quasi
Posts:
9,078
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 23:07:58 -0500, quasi <quasi@null.set> wrote:
>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
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.
quasi
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