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Topic: How does infinitesimal exist?
Replies: 21   Last Post: Jun 7, 2013 12:13 AM

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Peter Scales

Posts: 120
From: Australia
Registered: 4/3/05
Re: How does infinitesimal exist?
Posted: Mar 5, 2013 10:25 PM
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Taber,
Infinity is a tough concept!
> That's what I had always thought, until I saw a few
> proofs that seemed to say otherwise! Here's one:
>
> x = 0.999...
> 10x = 9.999... (Multiply by ten)
> 9x = 9 (Subtracts 0.999...)
> x = 1 = 0.999... (Divides by 9!)

The proof is correct, but you need ALL the nines, an infinity of them, which is hard to write on the page.
> So this is where I see a weird problem! Today,
> though, I was thinking and though that when you
> multiply by ten, aren't there infinity - 1 nines
> after the decimal? That would make 9x = 8.9999...1
> and x = (...Works it out on calculator) 0.999.... Oh
> so... wait what XD ! I guess I kind of just figured
> that out on my own!
>
> But am I right in saying there are infinity - 1 nines
> after the decimal?

Infinity - 1 = Infinity, so you still need all the nines.
> And now I'm confused about another thing. How can
> 0.999..., a rational number, be represented as a
> fraction?! 9/9 is 1 but anything over 9 is supposed
> to be that number repeating, (after the decimal
> place).

Decimal notation has limitations, especially when you deal with repeating sequences.
1/9 = 0.111...
3/9 = 0.333...
9/9 = 0.999...
These are all consistent representations of fractions as repeating decimals. But you need to keep reminding yourself that the sequences go on for ever.
If you try to omit the last digit you are changing from an infinite sequence to a finite sequence, no matter how many digits you are using. And then the rules are different. You need to keep your wits about you when dealing with infinite sequences. As soon as you start treating them as though they were finite sequences you are never far away from trouble.
> ...Help?
I hope this helps. It is good to think about these apparent inconsistencies, to clarify our ideas.

Regards, Peter Scales.



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