quasi
Posts:
9,080
Registered:
7/15/05
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Re: Proof 0.999... is not equal to one.
Posted:
May 31, 2007 11:21 PM
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On Thu, 31 May 2007 19:13:25 -0700, chajadan@mail.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
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.
quasi
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