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Statistical Independence in Probability, Analysis and Number Theory (Carus Mathematical Monographs)

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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!