Canadian mathematician
Albert Harold Lightstone (1926–1976) was a Canadian mathematician. He was one of the pioneers of non-standard analysis, a doctoral student of Abraham Robinson, and later a co-author with Robinson of the book "Nonarchimedean Fields and Asymptotic Expansions".
Biography.
Lightstone earned his PhD from the University of Toronto in 1955, under the supervision of Abraham Robinson; his thesis was entitled "Contributions To The Theory Of Quantification". He was a professor of mathematics at Carleton University and Queen's University.
Research.
Decimal hyperreals.
In his article "Infinitesimals" in the "American Mathematical Monthly" in 1972, Lightstone described an extended decimal notation for the hyperreals. Here there is a digit at every hypernatural rank rather than merely a digit for every rank given by a natural number. Such a hyperreal decimal is written as
formula_1
Here the digit formula_2 appears at rank formula_3, which is a typical infinite hypernatural. The semicolon separates the digits at finite ranks from the digits at infinite ranks. Thus, the number 0.000...;...01, with digit "1" at infinite rank "H", corresponds to the infinitesimal formula_4.
The difference 1 - 0.000...;...01 is 0.999...;...9, with an infinite hypernatural's worth of digits 9. An alternative notation for the latter is
formula_5
where "H" is an infinite hypernatural. The extended decimal notation provides a rigorous mathematical implementation of student intuitions of an infinitesimal of the form 0.000...01. Such student intuitions and their usefulness in the learning of infinitesimal calculus were analyzed in a 2010 study by Robert Ely in the Journal for Research in Mathematics Education.
Other research.
Lightstone's main research contributions were in non-standard analysis. He also wrote papers on angle trisection, matrix inversion, and applications of group theory to formal logic.
Books.
Lightstone was the author or co-author of several books on mathematics:
Awards and honours.
Queen's University annually awards the Albert Harold Lightstone Scholarship, named for Lightstone, to a fourth year honors undergraduate student majoring in mathematics or statistics. The scholarship was established by Lightstone's wife after his death.
References.
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