On Thu, 31 May 2007 23:07:58 -0500, quasi <firstname.lastname@example.org> wrote:
>On Thu, 31 May 2007 19:13:25 -0700, email@example.com wrote: > >>You cannot assert that students across the world are not taught >>that 0.999... is equal to one well before they are ever taught or told >>about limits. > >Limits are taught informally, very early. > >The formula for the area of a circle > > A=Pi*r^2 > >depends on the limit concept. This formula has been known for >thousands of years. But even then it was understood that the number Pi >exists as an exact number only in the limit. > >The concept of the sum of an infinite geometric progression, such as, >for example > > 1 + 1/2 + 1/4 + 1/8 + ... > >is taught early, possibly as early as elementary school, but certainly >it's taught to high school students. The concept of an infinite sum >_is_ explained to students by discussing limit concepts, at least >informally. > >By the standard limit-based definition of an infinite sum, > > 1 + 1/2 + 1/4 + 1/8 + ... equals 1, exactly
Correction: "equals 2, exactly"
>Once again, this is old knowledge -- thousands of years old. > >If you won't allows limits to be regarded as real numbers, you lose a >lot of math and science. Your conservatism is too costly in terms of >all the wonderful (and useful) mathematical and scientific theories >that you won't be able to accept. > >In a prior reply, you made the analogy that someone might reject >negative numbers for lack of understanding. You're doing just that >with respect to the real numbers. You don't believe .999... = 1 so >you are effectively rejecting the standard definition of the reals. > >I suggest you table your bias so as to least learn the standard >foundations. Here are some topics you should master before exploring >nonstandard versions ... > > (1) Sets, logic, proofs at an elementary level > > (2) Limits, sequences and series at an elementary level. > > (3) Learn the standard axioms and definitions for > > the natural numbers > the integers > the reals > the complex numbers > > (4) Mathematical Logic, Model Theory, Set Theory > >If you can master the above, then perhaps you'll know what you're >talking about when you discuss nonstandard ideas. > >But even before you start, let me caution you about a serious error of >logic you've already made. > >You are not allowed to change an existing standard definition to suit >your own prejudices. Of course, if you can show a definition is >somehow _inconsistent_, that's different -- then you can reject the >definition, but still, you're not entitled to replace it with your >own. You _can_ make up _new_ terminology for your version of a >concept. > >Thus, you don't have the option of saying that .999... is not a real >number. It is a real number, by definition, and it equals 1, also by >definition. On the other hand, if you want to study a system for which >.999... is strictly less than 1, fine, you can do that, but don't call >your numbers "real numbers". Call them something else.