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.

Get an introductory real analysis book.

I took several university mathematics courses, and none of them mentioned that shit.  They must save that for 2nd or 3rd year courses.  Therefore the assertion is bullshit.

Real analysis. Do you speak it? Decent one is "Understanding Analyis" by Stephen Abbott.

>>2
>>4

Should I wait untill I've got all the calculus stuff well under my belt, or shall I just jump in?

>>3

In the UK almost every degree is 3 years. Some are 4 or more.

>>3
They usually get to it in the second year.

>>5
Jump in. You've got nothing to lose and the calculus stuff is more fun when you have a better theoretical background.

>>5
Seconding >>6 , but have a calculus book handy to look stuff up in.  There's a little bit in even the most basic analysis books that you really can't get away from.  Still, nothing you can't teach yourself with a little effort.

>>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.

>>8
You mean Cauchy sequences, and all fields are not closed under division (1/0 is not an element from the field)

>>3

The original post has the flavor of suggesting what a goal of math education ought to be, and those of us in the last 20-30 years have presumably fallen far by the wayside, since everythign is continually, unendingly, going to hell in a handbasket.  I DID get from my own experience, however, that real numbers go more deeply than decimal expressions.  It all goes back to the notion of a line (Dedekind cuts).

tl;dr your course of education, like mine, sucked, because much of modern education sucks at emphasizing some finer points.  Or maybe it's just the new age and there's only so much information one can take in.....

First year university was calculus all year and linear algebra all year.  It wasn't about number systems, it was about solving hours of crap every week all course long for a lousy 10% max of final mark, and take two tests for the other 90%.

at any decent uk uni you start real analysis in the first year

>>3 at least, you should understand deeply Calculus 1 in order to full enjoy real analysis

I guess this real number fetish is a British thing

Different definitions, same real line.