123.3 + 0.1 is 123.3999999999 ?

[Andrew Dalke]

Erik Max Francis:
> There are all sorts of alternative definitions of the reals which have
> differing properties than the reals we've all come to know and love.
> They usually fall under the general category of "non-standard analysis."

Sadly, beyond a bachelor's in math I have only a lay knowledge of
math, which luckily includes Ian Stewart's excellent "The Problems of
Mathematics.", p74 in my copy

  The usual list of axioms for the real numbers is second order; and it has
  long been known that it has a unique model, the usual real numbers R.
  This is satisfyingly tidy.  However, it turns out that if the axioms are
  weakened, to comprise only the first-order properties of R, then other
  models exist, including some that violdate (2) above.  Let R* be such
  a model.  The upshot is a theory of non-standard analysis, initiated by
  Abraham Robinson in about 1961.  In non-standard analysis there are
  actual infinities, actual infinitesimals.  They are constants, not
Cauchy-style
  variables....

The (2) is
  if x < 1/n for all integers n then x = 0

This is in agreement with the statement you and Moshe mentioned.
Reals are reals, and only reals.  There are models related to reals
but which are not reals, and so while it may be that in some models
0.99999..... != 1, that isn't the case for R.

                    Andrew
                    dalke at dalkescientific.com