Re: Idle Thoughts

Bob Dawson wrote:
> "Jarle Stabell" wrote
>
>> What is meant by the expression: 1/infinity ?
>
> An infinitesimal. Let's call it the hypothetical last element of the
> infinite set [1/1,1/2,1/3,...].

Since that set/sequence *doesn't* contain a smallest/last element, I'm not
able to understand that definition. :-)

>
> Or in more practical terms, the value of x in the expression
>
> 1 - 0.999... = x

In standard mathematics, x = 0.

Do you have experience using non-standard analysis?

I've been told that some school (I think it was in Denmark) experimented
with letting students learn non-standard analysis instead of standard
(ordinary) analysis. In non-standard analysis, infinitely large and small
numbers "exists". If you use non-standard numbers in intermediate
calculations, but end up with a result not mentioning these "non-standard
numbers", there's a theorem of mathematical logic stating that the result
will hold true also in standard analysis. (Which is why it can be useful to
"temporarily" use these non-standard numbers).
I don't know how the experiments went, but I suppose that even though
non-standard techniques can simplify some tasks, it probably brought the
students problems when encountering "ordinary" mathematics.

Cheers,
Jarle


.

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