The Instigator
Singularity
Pro (for)
Winning
35 Points
The Contender
astrosfan
Con (against)
Losing
26 Points

.9999 repeating = 1

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Post Voting Period
The voting period for this debate has ended.
after 11 votes the winner is...
Singularity
Voting Style: Open Point System: 7 Point
Started: 9/19/2008 Category: Education
Updated: 8 years ago Status: Post Voting Period
Viewed: 21,858 times Debate No: 5466
Debate Rounds (5)
Comments (20)
Votes (11)

 

Singularity

Pro

As we all should know:

.9 = 9/10
.99 = 99/100
.999 = 999/1000 etc.

In this debate, I support the notion that an infinite number of 9's after the decimal place is simply a different notation for writing the number 1.

My claim is simply that .99999... (where the "..." denotes repeating) = 1 exactly.
astrosfan

Con

This sounds like it could be a fun debate and being con I negate that (.999 repeating =1 )

My case against this does require some complicated theory. The first thing that we have to understand that infinity is (despite its name) is a finite number as explained in this article http://en.wikipedia.org.... All infinity is the largest number and would end it would just be really big.

Now then with this idea down a will attack the theory that my opponent will try to use to prove his point. Most people would use this equation to try to prove this to be true.

Let x = .999...
10x = 9.999... (multiplying RHS and LHS by 10)
10x - x = 9.999... - x (subtracting x from both sides)
9x = 9
x = 1
.999... = 1

But the problem with this that if infinity did equal a finite number then the nines would have to stop a t some point. Now to use an example ( In this example I will say that infinity equals to 5 because I don't feel like writing a lot of 9's)

Let x = .99999 (.9 to the infinite digit)
10x = 9.9999
10x – x = 9.9999 - .99999
9x = 8.00001
Thus .99999…. wouldn't't equal 1
Debate Round No. 1
Singularity

Pro

I really didn't expect a response, let alone so quickly, so I appreciate you tasking up the challenge; good luck to you sir. Alright lets get down to it.

"The first thing that we have to understand that infinity is (despite its name) is a finite number..." - You

Seeing as infinity can be a property or a conceptual value, your introductory statement has this sentence structure:
~~ We have to understand that A (despite being A) is not A. ~~
This is a logical fallacy. See the Reflexive Property: http://www.mathwords.com...

Infinity is neither a number nor is it finite. The article you cited doesn't indicate infinity to be finite, but instead describes it's behavior for UNBOUNDED limits. On this note, I should stress that limits do not have to always apply to nor be finite numbers. Because, your entire argument is contingent on the point that infinity is not really infinite, and no reliable source has been demonstrated to back this claim, I would recommend a different approach. However, if you insist that your link does provide evidence to support your claim you will have to post quotes from it to better communicate the issue. Clearly, the burden of proof you carry has not yet been met.

That being said, my first proof for .999 repeating = 1 is as follows:
x/9 = .xxxxx repeating, if x <= 9

Example:
1/3 = .3333 repeating.
Therefore:
1 = 9/9 = 3/3 = 1/3 + 1/3 + 1/3 = (.333 repeating) times 3 = .999 repeating

Or you can always do the direct method:

4/9 = .4444 repeating
9/9 = .9999 repeating <---------- and 9/9 = 1.
astrosfan

Con

Lets get down to business. And to make it easier I will number the arguments

1.Is infinity a number?
Infinity is a finite number. All it is the largest number that you can have most people don't conceder it to be a number because you can never count to it because as soon as you the next number would become infinity, but this still means that even if you can't count to infinity it is a number and is finite. To use an example of how infinity can be a finite number, there are infinite number of decimals, fractions, and ratios that are in between the numbers 4 and 5. Now in this example 5 would act as infinity because if you counted each individual value in between these numbers you would never be able to count to 5 because the values would only get closer and close but never becoming 5. Now I hoped that most people got that example but it proves how infinity can exist even if you cannot reach it.
a.My opponent's first is attack is against my statement "The first thing that we have to understand that infinity is (despite its name) is a finite number...". Now first I would like to clarify that is statement was meant to ironic and funny not to be a real point. But I can still defend it, the reason that it is called infinity is because we think of it a something that can't be reached even though it is a real number.
b.Next my opponent tries to argue that my source doesn't support my claim but I think this quote should work just fine "In mathematics, "infinity" is often used in contexts where it is treated as if it were a number". Also you can cross apply my example from my overview that infinity can be represented as a number in sets. Next he try to say that my sources is so I decided to find some more sources that support what I am saying
this also explains the example I give in the overview
http://www.c3.lanl.gov...
http://www.mathacademy.com...

2.Right hand side left hand side is flawed.
With the fact that infinity is a finite number as I explained above this proves that the rhs lhs theory is flawed because when the decimals reach infinity they would stop and by multiplying it by 10 would cause it to have one less decimal causing it not to be equal to 1. And even if you don't believe that infinity is finite I can still prove that the same problem would happen in the rhs lhs theory with an infinite infinity.
a. Power of infinity- when it comes to infinity there are different values of infinity. Go back to the example I gave in the first overview about how there a infinite number of values in between 4 and 5, now there is also a infinite values between 4 and 6. But how can there be the same number of values between 4 and 5 and 4 and 6 if the second set contains the first set and should be bigger, that because it is the set of values between 4 and 6 is a higher power of infinity and is bigger the first set. Now when you apply this to the rhs lhs theory he you multiply the .999999… you will still have an infinite number of 9's after the decimal but it still is 1 less the what you originally had because it is a lower power of infinity

3.Why my opponent's math is wrong
In this part we have to understand one quick fact and that is that 1/3 does not = .333333…. repeating. This is because in our base 10 system we have no why to represent a 3rd thus when we try to divide 1 by 3 we get a repeating series of 3 but because it is never can divide evenly it will never = 1/3. Now with that said I will attack my opponent's theory, because as I stated above the 1/3 does not = .3333… it cannot be apply in the equation that my opponent uses, instead we have to look at from the only why that we calculate this equation is to use the only way to evenly divide 1 by 3 and that is by using remainders. (for the people who have forgotten how remainders work here is the wiki to remand you http://en.wikipedia.org...)

When we do the same equation with remainders this is what happens

1/3= .3333 remainder 1
1/3 x 3= .3333 remainder 1 x3
1= 1

Because when you multiply the remainder 1 by 3 it allows you to evenly divide it and would cause the last 9 in the sequence to become a 10 because of the added remainder which cause the next 9 to become a 10 because you would carry the one and so on and so on till you get one. This proves that not only is my opponent's flawed it only further proves that 9999… doesn't equal 1

4.More reasons that 9999… doesn't equal 1

a.The number line paradox- if you where to graph both 1 and .9999… they would not be in the same place even if you had the .9999's go to the infinite place it would never be in the same place as one there would always be an infinitely small gap between them that would prevent them from being equal
b.The law of the function- a function is a equation that has one value of y for each x. now if .9999… was to equal 1 it would cause 1 value of x being 1 to have 2 y values the first at .9999… and the second at 1 this would cause a problem because that would mean that our number system is not a function. This would prevent .9999… from being 1 because it violates the fact.

5.The burden of proof- earlier my opponent claimed that I have failed to meet my burden of proof this has one major problem and that is that I'm con I don't have a burden of proof the pro does all the con has is the burden of refutation. This brings us to has meet their burden my opponent has attempted to proved proof but have refuted it by showing how his math is flawed and how theory disproves the resolution with substantial evidence to minimal proof that my opponent has proved, because of this you as judges have only one option and that is to negate the resolution that .99999… =1 and vote con.
Debate Round No. 2
Singularity

Pro

DEBUNKING OPPOSING ARGUMENT:

First, I must insist on legitimate sources or mathematical proofs verifying that infinity is a finite number. My opponent example of defining infinity to be finite never addressed my proof of flawed conception (a != a), and uses an interesting thought experiment to try to support his definition. Yes it is true that there is an infinite amount of numbers between 4 and 5 but any one of those numbers in not infinitely large. By definition, there is an infinite of numbers between every pair nonequivalent numbers. None of these numbers are actually infinite, they are only mirroring the fact that they are on a continuum of infinite numbers.

On this note, my opponent states that .99999 repeating and 1 are not equivalent, therefore and an infinite number of numbers must be in between. Simply, name one of them.

The youtube video my opponent posted started off by stating it was a semantic debate rejected by mathematicians alike. Trying to redefine infinity as a number is flawed in that it is boundless, concept that can't be quantified in any situation unless it can be taken out of the equation. For instance, infinities can cancel one another under certain circumstances. To understand infinity while applying it to mathematics, we apply it as a rate of growth. A rapidly growing number that never ceases to end, hence IN-finite.

When my opponent stated that infinity was finite, he responded by saying it was meant as humorous and not meant to be a real point, despite the fact all of his attempts were based on the tangibility of infinity.

His defense was this statement:
"the reason that it is called infinity is because we think of it a something that can't be reached even though it is a real number."

Incorrect. Finite is an attribute assigned to anything that ends, or is bounded. Infinite is derived as NOT finite, or continuing without bounds. These are all the articles given to finite based on these all encompassing definitions:
http://i237.photobucket.com...
If I may direct your attention to the second bullet point which differentiates the obvious difference between infinite and finite.

When trying to support his claim, my opponent made a link to a reference which never stated infinity was a number nor that it was 'finite'. In response he posted a quote from Wikipedia:
"In mathematics, "infinity" is often used in contexts where it is treated as if it were a number".

The flaw is apparent:
"Treated as a number" is true because notion for continuing without bounds do appear in high level math. "Treated as" a number does NOT mean "is" a number.

My opponent cites two more links which supposedly support his claims, and AGAIN fails to direct any quotes from them. Giving my opponent the benefit of the doubt, I will assume he is not trying to just make it look like he has evidence and he believes that there is relevant information to our discussion. The first link seems to center around some thought experiences, but mostly on sets of numbers that span infinitely. I have never made an argument that sets cannot span infinitely, so I will assume he misunderstood my position. The second link clarifys my point quite nicely, that infinity does NOT end. Here is a quote conveniently close to the top:
"And then, dreadfully, "does it end?" And at last, the awful truth: it NEVER ends."

It appears my opponent wishes to disprove the right hand left hand concept in the second link. Here is the page again:
http://www.mathacademy.com...
His disproof states "[the] theory is flawed because when the decimals reach infinity they would stop". This is an extraordinary claim to say that one can reach infinity and that it ends, or simply even has an end. This is built on the discussion where he attempted to define 'infinite' as 'finite' and then dismiss it as unimportant/funny. Without proving this point, his disproof is lacking in every way.

Continuing his charade of extraordinary and unverified claims, he states that 1/3 != .333 repeating, because of a flaw in our based 10 number system. Again, this has yet to be demonstrated OR cited. He extends upon this conceived notion by trying to put a remainder after the infinite digits. Simply put, something WITHOUT AN END cannot have something AFTER it. He seems to be suggesting a number in the form of ".9999...1", which applies to the emotional response to the infinitude dilemma, rather than a logical mathematically derived form.

His next argument falls under the realm of limits which you can learn about in any Introductory Calculus class. He states that going to the infinite decimal place (once again dependent on his claim you can reach it) an infinitely small gap appears between the graph separating the number. The next claim that .9999 repeating would have a slightly different y value than the y value for 1 is equally illogical. The two numbers are one in the same and can be written in two different notations, and are not separate on the number line. Thus no conflict in x nor y values would arise.

Note: When I mentioned the burden of proof, I was referring to disproving the math I support. I agree that calling it burden of refutation might be more generally obvious. :D

ADDITIONAL PROOFS OF .9999 REPEATING

My opponent attacked the validity of 1/3 = .3333 repeating earlier in conjunction with all limits relevant to this discussion. Therefore, I will upload the mathematical limit notation for such limits with the original example of .9999 repeating:

http://i237.photobucket.com...
For clarification, the 10^n power represents the growing decimal places of continuing nines. A converging sums proof is ideal for our discussion of approaching infinity.

I should also recap my proof that there is no number between .999 repeating and 1. No number between means the difference is zero. A difference of zero means it is the same number. (a - a = 0)

REVISIT
I clumsily forgot to adress one of your problems which did too depend on a finite infinity. :P

_______________________________________________________________________________________________
<> <> <> <> <> <> <> <> <> <> <> <> <> YOUR POST WAS THIS: <> <> <> <> <> <> <> <> <> <> <>

Let x = .999...
10x = 9.999... (multiplying RHS and LHS by 10)
10x - x = 9.999... - x (subtracting x from both sides)
9x = 9
x = 1
.999... = 1

But the problem with this that if infinity did equal a finite number then the nines would have to stop a t some point. Now to use an example ( In this example I will say that infinity equals to 5 because I don't feel like writing a lot of 9's)

Let x = .99999 (.9 to the infinite digit)
10x = 9.9999
10x – x = 9.9999 - .99999
9x = 8.00001
Thus .99999…. wouldn't't equal 1

______________________________________________________________________________________________
<> <> <> <> <> <> <> <> <> <> <> <> <> <> <> <> <> <> <> <> <> <> <> <> <> <> <> <> <> <>

In addition to the dependence on a 'finite' infinity, 9.9999 - .99999 = 8.99991 (Not 8.00001)
A growning number of nines before the '1' recedes into the argument where I mentioned the ludicrous form of ".999...1". It's in the middle of my post.

I await your response.
PS. VOTE PRO yaya! XD
astrosfan

Con

To clarify before I start I'm going to be using the (%) sign to represent infinity and I'm going to us (~) as an approximant sign

1.One this first point I don't even have to prove that infinity is a number or if it even is finite all I have to do at this point is to prove that infinity has value and can be applied in an equation. Which I have done in 2 ways that my opponent has failed to attack 1st is the wiki definition, while my opponent does argue that it doesn't prove that it is a number but it does prove how it can be a value when it says that infinity acts as a number because it has value and can be applied in equations, this is further proved in power of infinity that my opponent has conceded as it states that infinity can have different values which I will use to latter on to further my arguments. The 2nd is from the youtube video when the person shows how infinity can be used in an equation which acts as proof to show how infinity can have value and it can be used in equations. Next my opponent tries to show to how the youtube video it trying to change the definition of a number but this completely untrue and all the video is doing is showing how infinity can be included in the definition of number, also it show that the mathematicians are not the final word on if infinity is a number not.
a. Next my opponent tries to attack my statement from the first round but ignores the fact that statement was meant to be ironic. And even if you are going to try to listen to my opponents none of it matters because as I stated above that I don't even need to prove that infinity is finite all I have to do is to prove infinity can have value and can be applied in an equation.
b.Next my opponent tries to attack my other evidence but 1st I would like to mention that my opponent attack my first new source which is enough to prove that infinity can have value and that value can change as is stated in the final section of this source. Next my opponent tries to attack my second new source by saying it shows that infinity is not a number because it cannot be reached, but this doesn't matter because as it states in the example about the fingers were it shows that even though you couldn't count to that final finger it was there and had a value this parallels the example I give about the infinite number of values between 4 and 5 how you could never reach it and acts a infinity thus proving my point.

2.The flaws in the rhs lhs equation happens because of the sub point A of the power of infinity which my opponent completely concedes that proves that infinity can have different values when use in a math equation. To show this in equation

%-1 = % now while most people would think that this would prove my opponent's point but the fact is that the % that you get out of the equation is actually less than the original infinity that you started the equation. This is caused by the powers of infinity that would allow for one infinity to be less than another. Now to apply this to the rhs lhs equation

The .99999… in the first step represents the first infinity in the example equation, while the 9.9999… has a number of nines equal to the second infinity in the example equation meaning that even if that there are an infinite number of 9's after the decimal in the second step it is still one less the original equation because it is a different power of infinity
a.Next with the power of infinity even if my opponent tries to say that my arguments are based on a finite infinity but this is not true this theory can work independently of a finite infinity because no matter how big infinity gets the second infinity will always be one less the original infinity.

3.My opponent's arguments
a..3333.. repeating- while most people think that 1/3 = .333…, but this is untrue, it is only approximately equal to .333…. To further prove I will work the problem out, if you divide 1 by 3 first you can't divide at all so you go to the digit behind the decimal then you 10 by 3 and you get a remainder 1 thus the answer isn't equal to one third, thus you do it again and divide 10 by 3 and get 1 remainder again and again the answer doesn't equal 1/3 this continues infinitely and you infinitely will get that remainder of one meaning that the answer will infinitely never equal 1/3 thus .3333… only approximately equal to 1/3. This means that .999… will only approximately equal 1. Then he attacks my use of the remainders to prove him wrong by saying that it would be .999..91 but this is not true because the 1 created by the remainder is added in on the last 9 meaning it changes to 10 which carries the 1 and makes the next 9 a 10 and so make it equal to 1
b. Next my opponent uses another equation to prove his point but there are some key flaws. The 1st is in between the 3rd and forth steps I would like my opponent to show the math behind how he was able to change the 9/10 became 1/10. The second is in between the 4th and 5th step is how the 1 goes from being inside the () to being out side of it because of the distributive property means that everything inside of the () is multiplied by what is outside of it meaning that that the 1 should change if it leaves the (). The third comes in the final step where I'm guessing that my opponent is saying that by dividing 1 by infinity you get zero, but this is untrue in that it only approximately equals zero as explained in this video at 1:50 . the final problem with this equation is that with you can prove that .9999…. can equal any number because at the point that if you don't believe the 3rd reason this is wrong then in the equation when 1 over infinity equals 0 then an thing that you subtract it from would = .9999….

4.Other reasons why it is wrong
On these points my opponent makes little to no arguments against them the only one he does make is that .999… and one are one in the same and that there is no problem. So to answer this I will this attack separately for each.
a.Number line paradox- to answer my opponent's attack I will answer a challenge that my opponent made earlier in the round when he ask me to show a number between 1 and .999…. To this I will answer with .9 to the % decimal place plus 1, now you may think this a ludicrous that there something larger then infinity but this has its support from my power of infinity sub point that shows how infinity can have different values and can be effected by math equations. This means that theoretically there are an infinite number of numbers between .9 to the infinite digit and 1 because of the different values of infinity that can be applied.
b.The law of functions- the problem with my opponent's attack on this is that both 1 and .9999… are in decimal form no if one was in fraction from while the other was in decimal I would agree that they could be in the same location but because the from they in it would prevent them from being in the same location.

5.Closing statements- after looking at the comment section of this debate I feel like it is needed to reiterate the fact that when you as judges vote on this round to weigh only what has been presented in this round and free from outside influence.
Debate Round No. 3
Singularity

Pro

I'll try to be concise for everyone's sanity. :P

"1. One this first point I don't even have to prove that infinity is a number or if it even is finite all I have to do at this point is to prove that infinity has value and can be applied in an equation."

Whoa there buddy. First of all, I agree that infinity can be used in equations. Heck I posted a mathematical proof using infinity. But... I have no idea where you got the idea that you no longer have to prove that something doesn't have to have a last digit (or end) in order to put something after it. If I understand your argument, having one less digit at the END of the string of 9's is what would theoretically offset it. The structure of your argument is still dependent on proving infinite digits have an end instead of being more like a ray. I can understand why you would want to abandon your attempts to prove infinity is finite, but redefining what you have to do without justification is going a little far.

If you are going to prove that infinity can be used in an equation, it has to be relevant to your point, such as using infinity in an equation the specific way you claim it can be used... just like I used infinity in an equation specific to evaluating .999 repeating here:
http://i237.photobucket.com...

"...mathematicians are not the final word on if infinity is a number not."

So who is? If the global common knowledge is in agreement with the mathematical community concerning an issue, and someone declares a semantic perception of the issue, they are they ones to give infinity its "new and improved" definition?

QUOTE: "I don't even need to prove that infinity is finite all I have to do is to prove infinity can have value and can be applied in an equation."

Explain why. Using it in an equation is uneventfully common and well accepted in the mathematical community. Mismatching 9.9999......90 with .9999......99 is not. Putting digits on the end of infinity is like trying to hold the end of a circle. There is no end, hence the definition 'increasing without bound'. A bound would be the end; no bound means no end; no end means no last number.

QUOTE: "% - 1 = %"
First, this equation only demonstrates the point that infinity is not a number, because no number satisfies the equation.

Second, different 'values' of infinity is a way of approximating the context of the equation. Rates of approaching infinity is simply a measure of say-- the coefficient.

For example: 2% > %
because it is a higher rate of increase without bound, this does not mean it is a higher infinity. For example, if you add up all the integers, you get infinity. If you add up all the even integers, you would get infinity but not at the same rate.

Stating that infinity has values is rather inaccurate as .5 infinity does not really exist, especially when you realize that there is no precise value of 1 infinity. I recommend reading this:
http://www.physicsforums.com...

QUOTE: "I will answer a challenge that my opponent made earlier in the round when he ask me to show a number between 1 and .999…. To this I will answer with .9 to the infinity decimal place plus 1, now you may think this a ludicrous that there something larger then infinity but this has its support from my power of infinity sub point that shows how infinity can have different values and can be effected by math equations."

This answer once again is dependent on a last digit in a repeating number. Your 'infinity can have different values' was imprecise, but more importantly didn't apply to your problem. simply put, manipulating the rate approaching infinity is VASTLY different than surpassing infinite digits.

QUOTE: "if you divide 1 by 3 first you can't divide at all so you go to the digit behind the decimal then you 10 by 3 and you get a remainder 1 thus the answer isn't equal to one third, thus you do it again and divide 10 by 3 and get 1 remainder again and again"

Thus it takes an UNLIMITED amount of 3/x, where x is a power of ten. I can see why you might think that because a remainder seems to always exist, an unlimited amount would also promote this. Its a common thought trap with a solution reminding me of Hilbert's paradox. The remainder is not unlike a taken room in a hotel. No matter how many rooms there are, if every one is taken, no more guests can be accommodated. Right? The answer is yes, until you consider the possibility of the oh-so-cool infinity. Since there is no limit, there is no reason why the people in the rooms can't be moved to their room number(n) +1, or maybe 2(n). This also demonstrates again, why infinity ISN'T really a number even though it can be used along side them.

QUOTE: "Next my opponent uses another equation to prove his point but there are some key flaws."

You have cited Wikipedia as a trustworthy source, so it might interest you to know that equation originally came right off of Wikipedia. I have some trouble understanding the apparent 'flaws' you see, but I will attempt to answer your questions. I cannot see any easy way to put an extra step in after evaluating the limit of a convergent sum, but your next question makes a little more sense. If you look up the properties of limits, it is perfectly legal to move certain numbers out of the limit notation. I wish I could help more but I don't know what it is that you don't know. :(
For the first question I recommend learning how to quantify converging sums. Here is a good example using .111 repeating:

http://en.wikipedia.org...
Here is the article citing the proof I used for .999 repeating = 1, among others:
http://en.wikipedia.org...
This link is also provides a good math example for your first question:
http://en.wikipedia.org...

Beyond that, you go to say that you can make any number equal .999 repeating using this formula and quite frankly you lost me. :(

QUOTE: "The law of functions- the problem with my opponent's attack on this is that both 1 and .9999… are in decimal form no if one was in fraction from while the other was in decimal I would agree that they could be in the same location but because the from they in it would prevent them from being in the same location."

It was discussed in comments that you jumped to the conclusion that 1 is in decimal form, so I decided to point out a similar errors instead. The decimal system is proven to be able to communicate the same numbers with multiple notions. For instance, you claim that because both are decimal notion and written differently, they can't be the same number. This is incorrect, using a number other than .999 repeating to prove this, I submit the notation 1.00 repeating.
Do you agree 1.00 repeating equals 1?

QUOTE: "I would agree that they could be in the same location"
Don't admit to that. ;J

A few references:
http://mathforum.org...
http://mathworld.wolfram.com...

uh.. that wasn't as concise as I planned. My apologies.
astrosfan

Con

To start this debate I will have an overview the go down the flow

First there is on key argument that my opponent has dropped though out this round and that is power of infinity. This is going to be the killing blow to my opponent's arguments because it shows how infinity can have multiple values and can be applied in math. My opponent tries to argue that infinity increases at a different rate but this only proves my point if % were = 1 the %+1=2 and while both answers to these problems are infinity the they way that they are used in an equation the second on would be bigger this can be applied to the rhs lhs equation because when you multiply the .9999… repeating by 10 it still has % number of digits behind the decimal but it acts as %-1 digits so it would be one less than the normal .999… this way I said all I have to prove is that % has a value and can be applied in equations .

The first point that my opponent makes is that it is illogical to have a 0 at the end of the repeating 9's but there is nothing illogical about this because using an example of the set of all real numbers we know that will end at infinity but the numbers in the between are the ones that continue to grow so the same can be applied to the 0 at the end while the 9's in between continue to grow and end in 0

Group my opponent's next 2 arguments and as I said above that these only prove my point because it shows how we can mortify the value that infinity has meaning that my opponent's arguments only support my point
My opponent's opponent again tries to make have to prove that I have to prove that there is a final digit in the repeating 9's but this creates a huge contradictions because for one because it is not he rate that it is approaching infinity it the speed at which is growing, the second problem with this if infinity constantly increasing they it would be imposable to use in an equations

Next my opponent tries to disprove the argument about the remainder with is example but this example is inadequate to describe to represent it would have to assume that 1 would increase to allow for the extra room to be created a better example would be that a hotel has 3 room and you have 10 people and you wanted to divide them equally up but o person would be left over and he goes to another hotel with another 10 people and they get 3 rooms and the same problem occurs even if you go to an infinite number of hotels there will always be that one left over and it will never go away. Also don't even listen to my opponent's argument because it is not specific to division and repeating decimals

Next on the new equation that my opponent has presented, first I would like to point out that my opponent conceded the argument that 1/% is only approximately 0 this means that his final step when he subtracts 1 by 1/% it would only approximately equal to 1 making my opponent's equation mute and don't let him bring this up in his next speech because this is his final speech and being new arguments in is abusive because there is no time to debate them, the next problem with the equation that I had was how he went from 9/10 k to 1-1/10n there is no way that that 9 becomes a 1 thus creating a huge flaw in the equation

Next I would like to point out my opponent's concession of the number line paradox that there are an infinite numbers between 1 and .999… this is proven by the power of infinity argument that shows how infinity has different values of infinity mean that there can be different number of 9's after the decimal. This point has be conceded don't let my opponent bring it up in his next speak because that is abusive and this is the reason that this debate round might as well be over because at the point that I have proven in a conceded point that 1 doesn't .999… then if it might in other situations it still doesn't equal it in this example proving that they don't equal

Next on the law of functions my opponent argues that there can be multiple examples of 1 in a decimal form that can be at the point of 1 but the example of 1.00 doesn't prove me wrong because we simply by removing the 0's behind the decimal making be 1 now the difference with .999… is that we can remove the 9's and simplify it like this like this. The next thing he says is that I conceded that they could be in the same location if only for that 3 letter word called "but" that is right after that statement saying how them being in the same locations is imposable.
Debate Round No. 4
Singularity

Pro

A side note: Please proofread your debate, as it is growing increasingly harder to understand you. :P
Alright lets get jiggy with it:

DEBUNKING THE OPPOSING ARGUMENT:
Q-"My opponent tries to argue that infinity increases at a different rate"

Perhaps I was unclear or simplifying. There is one infinity (the concept of boundlessness) and equations can make variables or other equations approach infinity at various rates.

Q-"It shows how infinity can have multiple values and can be applied in math."

I already told you that using infinity in math wasn't the problem... We both agree it can be used in math.
FOR EXAMPLE: Limit as x approaches %, for the function: f(x) = x is %.
This translates to, as x gets bigger it approaches no end and no boundary to become arbitrarily close to. It approaches infinity without reaching it, because infinity is boundless. X increases without bound, as does y or to hold the notation used earlier: f(x).
Because its used in equations, you illogically presume it has a definite value, with a definite number of decimal places.

Q-"this only proves my point if % were = 1 the %+1=2"

I'm assuming you meant "if .999 repeating = 1 then .999 repeating + 1 =2" Which I would agree with. Infinity does not equal 1. Otherwise, those people in stores who give out food samples with signs saying please take one, would get royally screwed. ;)

Q-"the they way that they are used in an equation the second on would be bigger this can be applied to the rhs lhs equation"

Uhh I'm still waiting on the example or source for using an infinite number of decimal places as opposed to in infinity. Besides, an increasing rate of approaching infinity, would not be quantifiable. There isn't even a variable for time. Have you taken any class covering limits? :X

Q-"this way I said all I have to prove is that % has a value and can be applied in equations."

Value is different than rate. Rate is different than notion expressing unbounded. Repeating decimal notation is not the same as coefficient rate determination. Read above please.

Q-"The first point that my opponent makes is that it is illogical to have a 0 at the end of the repeating 9's but there is nothing illogical about this"

There is no end to put the decimal after. Limitless, endless, boundless, I don't know how else to say it. :(

Q-"so the same can be applied to the 0 at the end while the 9's in between continue to grow and end in 0"

A number line has numbers grow, but your example assumes they are not the same number as its own proof.
[label a] Circular logic works, because [go to label a];
.9's adding is founded on the idea that we have numbers in between .999 repeating and 1. Your proof is invalid. Sorry.

Q-"it is not he rate that it is approaching infinity it the speed at which is growing"
My daily fill of lol's. I'm not so sure there is a distinct difference, but rate is the word mathematicians use. Speed tends to reiterate the distance / time relationship. See above (again). How is this a 'contradiction' or a by-product of you having to prove the finite properties (end/edge) of infinity?

Q-"if infinity constantly increasing they it would be imposable to use in an equations"
Your definition of infinity needs revision, but I think its my fault for being vague earlier. I will simplify: There is a positive and negative infinity that exists conceptually. Values of (or in) equations can try to reach these values at different rates.

Q-"Next my opponent tries to disprove the argument about the remainder with is example but this example is inadequate to describe to represent it"
When applied to the number of decimal places, and NOT the 'value' of infinity, it works fine. It merely demonstrates that there is no bound to the repeating 9.

Q-"Also don't even listen to my opponent's argument"

Classy. If you were truly correct in your disproof, you wouldn't need to direct people away from it.

Q-"first I would like to point out that my opponent conceded the argument that 1/% is only approximately 0"

I did no such thing! Libel I say! I specifically did a ctrl-f to find every "0" and went through and read every instance just in case I mistyped something. Not once did I say 1/% is approximately 0. I'm glad that the history is still open for viewing so that the voters can check this for themselves! (Be sure to make sure that within a post it is not a quote of another.) I'm kinda pissed now. >:(

Q-"don't let him bring this up in his next speech because this is his final speech and being new arguments in is abusive because there is no time to debate them"

You brought up a point, and said don't let me debate it because there isn't time to debate it?
1. You brought it up
2. You have another posting opportunity giving me the disadvantage EVEN if I do post.
3. It was either a VERY careless mistake or a blatant lie.
Why can't I mention it? Too late I did.

Q-"my opponent argues that there can be multiple examples of 1 in a decimal form that can be at the point of 1 but the example of 1.00 doesn't prove me wrong because we simply by removing the 0's behind the decimal making be 1"

You manipulated it. Have HONOR! Subtracting zero's doesn't do anything to the value but it does change the way it is written ...which is the point.

Q-"The next thing he says is that I conceded that they could be in the same location if only for that 3 letter word called "but" that is right after that statement saying how them being in the same locations is imposable."

I will answer this with another quote. See below this time.

Q-"both 1 and .9999… are in decimal form no if one was in fraction from while the other was in decimal I would agree that they could be in the same location but because the from they in it would prevent them from being in the same location."

Assuming 'from' was supposed to be 'form', I already demonstrated that 1.00 is another decimal form for 1.

Returning to this quote:
Q-"My opponent's opponent again tries to make have to prove that I have to prove that there is a final digit in the repeating 9's"

I would also like to post this quote. It was regarding your rhs and lhs experiment and number philosophy constructing the foundation for your entire argument:
Q-"But the problem with this that if infinity did equal a finite number then the nines would have to stop a t some point."

"the next problem with the equation that I had was how he went from 9/10 k to 1-1/10n there is no way that that 9 becomes a 1 thus creating a huge flaw in the equation"

I believe I understand you point more now, but since this is my last post I am not allowed to defend it. Are having trouble seeing how 9/10 = 1 - 1/10
Or 9/10 + 9/100 = 99/100 = 1 - 1/100
Get it yet? :
SUMMARY (to the voters):

My opponent has failed to present an argument that did not rely on adding another decimal on the end of and endless string of numbers. He failed to present mathematical evidence or citations for the form .999.....90. He failed to debunk any of my proofs (listed again below). He has abandoned the premise that he needs to prove that infinity has an end in order to attach something to it. He failed to respond to several minor challenges including 'who is an authority if mathematicians aren't'.

MAIN PROOFS:
PROOF 1:
1/3 = .3333 repeating.
Therefore:
1 = 9/9 = 3/3 = 1/3 + 1/3 + 1/3 = (.333 repeating) times 3 = .999 repeating

PROOF 2:
Let x = .999...
10x = 9.999... (multiplying RHS and LHS by 10)
10x - x = 9.999... - x (subtracting x from both sides)
9x = 9
x = 1
.999... = 1

(My opponents only attack against this was revealed in the last quotes I posted under the debunking section. It returns to the dependency on a 'finite' infinity.)

PROOF 3:
A simple Calculus solution that I hope I have cleared up for my opponent:
http://i237.photobucket.com...

VOTE PRO! XD
No letters lef
astrosfan

Con

Sorry about the grammar in the last few rounds I have had a lot of work to do and not much time to look over my arguments. So in this last round for the sanity of the people reading this I am going to be short and sweet.

1.1/3
There is one critical reason that you are not going to buy my opponent's argument and this he makes no response to my attack that .333… only approximately equals 1/3, thus they only way to find what 1/3 actually equals is to use remainders which when multiplied by the cause one to be added in on the 9's causing them to become 10's and carrying the 1 to the next 9 repeats the cycle to equal 1. This proves that multiplying 1/3 doesn't .999… it will equal 1. This argument goes dropped in my opponent's last speech meaning that there is no way that my opponent can hope to win this argument. The only argument that my opponent even attempts to extend is the example of the hotel, but this doesn't matter for 2 reasons 1 even if there were an % number of rooms there is always one left over meaning that you will never get an exact answer, and 2 you should prefer my example because it take into account for this flaw and is the only way to get an exact answer to the problem.

2.Rhs lhs
My opponent drops too many of my arguments in the last speech for him to have a hope to win this point. First my opponent seem to ignore all of the arguments I make, what I am saying is that even if % is not finite and is constantly increasing then when it is used in a math equation you change the rate that % is increasing, this is supported my example ,that my opponent miss interprets, that if % theoretically for the purpose of easy math equaled to 1 and if you added 1 to that % you would get 2 which acts as a higher power of infinity now even if infinity were to increase without bounds the answer to the equation would always be one above infinity. To explains further in the same situation if % increase to 2 adding one to % would equal 3 they same would happen for 4,5,6,7,8,9 even if infinity was constantly increasing the %+1 will always be bigger. This proves the flaws of the rhs lhs equation because it shows that even if there % number of nines behind the decimal when you multiple it by ten it acts as if there are %-1 9's behind the decimal but it will always be on less than the original .999… thus proving my point. The only other thing my opponent has to support his claim is that there couldn't be a 9 at the end of repeating 0's, but my opponent gives nothing to support his own claim I have proof of how we never able to reach infinity yet we know it is there and the numbers between 0 and % infinity as we attempt to reach it this proves how we could have a 9 at the end of the sequence of 0's because even if we can never reach it we know it is there. All my opponent's attacks are based on nothing he gives you no examples no base for his arguments all he says is that he is right yet I am wrong but I am the only one giving examples and proof to back my claims meaning you are going to prefer my arguments over his. Because these critical failures by my opponent he has no proof backing his rhs lhs theory meaning that there is no way that he can win.

3.His limit equation
On this argument my opponent drops one critical argument and that is that 1/% ~0 this shows that this equation could only prove that .999… can only be approximately equal to 1. Because in the final step of his equation where he takes 1-1/10 the % power, the 1/10to the % will be approximately equal to 0 and when you subtract this from 1 you get approximately equal to 1. This argument goes completely dropped by my opponent meaning that this completely disproves his attempt to prove his point

4.Number line paradox
This is going to be critical in this round because my opponent make no arguments against it. It disproves that .999… could equal 1 because they are located on different position on a number line even though they are very close together they are not in the same place meaning that they can't be equal. Because of the fact that my opponent drops this argument it proves that .999… doesn't equal 1 and is going to be this finishing blow because there is nothing from my opponent that attempts to disprove this meaning it is going to win me this debate.

5.Closing statements
Because this is the last speech I feel that I should reinforce the fact that when as judges vote on this round base your vote only on what has been presented in the round. Weigh the evidence that both sides present and look at the arguments and how each one proves or disproves the topic. Don't base your vote on what you know, but only the information given to you in the round.
Debate Round No. 5
20 comments have been posted on this debate. Showing 1 through 10 records.
Posted by ibstudent220 6 years ago
ibstudent220
Here's another proof that Singularity missed using.

1=.9999....
1=.9+.09+.009+...
1=.9(1+.1+.01+.001+...)
1=(9/10)(10^0+10^-1+10^-2+...)
Note that this is a geometric series, which can be expressed as
1=9/10 [&#8721;(1/10)^i ]

The sum of a geometric series is given as
k[&#8721;(r^i) ]=k/(1-r), for any 0<r<1

Therefore
1=(9/10)/(1-(1/10))
1=(9/10)/(9/10)
1=1
Posted by knick-knack 8 years ago
knick-knack
I read the debate and PRO won...
Posted by Mangani 8 years ago
Mangani
I voted PRO because the facts are on his side, but voted "con" under "most convincing arguments" because he made PRO discredit his own math by making a simple argument complex. The mathematical undisputable fact is that .999... represents .9 with an infinite number of nines behind it- infinity is not a number rather an expression meaning "without end". In that case, .999... equals 1 based on cons own math (disregarding his flawed example using infinity as a finite number). If you are still debating this, if you have any doubt in your mind about this, and if you voted con, you will fail math.
Posted by astrosfan 8 years ago
astrosfan
dude wtf what part of listen to the arguments actually said don't you get. you should at least read the debate before you vote. i mean i could understand vetoing against me if you have a reason from the debate, but just vote without even reading the debate is just dumb
Posted by knick-knack 8 years ago
knick-knack
Another one of these?
This has been done soooooooo many times...
PRO wins cause it's a fact.
Posted by Singularity 8 years ago
Singularity
Hahaha, it's cool.
I assume astrosfan wouldn't mind if you waited to post problems with our arguments after our last round though. ;)
Posted by Lightkeeper 8 years ago
Lightkeeper
Oh me and my big mouth.
I'll shuddup.
Sorry :(
Posted by Singularity 8 years ago
Singularity
Yeah, you saw it before I did, and now I feel hesitant to include it. :P
Posted by Lightkeeper 8 years ago
Lightkeeper
"The law of functions- the problem with my opponent's attack on this is that both 1 and .9999… are in decimal form no if one was in fraction from while the other was in decimal I would agree that they could be in the same location but because the from they in it would prevent them from being in the same location."

Sorry to butt in in the middle of the debate but I'm struggling to see how Con has assumed that "1" is in a decimal form as opposed to fraction form (1/1).....
Posted by Singularity 8 years ago
Singularity
Quote:
"who would tell you it isn't?! .999 repeating is one exactly. you can't get any closer to one without being one exactly or going over."
______
Whoever takes the position of Con...

Quote:
"you see, people who do these kinds of debates can only be doing it for the pure fact that they want to raise their win percentages."
______
A created a new history-free account so that my previous win/lose recored would not be impacted nor provide insight to my debate history.

Quote:
"Please cancel this debate. There is just no way CON can win. It's not even a debate!"
______
Then you should vote based on the integrity of the argument and its logical structure. A person with an inability to argue can be right but still not win and vice versa. The person who took the position of con obviously thought he could win. Can you give me another reason?
11 votes have been placed for this debate. Showing 1 through 10 records.
Vote Placed by bauerdude 8 years ago
bauerdude
SingularityastrosfanTied
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