First of all, in the usual model of the
real line
there are no
infinitesimal numbers (i.e. positive numbers that are
smaller than
1/n, for any positive integer
n). They don't exist.
Before the calculus was properly
formulated infinitesimals were very popular for a very good reason.
They make all the proofs a lot easier! For example Newton
and Leibeniz
used these "ideas". Eventually, after the work of Weierstrass the
familiar epsilon-delta arguments of mathematical analysis
that we use today were introduced,
and infinitesimals were relegated to the dustbin
of history.
Although, in high school mathematics infinitesimals are still often
employed in plausibility arguments (I will not call them proofs).
All that changed in the 1960s when Abraham Robinson invented the idea
of nonstandard analysis and hyperreal numbers. The basic idea
is that instead of working with the usual set of real numbers R
one adjoins some extra elements to R to form *R
the hyperreal line. This contains all the usual real numbers but it
also contains infinitesimals
(and their reciprocals which are infinitely large).
After doing this it is possible to set the clock back and give the same
kind of arguments that Newton gave but now in a completely
rigorous way. Those high school arguments also become possible to
justify.
This is quite a fun idea and I'm a bit surprised that it hasn't caught on
as a way to teach analysis. Probably the reason why it hasn't is that
although things are nice once you have the hyperreal line, its construction
is a bit abstract and nonintuitive and that perhaps scares people off.
See also smallest number greater than 0.