The book requires somewhat serious mathematical thinking.

A great strength of the book is the diverse mathematical concepts that it presents: homology groups, group theory, Turing machines, undecidability, Monte Carlo method.In a compact book, you learn a little about some important ideas in advanced mathematics.

Important!! This book is not written to popularize its subject matter, so it is different and definitely less entertaining than most popular science books.

why math is interesting, written for laypeople
Kac and Ulam were both strong mathematicians; this encyclopedia articlewhich they coauthored provides a nice counterweightto the answers-without-questions view of mathwhich can result from a little indifferentcoursework.

Easy and lively reading, written for everyone.

A bargain indeed
Just read the editorial review (Book News).It's all true (except for the publishing date,which is 1968, not 1986...) ... Read more