@tvrocks

yes, but if we were to use that number we get, and perform any mathematical function, it's measurable effect, would equal 1.

but yes, it is infinitesimally close.

This is all in the realm of theoretical mathematics so it doesn't really matter whther or. It it would be measurable. There should still theoretically be a difference between them, which is what was being discussed in the thread.

i say that there's a measurable difference between any two infinities, and we only lack the means/ability to fathom the difference.

apparently i do not understand how infinity works, rather than me having an opinion that is not popular/currently workable

@tvrocks

i'm only just now getting back into this thread, and i'm probably going to leave for good soon. Everything you've said so far i can agree with

Isn't .9 repeating the same as an asymptote?

@james:

I can agree with what you're saying. The ∞ symbol, as opposed to real numbers, doesn't stand for a specific value. It's unlike x in that it can't stand for any value, however, it stands for any generic quantity that's too large to be expressed in the standard number system. Both the number of positive integers and the number of reals between 0 and 1 would both be expressed as being infinity, however, they don't have the same value. Just as both 1 and 2 would be considered positive integers, they don't have the same value.

There are some infinities, such as x and y where x= some generic ∞ and y= that generic infinity, or x, +10 where it's possible to grasp the difference between them, however, with others, such as the 0 to 1 and all positive integers example, humans are unable to comprehend the actual difference between the two. that's why there are countable infinities, which are possible to comprehend, and uncountable infinities which are impossible to comprehend.

@fourofswords

1 would be an asymptote for an expression that resulted in .9 repeating.

In a system where 1/∞ is defined, sure, we can talk about 1-1/∞ being different from 1.

Taking for granted that people are using that system is another story.

I'm also uncertain as to whether 0.999... is complete notation in the hyperreal field. Does it refer to an equivalence class of all numbers differing from 1 by an infinitesimal?

*less than 1 by an infinitesimal

thank you tvrocks

I like this thread, it makes much more enjoyable reading than the political chit chat festival

@letholigica

If you're saying it's an equivalence class of all numbers that are less than one by an infinitesmal then yes.

I mainly argue about an infinitesmal as it seems like it would be defined in the system of mathematics as, if it wasn't, then mathematics would be broken when hyperreals were introduced. In the example of 0=1-x(1/x) if you plug in an infinite value for x, it should still work.

@tvrocks

are you saying that geometric series is wrong? The entire purpose of the theorem is to deal with sequences that converge, like .999...

facts:

.9999 is equal to 9/10+ 9/100+ 9/1000

{9/10,9/100,9/1000...} converges with a factor of 1/10

There is a proven method for determining the sum of this series (for the proof, check here, http://mathworld.wolfram.com/GeometricSeries.html)

why is {9/10+9/100+9/1000} and different than {1/2, 1/4, 1/8} or {3/10, 3/100, 3/1000}?

@tvrocks:
"I mainly argue about an infinitesmal as it seems like it would be defined in the system of mathematics as, if it wasn't, then mathematics would be broken when hyperreals were introduced. In the example of 0=1-x(1/x) if you plug in an infinite value for x, it should still work."

This reasoning doesn't make any sense to me. There is not a singular "system of mathematics" in which all constructs are defined, that gets "broken" if an undefined term is introduced. That's not how math works.

https://67.media.tumblr.com/75647e322f38cdeec701bfe09d4fe643/tumblr_o4ix14nE031s2wio8o1_500.gif

That we can construct a system where 0.999... =/= 1 means only that. It doesn't mean the system where 0.999... = 1 is broken or illegitimate.

That's why you have to specify the system you're working in before you can reach a conclusion about whether 0.999... = 1 is true or false.

-

@Yoyo: Your proof is correct for the real numbers. Indeed, one definition of the real numbers is as equivalence classes of convergent sequences of rational numbers. However, tvrocks is using a different number system where that proof doesn't work, and where 0.999... and 1 differ by an infinitesimal. His number system is legitimate, and his conclusion is legitimate within that system.

The bullshit part, and the part that makes this such effective troll bait, is where he uses that conclusion to argue against your conclusion. They're entirely separate conclusions in entirely separate systems with distinct assumptions. They have nothing to do with each other.

The system tvrocks is using causes simple multiplication and division to break down.
For instance, does this mean that 1/3 is not equal to .333...? Because .333... x3 would not equal 1 since there would be an infinitesimally small difference between the two. What is the decimal equivalent to 1/3 then?

The system tvrocks is using allows for multiplication and division. What you're talking about is not operations, it's decimal representation. I personally don't know how the hyperreal 1/3 is represented in decimal notation, but that doesn't affect the validity of the hyperreals. And 1/3 * 3 = 1 in the hyperreals just like it does in the reals.

It is operations. What is 1 divided by 3?

1 divided by 3 is 1/3.
1/3 is 0.1 in ternary notation (base 3).
1/3 is 0.3 in nonary notation (base 9).
We still know that 1/3 is a number and what number it is.

The only problem is how to represent 1/3 in decimal form. I brought this issue up earlier when I asked about what 0.999... represented in the hyperreal field. But it's not an issue of operations at all.

FWIW, with continued reading, it may be reasonable to conclude that 0.999... = 1 even in the hyperreals, and that additional notation is required to represent numbers that differ from 1 by an infinitesimal. See for example the discussion at the link:

https://www.quora.com/Is-0-999-dots-1-in-the-hyperreals/answer/Alon-Amit

so what you're saying is that fractions with repeating decimals don't exist. That's oddly specific and probably a failure of tvrock's system, not mine.

No, he's saying that decimal representations of those fractions don't exist (maybe). That's very different.

That's even worse lol. So everything else works out, but decimal representation for these numbers gets pushed out for a less useful, more confusing set of numbers.

I wasn't really saying either of those things. Mostly what I was saying is that *I don't know* the notation, not that the notation doesn't exist. Which is why I brought up the link, and why I asked about what 0.999... represented.

To be clear, after reading the material at the link, I think 0.999... is not an appropriate notation for a hyperreal number differing from 1 by an infinitesimal. Notation like 0.999...;...999 should be used instead. I suppose that won't stop tvrocks from defining his decimal representation however he likes (as noted at the link), but that's where I stand now.

Excellent contributions Lethologica. I wasn't being satirical when I commented that I enjoyed reading the contributions in this thread.