Although not have anything about hyperreal, My real calculus book says that infinitesimal means situation that something smaller than any positive number, but larger than zero, similar as that infinity means situtation that something larger than any number.
Is this definition correct in real calculus, not hyperreal calculus?
I heard that real number system doesn't have non-zero infinitesimal and wonder whether infinitesimal can be defined as 'situations'
So isn't above explanation flawed?
P S : sorry for bad english...
In mainstream calculus / real analysis, "infinitesimals" do not exist. Therefore there is no definition for the word.
Without seeing a quote from your book, is it hard to know what it means by bringing up the concept. Unless it's presenting non-standard analysis (but if it does that, it should have told you very explicitly), a reasonable guess could be that the book is trying to present some of the intuition that motivated the pioneers of calculus -- but that intuition was later replaced by precise definitions involving limits and no infinitesimals.