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# .999... is exactly equal to 1

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after 28 votes the winner is...
Biowza
 Started: 9/14/2008 Category: Science Updated: 5 years ago Status: Post Voting Period Viewed: 18,818 times Debate No: 5388
Debate Rounds (3)

 Pro I'm pretty bored so I was wondering to post a debate to see if anyone disagrees with this. Just to confirm the resolution is as follows: .999... = 1 Definitions .999... refers to .9 with recurring nines 1 refers to the real number, 1 = means is exactly equal to .999... is exactly equal to one I'll let my opponent start.Report this Argument Con hello, to everybody brave enough to read this argument, and thankyou to my opponent for this very entertaining topic. I about started laughing because of how interesting, and simply different of a topic this is, compared to most. I stand in negation to the topic, that .999, reccurring is exactly equal to 1 In it's own, the topic itself contradicts itself, and proves my side correct. this is a topic, that just requires common sense, and I'm positive that all of the voters have it. no matter how long you type it in, .9999999999999999999999999999999999999999999999999999999999999999999999999999.......... no matter how many nines you add to it, it will infinitely be less than 1. you may round the number up to one..... But this topic says it is exactly one. I just proved this wrong, .999...... will NEVER be EXACTLY 1. Vote with me in Negation. ThankyouReport this Argument Pro I'd like to thank my opponent for taking this debate, although admittedly this is not really a topic that is open to debate. There are many mathatical proofs for this, and I will go through one for you so hopefully my opponent and the voting audience can see why they should vote PRO. 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 This is a very simple mathematical proof for my position, which I am sure my opponent will find hard to poke holes through. Nevertheless, I wish him the best of luck.Report this Argument Con There are always and will always be errors in math. We live in an imperfect world. The resolution states that .999 is exactly equal to 1. Using mathmatics, multiplication and suptraction, make it so that it is no longer .999... Using mathematics goes against his resolution, so regardless what mathematics can do to it, .999 is not exactly equal to 1. It will come close to 1, but it never will be 1Report this Argument Pro My opponent has offered little to counter my resolution that '0.999... is exactly equal to 1'. As it stands now, the burdens stand as follows: The task of PRO (myself) is to demonstrate that 0.999... is exactly equal to one. The task of CON (my opponent) is to demonstrate that 0.999... is not exactly equal to one. All my opponent has presented is variations of the phrase 'It doesn't make sense', regardless of the fact that I have shown mathematically that it does make sense. As a guess, I would suggest that my opponent fails to recognise the notion of 0.999... it is not a finite number, it is infinite, it has no end at all. The statement 'it will infinitely be less than one' doesn't make sense because there is no end to this recurring decimal, it does not stop at one particular 9. If it did, then indeed I would be mistaken in my proposal that (as an example) 0.999 is equal to one. Allow me to demonstrate with my previous example, although this time showing why this can only work with recurring nines. x = 0.999 10x = 9.99 10x - x = 9.99 - x 9x = 8.991 x = 0.999 And I am sure from here you can see that indeed 0.999 =/= 1, the same is true for any finite number (except 1 lol), however for infinite recurring decimals it is indeed true that 0.999... is exactly equal to 1. To further expand on my point, allow me to show another example of what I just did. x = 0.9999999 10x = 9.999999 10x - x = 9.999999 - x 9x = 8.9999991 x = 0.9999999 You'll have to forgive me if I forgot a few nines (it was meant to be 7 nines), but the point should be clear by now. The case of 0.999... = 1 is only true for 0.999... Not only that, but the proof also doesn't work for other recurring decimals such as 0.888... x = 0.888... 10x = 8.888... 10x - x = 8.888... - x 9x = 8 x = 8/9 Only for 0.999... does it equal exactly 1. Now, allow me to present a simpler proof of the resolution. I'll do my best to explain each step. 1/3 = 0.333... [Any calculator can confirm this] 3*(1/3) = 3*(0.333...) [brackets added for ease of viewing, both sides are multiplied by three] 3/3 = 0.999... [I bet you can see where this is going] 1 = 0.999... [TADAAAAAA] This proof is much easier for our less mathematically minded voting audience, essentially what it is saying is that one third equals 0.3 recurring, when both sides are multiplied by three, the proof yields 1 = 0.999... some people may be wondering how I can simply multiply an infinitely recurring decimal by a finite number three. The answer is simple, as the 3's in 0.333... never actually end, the multiplication simply consists of making them all nines, of course if it was a finite number I could not do this proof either. It would fail at the very first step. 1/3 = 0.333 [WRONG!] The same proof works for 1/9, for instance 1/9 = 0.111... 9*(1/9) = 9*(0.111...) 9/9 = 0.999... 1 = 0.999... Now my opponent talks bout how 'math isn't perfect' and indeed there are still many paradoxes to be solved, however unless my opponent wishes to disprove rudimentary reasoning and the very basic algebraic processes which are the framework of most mathematical disciplines, then I suggest he attempt to highlight my arithmatic ineptitude. Ladies and Gentlemen, I have thoroughly acheived by burden. The only reasonable vote is PRO. I'll be happy to answer any questions by the voting audience in the comments section. Thank youReport this Argument Con as I stated before, mathematics are full of error. it always has to do with tricking the system. by using the equation, 10x-x, my opponent has tricked the system. it is an illusion saying that it is exactly equal to 1, but all you need is common sense to figure out that it isn't. my opponent tried to confuse you by changing the eqation on you when he mentioned .888 is not equal to 1, but if you use the equation exactly as my opponent did.... x = .999 10x = 9.999 (10x)-(x) = 9 9 is now a whole number, and is an equivalent of one. x = .888 10x = 8.888 (10x)-(x) = 8 8 is now a whole number, and is an equivalent of one. x = .777 10x = 7.777 (10x)-(x) = 7 7 is now a whole number, and is an equivalent of one. voters, my opponent is trying to trick you. he found the mathematic error, and he's trying to turn it on you guys. use your common sense is all I have to say. logically, the correct side to this is that .999... is NOT exactly equal to 1. if you were to try to use any other equation to make .999 reach 1, it is impossible. please vote in negation, this has been a fun topic. THANKS!Report this Argument
98 comments have been posted on this debate. Showing 1 through 10 records.
Posted by Hal314 1 year ago
An interesting website discussing this topic in great detail is at:

Posted by mjs1138 1 year ago
There is a common misconception that .999...is equal to 1, created by proofs such as the one below:

1/3 = .333...

3 (1/3) = 3 (.333...)

1 = .999...

The flaw in this proof is that 1/3 is not actually equal to .333... because division is never completed: part of 1/3 is always in a remainder and never appears in the quotient.
Posted by erik150x 1 year ago
Please excuse spelling below, I posted from my smartphone.
Posted by erik150x 1 year ago
Arguments which say .999... X 10 = 9.999... Are not neccessarly true, I would say wrong.

.99 x 10 = 9.90 and NOT 9.99

But people want to say:
.999... x 10 = 9.999...(9) instead of 9.999...(0)

Now you can say 9.999...(0) doesn't make sense or exist, I'll grant that.
But you also can't throw in an extra ...(9) or in other words 'infinitely small amount' and pretend like you didn't.

1/3 is not .333... It is .333... + ....(1/3000...)
Do the math your self, always a remainder.

Proofs that unvolve geometric series, make use of the limit process which in my mind basically assume 1/infinity = 0. This is the founding principle of calculus, but with fancy words not saying so. Basically calculus says... Let's assume 1/infinity = 0. Which for practical applications may be fine, but it doesn't make it an abdolute truth, you can't use calculus yo prove .999... = 1, because tje assumption is built in.

Posted by Qynze 2 years ago
Hehe...infinity minus 1 is still infinity, I suppose...
Posted by xiaotianZ 3 years ago
hahaha! nice! i like this math debate! as soon as the pro came up with solid evidence of math he won. nice! i love math and this debate gives me the shivers!
Posted by Metz 4 years ago
Ok I realize this is an Old debate but if .999=1 then

.999=99.9%
If a study is taken about whether 1000 people agree or disagree with X and 999 say they agree and one says they do not the numbers look like this: 999/1000 and 1/1000. In decimal Form that is .999 and .001
1=100% and 99.9%=100% as I demonstrated in the above example.... Thus .999 does not always equal 1
Posted by A.Roche 5 years ago
I am sorry, I retract my previous comment and I accept that 0.9 recurring is in fact equal to 1. After actually thinking about it a little, I now concur with the pro. The moral of this story is "think before you speak" haha.

haha, I'd like to thank Logical-Master, Ragnar_Rahl and CoronerPerry for their sharp, albeit humbling comments in response to mine, plus they are correct in this case, so please listen to them!

Quite frankly, I wish comments could be deleted from these debates, as I'd certainly like to delete mine!
Posted by Ragnar_Rahl 5 years ago
"A.Roche, then how come 1/3 multiplied by 3/1 = 1 1/3, as we know is represented by .333333..."

The assumption that .33333... =1/3 is fundamentally the same assumption as that .999999.... = 1. As such, that "proof" is circular, though valid ones exist :D
Posted by Logical-Master 5 years ago
"I am currently studying Electrical Engineering at university, so I know what I'm talking about"

Appeal to Authority. Just because you're studying Electrical Engineering, it does not lead us to conclude that you're an expert or know what you're talking about when it concerns whether or not .999 is equal to 1.
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