…and the point is?
The point is that there are numbers larger than 0.999~ but smaller than 1. For example, 0.FFF~ (in hexadecimal system) is larger than 0.999~ but smaller than 1. In base-20 system, we have 0.JJJ~ which is larger than both hexadecimal 0.FFF~ and decimal 0.999~.
…and the solution would be?
Solution to what?
I thought you posed a dilemma of sorts?
This thread is only about base I0 Magnus and so talking about other bases or systems is not relevant here
Also all the relevant arguments have already been made which is why the thread stopped two years ago
Yes, but bases can be consolidated, no?
I don’t agree.
I don’t agree either. Other bases are fair game in mathematical debates. (Btw… Magnus is right)
When I was reading through this, I noticed the following that seems to bring it all to a salient conclusion.
It makes sense that “0.999…” is just an expression signifying that some ratio cant be expressed by a fixed number of digits (unlike 1). It isn’t actually a number. And neither are all of those other expressions that end with “…”. And apparently that is why the Wikipedia proofs are misleading.
To me, that seems like a “game over”.
Well, James was very firm that infinitesimals were useful, and usefully different than the “convergence”…
Hence, his InfA thing
This seems to explain why he would -
I’m no maths genius but isn’t calculus just the sum of the infinitesimals? Where would science be without calculus?
Obsrvr,
I’ve been in many threads with James …
So I’m certainly putting my own words in his mouth:
James would say that both infinitesimals and approximations are useful in their own right and for different reasons.
Calculus is awesome… it spans multi-disciplines… where would industry be without it.
I am not sure why you think it’s not a number i.e. a symbol representing some quantity. It appears to me that it clearly is. It has many properties that numbers have e.g. it’s greater than some numbers and less than others.
What it isn’t is a finite quantity, that’s for sure, and that’s why it can’t be 1.
Not really. I just expanded upon Gloominary’s post.
If it isn’t a “finite quantity” I imagine that it isn’t a number.
The whole question boils down to a confusion between the qualitative and the quantitative.
“1” is clearly a precise quantity, but as soon as you profess 0.(9) you add in the quality of “endlessness” to describe the repetition of the quantity of “9” for each decreasing power of 10 (or whatever base you’re using).
0.(9) is an attempt to restate the quantity “1” in a way that involves endlessness. As is 0.(3) to restate 1/3 when one divides 1 by 3. It’s an admission that one cannot denote 1/3 etc. entirely quantitatively without the use of the quality of endlessness. Multiplying 1/3 again by 3 is obviously 1 (3/3), yet multiplying 0.(3) by 3 is not so obviously 1 (0.(9)) precisely because of the injection of the qualitative into the otherwise entirely quantitative.
Subtracting 0.(9) from 9.(9) to get the exact quantity of 9 requires the same confusion.
As soon as you allow the notion of the qualitative into the quantitative you invite possibilities such as ε as an epsilon number and so on.
This is the same kind of mistake that every extended or “new” number set allows - much to the advancement of mathematics and other utilities… but not truths. Experientialism highlights the distinction.
So we see how useful it is to make particular types of mistakes that are not true, but are useful: such as the notion that 1 =/= 0.(9)
Is it really? No.
But that’s the wrong question.
The more useful question is whether any new knowledge can be gleaned from the possibility that 1 =/= 0.( 9)
That would be “1.000…”, not “0.999…”
No, I meant 0.(9)
The recurring 9 necessarily must recur endlessly without defined quantity (infinitely) for 1 to divide 3 times into 0.(3) and multiply back to 0.(9)
For 0.(9) the 9s must recur with the quality of endlessness for it to be subtracted from 9.(9) to equal 9 exactly. If they didn’t, you couldn’t get 9 exactly.
By contrast the quantity of 0s that you put after “1” doesn’t matter, whether it’s 1.0, 1.00 and so on - it doesn’t affect the quantity of 1 even if you try to impose the quality of endlessness with 1.(0)
The quantity of 9s that you put after “0” most definitely matters because 0.9 is different to 0.99 and so on with any non-endless (finite) quantity of 9s after the 0 different to 0.(9) with its quality of endlessness.
So 0.(9) is 1 only with the quality of endlessness, where 1.(0) is 1 with or without the quality of endlessness.
Endlessness is irrelevant to 1 when denoted as 1 so endlessness isn’t necessarily involved.
Endlessness is essential to 1 when denoted as 0.(9) so endlessness is necessarily involved.
There’s no issue dividing 9 by 3 to get 3, and then multiplying it back by 3 to get 9. The 9 is the same before and after the operations.
So why would dividing 10 by 3 to get 3.(3) and multiplying it back by 3 to get 9.(9) be any different? The 10 is the also same before and after the operations.
There’s no issue subtracting 1 from ten times that i.e. 10 to get 9.
So why would subtracting 0.(9) from ten times that i.e. 9.(9) not get 9?
0.(9) not being 1 requires a double standard for (mod 0) versus some other modulus, which removes a fundamental necessity that’s essential to mathematics: that it’s consistent.
But that’s my whole point: introducing any notion of the quality of endlessness to quantities confuses everything. That’s why infinities are such a minefield.
Hence why “1 = 0.(9)?” is the wrong question - the more useful question is what happens if you go against the truth that they are equal and say they aren’t. It’s what we did for complex numbers - there is no square root of minus one in truth, but what if there was? What usefulness can “i” provide? Turns out it provides a lot of usefulness even though it’s not true that “i” exists any more or less than it’s true that epsilon numbers and hyperreals exist.
The whole debate behind this thread isn’t looking deep enough - and as always, Experientialism puts it all into perspective.