I has to be equal to 1 in the Real number system because it can’t be anything else. Did you not understand the explanation in terms of integers?
Are we saying that the series of .9r is a complete set? Therefore = 1.0
so the infinity remaining is [.1r] a non-value, why?
I would not word it that way but essentially yeah.
The difference between .999… and 1.0 is so small that there is no real value for it. Therefore, .999… gets bumped up to 1.
Perhaps one could visualise the integers as cubes/volumes/spaces filling up the set, as opposed to thinking of them as points and there always being a gap. Then there is nothing remaining.
Is there a reason why integers are thought of as points? I mean, we have to first consider what the elementary rule of division/cardinality should be. easier to get from finite to infinite if we are using spaces, and possibly a key to understanding how qm works underneath everything.
The remaining value is .00000… James is saying something like “there’s a 1 at the end”, but I’m arguing that there’s no end, so there’s no 1, it’s just 0.
James, would you agree that 1-.999… = .000…1? Phyllo?
The idea of doing that is not really related or relevant. We are not dealing with a limit on how small a number can be before it is no longer divisible. We are dealing with whether one number is precisely equal to another regadless of how small the difference between them might be.
That is not quite right. I often use that as an example to relay the basic idea, but there is no actual 1 or 0 at the end. Note that we agree that there is no end, yet you claim that the end is a 0 not a 1. I am saying that because there is no end, the end cannot be 0, which is what it must be in order to have 0 difference between them.
In my own personal notation, I would write it like this:
Where R = “the ever Remaining infinitesimal amount”,
[1.000…:0R] - [0.999…:9R] = [0.000…:1R]
…to be read as; “1.00 with no remaining infinitesimal amount minus 0.999… with an ever .9 remaining amount equals .1 infinitesimally small remaining amount.”
And conversely:
[0.999…:9R] + [0.000…:1R] = [1.000…:0R]
“.9 ever remaining infinitesimal amount plus .1 ever remaining infinitesimal amount equals 0 ever remaining infinitesimal amount.”
Two points:
-
There are no infinitesimals in the Real number system.
-
If .999… is not equal to 1 then some pretty simple math breaks.
9 * 1/9=1 but
9 * 0.111… would not be equal to 1. Where did the missing amount go in that case and how did it disappear?
No matter how hard you try to get to the amount which separates 0.999… from 1, “you can’t get to it”. That indicates that it’s too small to be a Real number.
(This is only a way of visualizing the situation. There is no actual moving towards an end or some kind of growth.)
Even if, its not a real number, it is virtually impossible for it not to be differentiable within that absolute space,since then it would violate the form of the numerical progression, through it’s value.
It is beginning to feel a bit like the classical Meno, where the intuitive basis of higher transformed calculus is demonstrated, perhaps through other forms of value. If so can the pro genitor of this forum explain, using inductive analysis, without symbols , if it can be clarified that way,? Because we may have hit a brick wall here.
r
,
That difference would have to be a number on the real number line.
I get it - it “feels” like there is or should be a difference.
The progression form would have to be altered to account for the difference, then, to create a pseudo number-value, if the form of the progression is to be sustained. Say, after the thrillionth, value of the sequence start reducing toward 0 ,
As the case with i believe pie since the circle is closed virtually.
It is much more then a feeling, it is going against the grain, not
to think otherwise, it’s more a cry for sense, common sense.
based on games between logic, sense and math.
Just as i suspected: a prof.Canada in Tokyo found a final value to pie: after running a computer 1.3511trillion times in a supercomputer :
I had no prior knowledge of this, and tends to point to the value of an intuitive basis to common sense.
But if there’s no end, there’s an infinite string of 0s. And 0, like any integer, can be written as 0.000…, i.e. with an infinite string of 0s.
So you want to say that 0.000… is greater than 0.000…, even though they each are infinite strings of 0s?
It’s more than notation, you’re using your own personal math (it seems similar to hyperreal math). In the standard reals, two numbers whose every decimal place is the same are the same number, there are no additional hidden values.
It’s a little funny that in asserting that there can’t be two decimal expansions for the same number, you’ve landed on the idea that the same decimal expansion can represent two different numbers. I think this makes it pretty clear that, whatever the validity or usefulness of your mathematical system, it is not the standard reals.
Phyllo, I didn’t like your integer analogy at first, but I’ve come around to it. There’s agreement in this thread that it’s possible to create a system where the difference between 1 and .999… has a non-zero value, but that in real numbers, the difference is zero.
1/10
1/100
1/1000
…
gets smaller and smaller
1 divided by infinite is not close to zero, it is zero.
Because infinite is not a large number, it is infinite.
0.99999999… is not almost 1, it is 1.
You stated the “proof” backwards:
1/9 = 0.111…
9 * 1/9 = 1
9 * 0.111— = 0.999…
ergo
1 = 0.999…
Of course the problem is that 1/9 was never really equal to 0.111…, but rather “0.111…” was the only way that the decimal system could relay the message.
The “missing amount” disappeared in your imagination.
No. It “feels like” there is an end “at infinity”, when there isn’t.
Don’t count your chickens…
… until the fat lady sings:
Have you been sleeping? The hyperreals are part of the real number system and are on the real number line. And there is no bottom to the hyperreals.
Not at all.
The difference:
0.1
0.01
0.001
0.0001
.
.
.
Infinitely and forever, the last digit is ALWAYS a 1.
You haven’t been keeping up.
That series of
1/10
1/100
1/1000
.
.
.
has no end to it, thus there is ALWAYS a “1000…” remaining to be summed.
It NEVER gets to “infinity” because “infinitely” MEANS “ENDLESSLY”.
“0.999…” means “never quite equals 1.000”.
That list doesn’t matter, it’s only a mental crutch.
Infinite is not a number, not a real or natural number anyway (and with real and natural I’m talking about those number spaces).
What’s the number last before infinite? There is none.
Infinite is an abstraction and is not in the real or natural numbers category in mathematics.
Someone who gets that 1 and 2 and 3 do not exist in reality but are abstractions and only exist as concepts in the mind (they aren’t even referring to anything in reality) also gets or better, can accept the abstraction called infinite.
You are behind in the discussion. It doesn’t matter if numbers are a part of the real universe. And thinking that “AT infinity” a series is equal to something is the “crutch”.
If there is always a 9 at the end of an endless series, then “at infinity” there would still be a 9 there too. Because there really isn’t an end, there are only 9s listed forever which is not equal to 1.000… with its 0s listed forever.
What number can you divide into 1 and get 0? There isn’t one.
If that’s your argument then it is the argument that 0.9999999… is not equal to 1 because the arrangement ‘0.9999999…’ looks different than 1.
It’s the same argument as ‘one’ differs from ‘1’.
Is there a difference between 0.99999999… and 0.999… (as a number)? No? It’s referring to the same number, yes? That number happens to be 1.
That’s according to the mathematicians.
But you can attach some other meaning onto their abstractions.
0.111… is a symbol for the number - it’s exactly equal to 1/9. How do I know that it’s a symbol rather than the number itself? Because .111… does not conform to the standard system of representing decimal numbers.
Symbols can be manipulated without any loss of precision. If you split up the number 1 into nine parts represented by symbols and then recombine the symbols … you must get back the number 1. If you do not, then symbol manipulation would be useless and math would not “work”.
No they are NOT. Carleas has been right all along. Reals are a subset of hyperreals.