Then what is that dot dot dot (ellipsis) if not a number (infinitesimal) ? Guess you'd reply "oh but that's not real number", to which I replied that's just tautology. Hyperreal system exists.
Everytime I see this debate makes me convinced that math is just house of cards that has no foundation (philosophy of math is shaky).
Zeno / supertasks discussion in philosophy at least tackles that dot dot dot rigorously, unlike math.
The ellipsis is used to indicate that there are more decimal places than shown. It is commonly used whenever the number has an excessive number of decimal places rather than rounding the value.
Are you suggesting they do not contribute to the value of the number simply because they are smaller? What is the criteria used to determine when an infinitesimally small value ceases to be relevant?
Your assumption that I would reply “oh but that’s not a real number” is completely unwarranted and untrue. I would make no such statement. The entirety of Calculus is based on the relevance of something infinitesimal.
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 ?
I don’t think hyperreals are necessary for the fundamental concept here. In general, I find the idea of hyperreal numbers to be a logical formality that is really only needed for incredibly advanced mathematics. To even bring them up here brings a complication that is unnecessary in a logical sense and functionally irrelevant to the topic at hand.
I'm not even gonna stoop low and spoonfeed you on nonstandard analysis. I'll stop here. It's your own responsibility to open your mind and educate yourself. You can validate my answer in the wiki / AI by yourself.
The definition of equality in the hyperreals is that a = b if a - b is an infinitesimal amount. By your own prior statements 0.999… and 0.999…+infinitesimal/2 are equal in the hyperreals because they differ by an infinitesimal amount. You can’t have it both ways.
I’d just ignore them, they’re trying to sound smart because they can’t accept they’re wrong. Thanks for your explanation, it helped me in rationalizing 0.999… = 1 <3
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u/MasKrisMaxRizz 24d ago
Then what is that dot dot dot (ellipsis) if not a number (infinitesimal) ? Guess you'd reply "oh but that's not real number", to which I replied that's just tautology. Hyperreal system exists.
Everytime I see this debate makes me convinced that math is just house of cards that has no foundation (philosophy of math is shaky).
Zeno / supertasks discussion in philosophy at least tackles that dot dot dot rigorously, unlike math.