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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