### Read É PDF, DOC, TXT or eBook É James M. Henle

School mathematics Contents Introduction Language and Structure The Hyperreal Numbers The Hyperreal Line Continuous Functions Integral CalculusDifferential Calculus The Fundamental Theo. Many books on calculus using hyperreals focus on the strict formalism For the hard core mathematician this might be the adeuate approach however for us normal people with normal uestions eg WTF ARE hyperreals this short and concise book appears to be the first choice Easy reading

### James M. Henle É 0 characters

Infinitesimal CalculusRigorous undergraduate treatment introduces calculus at the basic level using infinitesimals and concentrating on theory rather than applications Reuires only a solid foundation in high. The calculus was created as many know by Newton and Leibniz Newton's concept of calculus was based on continuity while Leibniz used a conceptual framework based on infinitesimals numbers smaller than any real number but less than zero In the 19th century a rigorous basis was established for Newton's conceptual framework but it became an article of faith that infinitesimals could not be rigorously used as a basis for calculus However in the 20th century a rigorous basis was established for an infinitesimal based treatment of the calculus as a result of Abraham Robinson's nonstandard analysis This involves expanding the real number system to a much larger number system the hyperreal number systemIn the physical sciences it is common to use an intuitive treatment of calculus that includes infinitesimals; however nearly all books on basic calculus avoid them and ignore Robinson's ideas I only know of two exceptions a book by H J Keisler who edited Robinson's papers and this one Each has its advantages and disadvantagesKeisler's book is unfortunately out of print and nearly unobtainable It is a complete textbook of calculus using the approach through nonstandard analysis Its treatment of the hyperreal number system however I find hard to understand By contrast this book has a very much clearer treatment of the hyperreals; I think I finally understand how they are constructed after reading this book But this book is not a complete textbook of calculus It covers the theory and covers it extremely well but does not even attempt to teach how to use calculus Therefore it would not be appropriate as a sole textbook in a calculus class for exampleI have read other work by Henle and it is clear that his forte is explaining unusual number systems He does a great job in this book at what he does I just wish he had added material on how to actually use calculus Unfortunately the reader will have to augment this book by another and since no other in print book that I know of uses this nonstandard analysis based approach there will be a disconnect if anyone tries to combine it with another book