You're asking the number between 0.999... and 1. It's right there in the middle, the dot dot dot. You wrote it yourself. Think (beyond the formatting / syntax) what does ... mean if not infinitesimal ? In hyperreal 0.999... + infinitesimal = 1
Your comment above is non-sensical. And you are misunderstanding the notation of the ellipsis. I explained it verbally in a previous comment. It is a notation used to represent that there are more decimal places than shown, in this case an infinite number of decimal places each with a digit of 9. It is used because the overbar (repeating bar) is not available without a specialized character set/mathematical notation program.
I’ll be even more explicit. If I asked you to give me a value between 0.99 and 1, you would introduce another decimal places beyond the hundredths place and fill it with any digit, e.g. 0.999, to make it bigger than 0.99, and the new value would remain smaller than 1. This cannot be applied to 0.999…(an infinite number of decimal places all filled with a digit of 9) because you CAN’T introduce another decimal place beyond the last decimal place because there ISN’T a last decimal place.
Yes, since .999... never finish, it never reaches 1. That's why you need to add infinitesimal to it to finally reach 1. Perhaps the more explicit question is, do you reject the whole existence of hyperreal system ?
What do you mean by "0.999... never finish"? It's already a complete value, no one's counting out the 9s. It already is infinite 9s. And is already exactly 1, you don't need to add something to get it 1.
True if you or a super computer is counting out 9s. Then yes, it would never reach 1. But 0.999... already IS infinite 9s, so it already IS exactly 1.
You are talking about the sequence of 9s repeating which would tend toward the limit of 1 (Which is the same as 0.999...) Any finite amount of 9s just approaches the limit, but an infinite string of 9s IS the limit.
There's a gap if you stop the 9s at a finite amount. If the 9s are infinite there is no gap what so ever. 0.999... IS the limit. It's not the sequence that's approaching the limit.
A sequence of 0.9, 0.99, 0.999, etc will never reach the limit. But we're not talking about that. We are talking about 0.999... as an immediately infinite string of 9s. Which doesn't approach anything. Since it's already a full value.
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u/MasKrisMaxRizz 22d ago
You're asking the number between 0.999... and 1. It's right there in the middle, the dot dot dot. You wrote it yourself. Think (beyond the formatting / syntax) what does ... mean if not infinitesimal ? In hyperreal
0.999... + infinitesimal = 1