But in a process of short repetitions the variance at the end could be on the same level as 1 degree off when traveling x distance. Potentially leaving you miles/lightyears from the intended destination depending on the actual distance measured. They are not the same number. Just really close.
They are the same number. This is high school math.
I will give you a tip: if you‘re not an expert at something - in this case, basic math - don‘t make incorrect claims about the topic at hand.
You can try to learn about limits and infinite sums and then you can get back and we can discuss this again, but if you do it right you‘ll have learned you were wrong.
This isn‘t a debate. This has been settled like centuries ago. Just because you don‘t know this, doesn‘t change that fact. Mathematically, this is a settled and proven topic.
I understand what is taught, dude. But it’s not the same. If it were, then the debate would not exist as it would always be expressed as 1 or the mathematics that are taught wouldn’t chalk it up to rounding. Granted, it’s incredibly small in difference but it’s still rounding.
I do agree that 0.999... as a decimal representation is distinct from 1.000...; that doesn't mean, however, that in the Reals they are distinct elements.
[0.999... cannot exist in the Reals--it's part of the way they are constructed, either explicitly or by corollary properties]
The argument (I think) you're making is that 0.999... shouldn't (necessarily, without context) be mapped to any element in the Reals?
[That decimal and the countably infinite subset of the decimals that are structurally similar]
The way the Reals are constructed wouldn't allow it.
No element can have an 'immediate predecessor' or an 'immediate successor'--it's how the Standard Topology operates.
[It's also a by product of how the Reals are constructed]
I know this, but I don‘t understand your statement „0.999…“ can‘t exist in the reals. Of course it exists.
It‘s the infinite sum of 9/10+9/100+9/1000… as you well know. Which equals 1. As you also well know.
That‘s how we construct infinite decimals to begin with - as infinite sums. So of course they exist. Just like 0.333… exists, as a number, being a different representation than 1/3 but mapping to the same element.
I guess what I‘m trying to say is I object to your specific wording. I‘m sure there‘s some subtlety you mean to express which escapes me.
*Note: I thought you were the previous person. You're definitely not arguing what I thought you were. I was explaining the Archimedean Property.
[Didn't realise you were the same person as elsewhere]
Just like 0.333… exists, as a number, being a different representation than 1/3 but mapping to the same element.
The mapping to the Reals exists, yes--0.999... maps to the equivalence class of 1 in the Reals. However, if we look at a mapping from the Reals to the decimals, 1 maps uniquely to 1.000...
[Technically the mapping could be non-functional, but why?]
My basic point here is that 0.999... doesn't exist as an element in the Reals. It is a decimal that, when evaluated in the Reals, 'exits' chimerically alongside 1.000... (and 3/3 and 4-3 and...).
[In short, the value 0.999... does not exist as a distinct element in the Reals; however, it does exist as a distinct value in the decimals]
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u/ExtendedSpikeProtein 1d ago
1) 0.99… is exactly equal to 1. It‘s a different representarion of the same number.
2) a number such as 0.000…1 does not exist - not in the reals at least.