Instigator / Pro
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1443
rating
11
debates
22.73%
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Topic

0.999... = 1

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Finished

All stages have been completed. The voting points distribution and the result are presented below.

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0
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After not so many votes...

It's a tie!
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Category
Philosophy
Time for argument
One week
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Open voting
Voting period
One week
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Four points
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Unrated
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10,000
Contender / Con
0
1672
rating
283
debates
66.96%
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Description
~ 531 / 5,000

0.999... = 1: Prove that 0.999 (repeating to infinity) isn't equal to 1.

Proof 1:
x = 0.999...
10x = 9.999...
10x - x = 9.999... - 0.999...
9x = 9
x = 1

Proof 2:
0.333... = 1/3
0.666... = 2/3
0.999... = 3/3 = 1

Proof 3:
All numbers that aren't equal to each other will have a number(s) that comes in between.
x < x+1 --- The number that could come in between of x and x+1 could be x+1/2.
So, give a number that comes in between 0.999... and 1.

You (as Con) will try to prove 0.999... (repeating to infinity) IS NOT EQUAL TO 1.

Round 1
Pro
0.999... = 1: Prove that 0.999 (repeating to infinity) isn't equal to 1.
Proof 1:
x = 0.999...
10x = 9.999...
10x - x = 9.999... - 0.999...
9x = 9
x = 1
Proof 2:
0.333... = 1/3
0.666... = 2/3
0.999... = 3/3 = 1
Proof 3:
All numbers that aren't equal to each other will have a number(s) that comes in between.
x < x+1 --- The number that could come in between of x and x+1 could be x+1/2.
So, give a number that comes in between 0.999... and 1.
You (as Con) will try to prove 0.999... (repeating to infinity) IS NOT EQUAL TO 1.

Con
I am here to expose a lie. This is taught in official school math but is an absolute lie.

1/3 is not 0.3333.... with a '3' at the end. It is 0.333... that is eventually supposed to change into a value that is a 'third' but doesn't exist. So it will be 0.333...0333... over and over again. 

The idea that because 0.999... is going on forever it's somehow less impossible or stupid than a number which doesn't have the identical digit throughout its infinite series, is nothing but pseudointellectual deception. 0.999... is meant to end in a 0 theoretically. 0.000... with a '1' at the end of 0's is not less possible to exist or less 'real' at all.

The difference between 0.999... and 1.000... either exists or 0.999... doesn't exist at all numerically and therefore it can't be equal to 1.
Round 2
Pro
Okay,
You still haven’t truly disproved proof 1 & 3 as they’re the best ones.
If 0.999... isn’t 1 then what is a number that goes in the middle to prove it. 
And don’t even try to disprove 1 because it’s impossible. 
0.999... = 1
Con
The number in the middle is 0.999... with a '05' at the end. Similarly 0.999... itself has a '9' at the end that is meant to also theoretically be followed by infinite 0's (since all numbers have that following and the infinite 9's must end in a 9) so it's just as impossible of a value as the midpoint between it and 1.0000...


Scenario 1 is HOCUS POCUS BBBBBBBBBOOOOOOOOOOOOOOOOOOOOOOGGGGUUUUUUUUUUUUUUUUUUSSSSSSSSSSS
What you did with 'x' in the scenario is nothing more than a magic trick due to limitations of algebra and the fact that 0.999... is a fake value that doesn't even exist as you never reach the last 9. Try and reverse engineer it. Do it with x = 1 at the start and get me 0.999... at the end. You can't, meaning it's a pseudointellectual hoax.

Let's see Scenario 3...
All numbers that aren't equal to each other will have a number(s) that comes in between.
x < x+1 --- The number that could come in between of x and x+1 could be x+1/2.
So, give a number that comes in between 0.999... and 1.
oh. I handled that at the start of this Round.
Round 3
Pro
Okay, I will reverse engineer it.
x=1 
10x=10
10x-x=10-1
9x=9
1=1 
There. And since 0.999... is equal to 1, I have reversed engineered it. While it may not look like 0.999... at the end take a look at the next set of equations
x=3/3
10x=30/3
10x-x=30/3-3/3
9x=27/3
9x=9
x=1 
Now reverse engineer that to get 3/3, exactly, you can't. So are you going to say to me that 3/3 isn't equal to one? If you really wanted to you could simplify 3/3 and 0.999... to 1 in the second step, but I didn't show you the full extent of the proof.

Exactly, since there is no midpoint on the number line between 0.999... and 1, there is no difference between 0.999... and 1. 
And to all the people in the comments section, my point about that is, in fact, correct, plug-in random numbers and see it yourself, and by the way, it doesn't matter that 0.999... is irrational.
0.888... and 1
0.888... < 0.888...889 < 1 
0.888... is not equal to 1

0.777... and 0.888...
0.777... < 0.777...778 < 1
0.777... is not equal to 1

0.999... and 1
0.999... < x < 1
x=?
Therefore, 0.999... must equal 1.

You all stated that... 
The number in the middle is 0.999... with a '05' at the end. Similarly 0.999... itself has a '9' at the end that is meant to also theoretically be followed by infinite 0's (since all numbers have that following and the infinite 9's must end in a 9) so it's just as impossible of a value as the midpoint between it and 1.0000...
How does this make any sense? Let's go through it.
The number in the middle is 0.999... with a '05 at the end.
     Okay, 0.999...995 <0.999... < 1 ---- Irrelevenet to 0.999... equalling 1
Similarly 0.999... itself has a '9' at the end that is meant to also theoretically be followed by infinite 0's (since all numbers have that following and the infinite 9's must end in a 9) so it's just as impossible of a value as the midpoint between it and 1.0000...
     Similarly? 0.999...995 is not equal to 0.999... No, 0.999... is not theoretically followed by an infinite amount of 0's because 0.999... isn't equal 0.999...999000...
     If there is a zero at the end of any decimal it means that it is not irrational, however, 0.999.. is rational, so that makes no sense. You said it yourself, the midpoint is impossible, just like the midpoint of 3/3 and 1 is impossible, proving that 0.999... = 1. You don't really understand infinite numbers and series, that is the problem. 

Regarding your first round argument.
Arguing that 0.333... isn't equal to 1/3 simply shows you have no knowledge about infinite series, hyper numbers, surreal numbers, and Zeno's Paradox and how these things work. A quick example of why 0.333... = 1/3 (this proof also works to show why 0.999... = 1 if you multiply it by 3).

1/2 + 1/4 + 1/8 + 1/16 + 1/32 + 1/64 ... = 1 

Now with 0.333...
3/10 + 3/100 + 3/1000 + 3/10000 = 1/3 

Now if you multiply it by 3, there is another proof 0.999... equals 1.
9/10 + 9/100 + 9/1000 + 9/10000 = 1
All I can recommend is to study hyper and surreal numbers and infinite series/Zeno

You stated in round 1,
The difference between 0.999... and 1.000... either exists or 0.999... doesn't exist at all numerically and therefore it can't be equal to 1. 
Okay, well then if you truly believe this, let's put it to the test with other numbers that have similar properties.

The difference between π and 4 wither exists or π doesn't exist at all numerically and therefore it can't be equal to 4.
Does this make any sense? No, not at all. 

Let's go through it.
1. The difference between 0.999... and 1 does exist, it's called 0!!!
    The difference between 4 and π does exist, it's called (whatever 4-π equals) ~0.86... however it never terminates 
There is no such thing as two real numbers (which 1, 4, 0.999... and π all are) that cannot be subtracted from each other so saying that the number wouldn't exist at all numerically makes no sense.

At this point in the debate, I find that all of my three original proofs still stand (unproved by you) and that you making completely untenable arguments that anyone who knows anything about complex maths could easily disprove as I have done. I hope I've changed your mind by now because I don't want to see such silly arguments and untrue assumptions and statements without any proofs next round. I expect you realize right about now that I'm right, 0.999... does in fact equal exactly 1. So if you do believe it, please announce in the next round that You forfeit and I'm correct/right, 0.999... = 1.

P.S. I know it might sound quite condescending, but I simply don't want to waste my time (45 minutes to write this round) with someone who makes arguments and proofs so untrue and as I said untenable.  I apologize I was a bit harsh, but I'm simply warning you that arguments like the last two rounds will not win you this debate. I also apologize for skipping around between your rounds one and two, I understand if it's a bit disorienting, I got a bit carried away, ;) 
Thanks,
Aden

Con
I'm simply warning you that arguments like the last two rounds will not win you this debate
They have and will. I am right. You should warn yourself before talking to me in such a tone.

I am most amused by this excuse by Pro: 

There. And since 0.999... is equal to 1, I have reversed engineered it. While it may not look like 0.999... 
He then proceeds to have '3/3' which is a division action not a value. 3/3 actually is 1/1 if you transcribe the fraction and divide it by 3, so he had '1' at both the start and end of that equation, never ever having a value other than 1. What he tried to prove was nothing at all other than that 1 = 1 and that if we wrongly assume 1 also equals 0.999... that we can mistakenly replace it with 1 in equations and at some point an error due to limitations of algebra will occur.

The fact that you never ever will get x to equal the '.999...' value at the end means that there is something wrong and I will tell you what's wrong; infinite series are illusions, they don't actually exist as anything other than 'values that should be real'. You probably can have whatever 0.999... is as an actual number in another system of counting (so not denary which is the one most humans count in and has 9 whole number values before reaching 10 from 0) and in that other system of counting, you'd see that the '1' in the denary system is perhaps a differently put and even perhaps unattainable value that is forced to be represented by a decimal value that goes on infinitely. You see, in base-9, '1' in denary is actually '1+1/9) to them. Their '1' is 0.9 in denary. They don't have a '9' it's their '10'. So if you start exploring other ways of counting, you'd see even more blatantly that '0.999...' in denary isn't '1' in denary, since the former may actually be the whole number value.

I have covered everything thrown my way in this debate including finding a midpoint. the reason the midpoint can be confusing is because I say '05' at the end of all the 9's because the different is '0.000...1' That '1' at the end will become '05' while in reality the '05' would end up needing to push forward and become a '5'. So the 'real midpoint' ends up being '0.999...5' even though the usual way to find midpoints in numbers is to find half way between the difference. This is again due to the nature of infinity and how impossible it is to ever reach the last '9' in the series.

Throughout this entire debate we have seen Pro avoid and deflect the fact that the last '9' in 0.999... is no less impossible to attain or more possible to attain than the '1' that is at the end of all the '0.00...' which is the different between 0.999... and 1, as well as the '5' that is at the end of the midpoint between them.

Either 0.999... is a fake value that can't be equal to a real one or it is a real value and the difference between it and 1.000... is valid too. There's not third way to wriggle out of this.
Round 4
Pro
What 5?
0.999... goes on infinitely with nines so there are no 0’s nor your 5 midpoint
Well what happens to the zero?
It disappears. Since it infinitely foe son forever we can conclude that you just add 0 to 9 an infinite number of times which still is 9.
And as I said before even now it’s still through. 
0.999...995 < 0.999...
And now
0.999...905 is still < 0.999... 
How does your number with a 5 become greater than 0.999... but less than 1? 
You also seem to believe that it’s to do many things with 0.999... because it never ends but that is simply incorrect.
As I said, surreal, hyper, and Zeno’s paradox will pretty much explain how to do most operations regarding infinite series, 
In another sense, think of 0.999... as another way to write 1, like 5/5, or the square root of 1
Zeno’s paradox further proves that 0.999... equals 1.
0.999... is not a fake value, it’s value is 1.
Want another super easy but less intellectual way,
Type calculator in google and type in 0.999... with about 20 9’s, then if you push equal it will spit out the number 1.
Let me show it to you in BABY steps so your non-maths brain can understand. 
X = 0.999... = 0 + 9/10 + 9/100 + 9/1000  and so one multiply the denominator by 10
10X = 10(0.999...) = 10(0 + 9/10 + 9/100 + 9/1000 ...) = 10 (0) + 10(9/10) + 10(9/100) + 10(9/1000) ... = 0 + 9 + 9/10 + 9/100 ... = 9.999... 


Con
There is a lie happening here. 0.999... is not greater than 0.999...05 rather the '5' is just as impossible to ever get to as the '9' and of course '0's that will follow the 0.999 recurring (as all numbers that exist eventually must have infinite 0's following them after the last placeholder). The midpoint between 0.99... and 1 is obviously greater than 0.999... and because 0.999... is a fake value that doesn't really exist in the denary counting system, the midpoint between it and 1.000... is also pretty impossible to actually attain. The last '9' is just as valid and invalid as the last '5' in the midpoint.

Either 0.999... never ever reaches the last 9, negating it as existing at all as a valid value. 

Or it does and the '1' at the end of the 0's in the difference between it and 1 is reached as well as the '5' a the end of the midpoint.

It's as simple as that.

Since Pro randomly decided to use sources in this last Round, I will too.


.999=x
10x=9.999
10x – x = 9x
9x=9
1x=1.
.999 = 1.

The hand-waving is obvious. How does one multiply an infinite series of 9s by 10? What happens to the zero you’d get when multiplying by ten? Are we to believe it just disappears?
Of course not. The proof is invalid. It’s just an optical illusion relying on tricking the reader by hoping they don’t notice the hand-waving. The reality is no “proof” can address the issue better than simply looking at the two values. If the two are equal to one another, subtracting one from the other must give an answer of 0.
When we do that, we see there is a difference – 0.000…1. It’s infinitely small, but it is real. Therefore, the two numbers aren’t equal. Why then do so many people believe they are? Because they say so.
Literally. They say so, so it’s true. That’s it. You see, there is a thing in math called an axiom. It’s a statement assumed to be true without proof. One axiom underlying the real number system basically says:
Non-zero infinitesimals do not exist.
Which means 0.000…1 does not exist. Why? Because we say so.
Only we don’t say so. A person who thinks 0.999… doesn’t equal 1 obviously believes infinitesimals exist. They don’t accept that axiom. They’d use a different one, like many mathematicians who work with infinitesimals on a regular basis.
That’s right. There’s an entire field of math which uses infinitesimals. It’s just as valid as the real number system. Which one you use is merely a matter of preference. Whether 0.999… and 1 are equal is based on the completely arbitrary choice of whether one uses the real number system or a different one.
All the “proofs” the two are equal are meaningless. Implicit in all of them is the statement, “Using the real number system.” That’s begging the question. It’s tricking people by assuming there could be no difference between the two numbers then concluding there is no difference between the two numbers.
Anyone who understands how math works should know 0.999… equals 1 only if you choose for it to. “Proofs” the two are equal tell us nothing about the subject but everything about the speaker. Namely, they don’t know what they’re talking about.

If you feel 0.999… does not equal 1, you’re right. If you feel it does equal one, you’re right too. Which answer is “right” just depends on which type of math you feel most comfortable with. It’s purely a matter of personal choice.