Hyperreal numbers, denoted by 
, are a similar
extension of real numbers [16, 20, 63].
Hyperreals extend the reals
by adding both infinite numbers, such as 
, as
well as infinitesimal numbers, such as 
.
Since satisfies:
is not a real number. Infinitesimals and infinities
are very useful in presenting non-standard analysis which,
for many, is more intuitive than standard real analysis.
Some argument can be made for using as an idealized model
of 
since contains two distinct numbers
+0 and -0 which would correspond to and 
.
IEEE 754 does not, however, contain a third number 0 distinct
from +0 and -0.
The hyperreals are an extension of the reals; they are constructed so that all statements which are provable over the reals are provable over the hyperreals, using a classical proof system. There is another, substantially different, approach to non-standard analysis [54]. With this ``smooth non-standard analysis'', all functions are infinitely differentiable.
| Jeff Tupper | March 1996 |