Real numbers as elements of the unique...

"One of the early objectives of almost any university mathematics course is to teach people to stop thinking of the real numbers as infinite decimals and to regard them instead as elements of the unique complete ordered field, which can be shown to exist by means of Dedekind cuts, or Cauchy sequences of rationals."

http://www.dpmms.cam.ac.uk/~wtg10/decimals.html

Could someone explain this to me using laymans terms? I'm doing basic calculus and probability atm, probably the stuff you do in the last years of school in the U.S. Maybe there are other degree level math that I could learn in my spare time that'd be abit of a leg up? If you know what I mean.

>>1
Go look for that "A proposition..." thread from a few weeks ago, if it's still on here.  That was about more or less the same thing.

The short, short version:  You call a space "complete" if ever single sequence you can ever construct in that space can only ever converge to something in that space.  You call a space "closed" under an operation if applying that operation to any two elements of that space give you a result also in that space.  The integers (...-4, -3, -2, -1, 0, 1, 2, 3, 4...) aren't closed under division, since for instance 1/4 isn't an integer.  Add in all the possible results of division into the natural numbers and you get the rational numbers.  These are complete under addition, subtraction, multiplication, and division, which is good.  The rationals aren't complete, though, since you can construct sequences (3, 3.1, 3.14, 3.141, 3.1415...) that don't converge to any rational number.  If you add every possible irrational number -- or every possible value that a sequence of rational numbers could converge to that's not itself a rational number -- to your set of rational numbers, you get the reals. 

Then there's a lot of math to show that the real numbers are the only possible set that has all the nice properties we want from the real numbers, and really nobody who's not going for a PhD really gives a shit about that.