your theory can't be used to do actual math, only speculative discussion on infinity. Therefore it's useless in this scenario.

I would, but it's been too long since I took freshman calculus.

The real controversy is people who use more than three dots for recurring decimals...

Of course I'm not entirely wrong, James. I'm entirely right--as long as I'm working with Q's axioms instead of R's. Similarly, tvrocks can choose the axioms of the hyperreals and be equally correct.

What you *can't* do is state the claim independent of the axiomatic system it belongs to and pretend to be saying anything meaningful.

Similarly, I don't know if you're working with some definition of infinities I'm not familiar with, but I do know most people would interpret you as talking about Cantor's infinities, and in that logical framework your claims are straightforwardly false, as infinities differing only by a finite number are equal in that system. It's that or you're simply refusing to state your axioms, and therefore not saying anything at all.

I don't understand most of what's being said, but I'm pretty sure I can recognize a smackdown when I see one.

@Yoyoyozo

"your theory can't be used to do actual math, only speculative discussion on infinity. Therefore it's useless in this scenario."

umm... what? the scenario that i was arguing against was when Vashta used Algebra, and i said that in his equation the number of infinite 9s in his .9 repeating and the number of infinite 9s in his 9.9 repeating his different by some degree, and that degree, would be: 1-(that degree)=.9 repeating (the original .9r)

so when you say it's only usable in "speculative discussion"

yeah, that's what i'm doing right now

@lethologica

clearly i missed something

ok... so first of all define your terms "Q's axioms" and "R's" I don't know what those are. They do not apply to the argument i'm trying to make

"What you *can't* do is state the claim independent of the axiomatic system it belongs to and pretend to be saying anything meaningful."

so i have to state something that is true, and unless i do, i can't say anything that is meaningful? i can't just be abstract?

"hey do not apply to the argument i'm trying to make"

and if they DO apply to the argument i'm trying to make, then please elaborate.

Btw, yes it does. It's really basic too.

The whole point of this thread is that at a basic level, they are completely equal, and only if you are a math major do things get complicated enough that it comes into question.

Okay, here is a statement: infinity, while it may never be measurable in length, can in fact be larger or lesser than other statements regarded as infinity, though the measurement between the two is impossible to determine.


on a separate matter, what Vashta said was

"x = .99999999999999999..........
10x = 9.99999999999999999............

10x - x = 9x = 9

x=1

therefore .9999..... = 1"

but i say
there is a difference >9 between .9r and 9.9r in this example. we can not directly measure it independent of context, however, 1-thie difference=.9r (in equal length of x)

Vashta's argument is circular anyway--it depends on 0.999... = 1 to begin with.

Q = rationals
R = reals

You can't make claims about infinity when you haven't defined infinity. It doesn't matter how abstract you think you're being.

If I say
A(x)=zxcjkvhlk(x) + 1
&
B(x)=zxcjkvhlk(x)

and i plug in x=1

then I can't say anything about whether A(x) and B(x) are different because I don't know wtf zxcjkvhlk(x) is.

@Zybodia

yes! they are at a basic level equal, we can not measure them to be unequal, it is unfathomable, the difference between them.

however, I still believe they are not equal, regardless of our ability to measure their difference.

i'll be back in about 20 minutes

Don't rush, take an infinite number of minutes

I like Ghug's contribution, the countable an uncountable infinities and numbers. So are there countable countable and uncountable countable as well as countable uncountables and uncountable uncountables ?

I like this example of different infinities.

Say there's a list of all the natural numbers starting with 1, in order. An infinite number. Then you have a list of non-terminating decimals between 0 and 1, of which there are also an infinite number. So you pair them up in no particular order. Like so:

1 0.6381727...
2 0.7482759...
3 0.9265928...
4 0.5291858...

And so on. Now, take the first decimal place of the No 1 decimal (in this case 6) and add 1 to it. Do the same for the second place of No 2 (4), the third place of No 3 and so on all the way down the list.

So this number would be 0.7572...

This number was not in the original list. Why? The first decimal place is not the same as the first number, so it can't be the first number. The second decimal place is not the same as the second number, so it can't be the second number, and so on. Hence, there are more non-terminating decimals between 0 and 1 than there are natural numbers between 1 and infinity.

Every two numbers on the real number scale has infinitely more numbers between those two. Name 1 number that is between .9 recurring and 1.

No matter what they cannot be the same number as there will be a 0.0001 (with an infinite number of more 0s) difference anywhere in the infinity of 0.999r

But if there's an infinite number of 0s before that 1, can there ever actually be a 1 at the end of the chain? (The answer is no, by definition of infinity)

Exactly, there is no such number, it's just zero. You don't get to just make 0.(0)1 - doesn't work that way.

Furthermore, lol to those who say that 10 x 0.(9) = 9.(9) has less 9s in the decimal than 0.(9). If you think that you absolutely failed to grasp the concept of infinity. There are different infinities, but those two are exactly the same.

A decimal is really just notation for an infinite sum. 0.9 recurring is the sum of 9/(10^i) for all i from 1 to infinity.

It's fairly easy to look up the rules for figuring out what the limit of an infinite sum is.

Exactly, in which case the 0.(0)1 would be lim n \to \infty 1/n, which is zero.

@yassem:
"Exactly, there is no such number, it's just zero. You don't get to just make 0.(0)1 - doesn't work that way."

Well, it *can* work that way, but you have to be explicit about it, and then you're no longer working with the set of real numbers (see: hyperreals).

Just like how you *can* define a point at infinity where parallel lines meet, as long as you're clear that you're not working in the Euclidean plane anymore (see: projective plane). Or how you *can* define the square root of negative one, as long as you're clear that you're moving outside the set of real numbers (see: complex plane).

Generally, the answer to "There is no such number" is both "That's true in the system you're using" and "Hey, what if there *was* such a number?"

https://betterexplained.com/articles/a-friendly-chat-about-whether-0-999-1/

We can literally do this for 1 + 1 = 2, too, if you like (see: tropical math).

Shock trooper (or anyone). What's 1-(1/∞)?

by all measurable standards, 1

The thing about hyperreals is that they're, by definition, immeasurable.

The thing about it is that it wouldn't actually be 1, despite it being infinitesimally close to 1. (Which would make sense as it was divided by an infinitesimal.) If you do 1- an x number of 1/x expressions then, no matter the value of x, it would end up being 0. for example: 1-1/x where x=3 would be 1-1/3-1/3-1/3 which would be 0. This could also be written as 0=1-x(1/x) which should be true for all values/ quantities. If you were to do x=∞ Then it should still be true in the end as an infinitesimal is just the inverse of some infinite value. The word infinitesimal comes from a 17th-century Modern Latin coinage infinitesimus, which originally referred to the "infinite-th" item in a sequence. https://en.m.wikipedia.org/wiki/Infinitesimal