## The hyperreal line

The hyperreal line is an extension of the real line. It contains all real
numbers, and some infinite and infinitesimal numbers.

We say that an element of the hyperreal line x is finite iff |x| < n for
some integer n. x is infinite iff it is not finite. x is infinitesimal
iff |x| < 1/n for all integers n. (Note that 0 is trivially
infinitesimal.)

In the normal real scale without magnification, the hyperreal line
(nonstandard analysis' version of the real line), looks like the real
line, thus:

/ -1 0 1 2 \
/ ........... __________|___|___|___|_________ ..........\
\ negative finite positive /
\ infinite infinite/

What makes the hyperreal line different from the real line is that it
contains an infinite number H, and an infinitesimal 1/H = d.

If we look at 1 through an "infinitesimal microscope", and magnify by
the infinite amount H, then we can distinguish points which are an
infinitestimal distance d apart, thus:

1-2d 1-d 1 1+d 1+2d 1+3d
____________|______|______|______|______|______|______________

If we look at H through an "infinite telescope", then we can see infinite
numbers on the ordinary real scale, thus:

H-2 H-1 H H+1 H+2 H+3
_____________|______|______|______|______|______|______________

The ideas of the infinitesimal microscope and infinite telescope are due
to Keisler. A more detailed version of this page can be found in his
paper, "The hyperreal line", in Real numbers, generalizations
of the reals, and theories of continua (ed. Ehrlich P) Kluwer Academic
Publishers, Netherlands
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