"...he defined the Dedekind cut (German: Schnitt), a new idea to represent the real numbers as a divisions of the rational numbers. An irrational number is a cut separating all rational numbers into two classes, an upper and lower class (set) For example, the square root of 2 is a cut putting the negative numbers and the numbers with square smaller than 2 into the lower, and the positive numbers with square greater than 2 into the higher class. This is now one of the standard definitions for the real numbers."

http://en.wikipedia.org/wiki/Dedekind A further link, a little more formal language --

http://en.wikipedia.org/wiki/Dedekind_cut Whoa -- what's this?? I never heard of surreal numbers before...

"The surreal numbers are a class of numbers which includes all of the real numbers, and additional "infinite" numbers which are larger or smaller than any real number. They also include "infinitesimal" numbers that are closer to zero than any real number, and each real number is surrounded by surreals that are closer to it than any real number. In this, the surreals are similar to the hyperreal numbers, but their construction is very different and the class of surreals is larger and contains the hyperreals as a subset. Mathematicians have praised the surreal numbers for being simpler, more general, and more cleanly constructed than the more common real number system.

"Surreal numbers were first proposed by John Conway and later detailed by Donald Knuth in his 1974 book Surreal Numbers: How Two Ex-Students Turned on to Pure Mathematics and Found Total Happiness. This book is actually a mathematical novelette, and is notable as one of the rare cases where a new mathematical idea has been first presented in a work of fiction."

http://en.wikipedia.org/wiki/Surreal_number You know, I can't really do much math beyond what they taught me in college, (though I try to teach myself abstract algebra once in a while to keep the brain cells happy,) but I can follow links on wikipedia and read *about* math for hours...