Re: Definition of real number ℝ --infinitesimal--

On 2024-04-04 13:57, olcott wrote:

On 4/4/2024 1:08 PM, André G. Isaak wrote:

On 2024-04-04 08:55, olcott wrote:
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Different enough to not me equal.
[0.0, 1.0] - [0.0, 1.0) = 0.0...1
0.000...2 - 0.000...1 = 0.000...1
*A good notational convention for infinitesimals*

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And that's supposed to mean what exactly? That you take an unending sequence of zeros and once that unending sequence ends you tack on a 1?

Your complete lack of response is noted. What does ... mean in the above? I know what it means in standard decimal notation, but can't make heads or tails of it in your notation.

0.999... + 0.000...1 = 1.0
0.999... + 0.000...2 = 1.000...1
0.999... + 0.000...3 = 1.000...2
0.999... + 0.000...n = 1.000...n-1

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So what's 0.999... + 0.000...05 ? Is that half an infinitesimal shy of 1?
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 That is 5 infinitesimals.

So then what's 0.999 + 000...5 ?

Do you have a better way to encode them?

It's your proposal, not mine. I'm working with the reals which do not admit infinitesimals, so I have no reason to encode them.
You're the one who wants infinitesimals. But if you want to propose a system which uses them, you need to actually define it. Just inventing a dubious and unexplained notation is not enough.
How are numbers constructed in this system?
How do infinitesimals work with respect to addition, subtraction, multiplication, and division? What, for example is 0.000...1 × 0.000...1 ?
If you can't explain the above, you don't have a system.
André
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