## Lester R. Ford Awards

The Lester
R. Ford Awards, established in 1964, are made to authors of
expository articles published in *The American Mathematical
Monthly. *The Awards are named for Lester R. Ford, Sr.,
a distinguished mathematician, editor of *The American
Mathematical
Monthly, *1942-46, and President of the Mathematical
Association
of America, 1947-48.

## Greg Martin

Absolutely Abnormal Numbers, Amer. Math. Monthly 108 (October 2001), pp. 746-754.

Normal numbers are those whose decimal (or other base) expansions contain
all possible finite configurations of digits with roughly their expected
frequencies. It is known that almost all real numbers are normal to any
given base. Thus, the set of absolutely abnormal numbers, those
that are not normal to any base, must be a set of measure zero.
The rational numbers provide easy examples of such numbers. However, the
set of absolutely abnormal numbers is an uncountable dense subset of the
real numbers, hence must contain irrational numbers. This article is
unique in that the author actually constructs an example of an irrational
absolutely abnormal number. In an exceptionally clear, straightforward
fashion the reader is shown the motivation for considering Liouville
numbers, followed by the description of one particular such number, and
finally the main result, that this number is absolutely abnormal.
Moreover, as pointed out by the author, this technique can be generalized
to give uncountably many irrational absolutely abnormal numbers. This
paper provides some very satisfying results in an area of mathematics
where one can prove the existence of some types of numbers but actual
constructions are rare.

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Last updated 9 Sept, 2007