1 is 0.9999999999999............

C

chinglu

Valued Senior Member
This has been discussed over and over.

However, there are some proofs in set theory in which this does not hold.

For example, use transfinite induction on only natural numbers.

So, we seek the least index n such that .9(n) = 1. We find it cannot be decided since for all indexes n, .9(n) != 1. This is a problem.

We also also can take an approach of considering a neighborhood around 1. We also see for all n, we can find a number between .9(n) and 1.

We can't find any natural number in which this relation is false. So, if we could actually exhaust all n, we are left in the state there is a number between .9(n) and 1. This actually means we cannot exhaust all n.

I hope we can see discussion on this issue.
 
chinglu said:
This has been discussed over and over.

However, there are some proofs in set theory in which this does not hold.

For example, use transfinite induction on only natural numbers.

So, we seek the least index n such that .9(n) = 1. We find it cannot be decided since for all indexes n, .9(n) != 1. This is a problem.

We also also can take an approach of considering a neighborhood around 1. We also see for all n, we can find a number between .9(n) and 1.

We can't find any natural number in which this relation is false. So, if we could actually exhaust all n, we are left in the state there is a number between .9(n) and 1. This actually means we cannot exhaust all n.

I hope we can see discussion on this issue.
Click to expand...


Just about to log out, so briefly, mate...

From past discussions I've been privy to, it appears that 0.99999999999..... is merely a TRIVIAL alternative NOTATION FORM for the 'unitary' output/symbol "1" (or "1.0" etc).

In short, 0.9999999... seems NOT to be a number OR a function as such, but a convenient expression indicating certain 'properties' (in certain contexts?) which may not be as evident if just the "1" or "1.0" was used.

Anyhow, that's what past discussions seemed to imply. But I will read-only this further discussion with interest just in case further insights result from it! Thanks for your interesting discussions elsewhere, chinglu, everyone! Enjoy them and good luck to you all. Bye for now. :)
 
Tach said:
Elementary arithmetic says that it is a number.
Click to expand...

It stands for "1.0" or "unitary" term. The form/notation is purely a matter of preference depending on the context which the notation/form arises in. That has been agreed upon long ago. It is also NOT a function; since you did not question that then I must assume you agree with that? :)

Bye.
 
This is a subtle topic. It is best to ignore decimal notation & deal with an infinite geometic sequence.

.99999 . . . . can be represented by the following geometric series.

9/10 + 9/100 + . . . . + 9/10[sup]n[/sup].​

The limit of the above series is one. Some of the more pedantic mathematical texts texts avoid saying that the above series is equivalent to one. Instead, they make statements similar to one of the following.
The limit is one, providing no further discussion.

Treating the sum of the series as equivalent to one does not result in a paradox. The actual statement (which I do not remember) is semantically equivalent to this statement.
Click to expand...
Less formal texts state that the series is equal to one.

Attempts to prove the limit using decimal notation are flawed. Using hexidecimal arithmetic, (A thru F being 10 thru 15) .FFFFFFF . . . . is always less than one & greater than .99999 . . . . which refutes the notion that the limit of .99999 . . . . is one. Using a proof based on the sum of geometric series results in a valid proof that .99999 . . . . & .FFFFFFF . . . . both have the same limit (one).
 
Undefined said:
It is also NOT a function; since you did not question that then I must assume you agree with that? :)
Click to expand...

It would be a waste of time for me to address all your fringe ideas.

It stands for "1.0" or "unitary" term.
Click to expand...

Hopefully, by now, you figured out that , contrary to the fringe misconceptions that you espouse, it is a number. Just like 0.
 
I remember being involved on a thread long ago on this topic, and came away with the understanding that mathematicians are clear, the notation .999... represents 1. That notation means that you carry out the 9's infinitely.

You cannot truncate the infinite series of decimal places because then you have a finite value that is less than 1, and you are not representing 1 unless you add the "..." at the end to desigate that the notation is invoked to represent 1.
 
Tach said:
It would be a waste of time for me to address all your fringe ideas.



Hopefully, by now, you figured out that , contrary to the fringe misconceptions that you espouse, it is a number. Just like 0.
Click to expand...

If you don't bother to fairly and properly read and understand in full context, then how can you judge anything at all, except from your ingrained prejudices and beliefs?

The conversation is ongoing. New approaches have implications for current partial axiomatic systems which need to be 'bridged' somehow and circumvent the incompleteness theorem 'barrier' which each such partial system faces in isolation from overarching context.

And I already made clear (in my posts to rpenner et al in the other thread) what "0" may be fundamentally and non-trivially. The trivial 'undefined' states for some "0" instances arising in current partial systems is the problem for those systems. My suggestions would have those problem situations forestalled from the outset axiomatically and contextually in reality, rather than let it become a source of 'undefined' situations later on in those current/conventional partial systems. Zero is not a number in some contexts. It is a number in other contexts IF the problematic expressions involving zero are kept isolated in parentheses AND the 'rules' are not trivialized to make "0" become "undefined" just because the partial system IS partial and not complete because of those very axiomatic rules to begin with.

But you won't due fair and due diligence to try to understand the thrust, subtleties, implications and arguments/suggestions as already presented; so you will keep on kneejerking and insulting from personal/dogma/ego while missing the whole point/discussion involved

Rather than just kneejerking and preening your ego and prejudicial beliefs ad nauseam, you should act like a real scientist and actually read and understand all the context/subtleties surrounding new discussions before posting anything at all.

Ego is no substitute for the scientific method and free and fair discussion without prejudices personal or dogmatic, Tach. Good luck. :)
 
Undefined said:
New approaches have implications for current partial axiomatic systems which need to be 'bridged' somehow and circumvent the incompleteness theorem 'barrier' which each such partial system faces in isolation from overarching context.
Click to expand...

You haven't provided any "new approach". Just plain misconceptions.

Rather than just kneejerking and preening your ego and prejudicial beliefs ad nauseam, you should act like a real scientist and actually read and understand all the context/subtleties surrounding new discussions before posting anything at all.
Click to expand...

There is no subtlety, contrary to your misconceptions both 0 and 0.(9) are numbers.
 
0.999 . . . does not equal 1.
it might approach 1 or 0.999 . . . might have a limit of 1 but it doesn't equal 1.
we can get extremely close to 1 by adding more 9s after the decimal point, but we will never get there.
only 1 equals 1.
- the opinion of a non mathematician.
 
The most interesting thing about this discussion is not whether 0.999...=1 (I mean really, who cares? What difference does it make?)
It's why our minds rebel from the notion.

Intuition and ambiguous teaching lead students to think of the limit of a sequence as a kind of infinite process rather than a fixed value, since a sequence need not reach its limit. Where students accept the difference between a sequence of numbers and its limit, they might read "0.999..." as meaning the sequence rather than its limit.​
- Tall and Schwarzenberger, quoted in Wikipedia

The lower primate in us still resists, saying: .999~ doesn't really represent a number, then, but a process. To find a number we have to halt the process, at which point the .999~ = 1 thing falls apart.
Nonsense.​
- Cecil Adams, also quoted in Wikipedia
 
someguy1 said:
Yes, 1/2 + 1/2 = .999...

since the RHS is 1.
Click to expand...
True, if you invoke the meaning of .999... notation as being equal to one, but that is not in the spirit of the Captain's question, of course ;). Is there a situation where the results of the math describing it ends up in the infinite series .999...?
 
@ Leopold,

1.999... continued equals 2, yet 1.99999999 without continuation does not equal 2 from my understanding.

a) Divide a pie into 3 equal parts. 1 / 3 = 0.333..... Continued.

@ Captain Kremmen, (this will result in 0.999...)
b) Now try to get the three parts to equal a whole. 0.333... + 0.333... + 0.333... = 0.999... = 1


If 0.333... equals 1/3 of the pie then 0.999... must equal the whole pie.
 
That's it.
So in that case, the 0.999..... isn't equal to 1, it is the wrong answer.
Wrong, even though it is the number that you get from the calculation.
The result is always inaccurate, no matter how many digits you put after the zero, even an infinite number.
 
You could consider the difference in complexity between the two 'equivalent' representations.

Is 1.0 more or less complex than 0.999[sup]r[/sup]? Is an algorithm that ouptputs "1.0" less complex than an algorithm that outputs "0.999..."? Intuitively, the second would never halt, the algorithm has to output an infinite string.
 
arfa brane said:
You could consider the difference in complexity between the two 'equivalent' representations.

Is 1.0 more or less complex than 0.999[sup]r[/sup]? Is an algorithm that ouptputs "1.0" less complex than an algorithm that outputs "0.999..."? Intuitively, the second would never halt, the algorithm has to output an infinite string.
Click to expand...
Yes, there is the matter of complexity, but 1 and .999... are defined as being equivalent, and thus we have a means to replace the complex .999... with the simple 1.
 
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