On Thu, 31 May 2007 19:13:25 -0700, firstname.lastname@example.org 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
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
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.