0.999... is equal to 1
Voting Style:  Open  Point System:  7 Point  
Started:  9/13/2016  Category:  Education  
Updated:  10 months ago  Status:  Post Voting Period  
Viewed:  889 times  Debate No:  95363 
New to the website, thought I`d give a topic I`m pretty familiar with a try for my first debate. Accepting this debate means you will argue that the two values 0.999... and 1 are not numerically equivalent (i.e. they have different values). I am arguing they have the exact same value, no rounding required. You may use any logic or reasoning you wish, the rules are very lax for this debate.
Definitions: = (Equal) : Having the same value 0.999... : a repeating sequence of nines, read as“zero point nine repeating,” “zero point nine reoccuring,” etc. Round 1: Accepting debate Round 2: Pro and Con make their opening statements and arguments, providing proofs to support their claims if able. Round 3: Pro and Con rebut each other`s claims, providing any new information/evidence to the contrary if necessary. Round 4: Conclusions (any final and closing statements) Happy debating! I'll accept. I hope you stay on the site, since you look like a nice debater. 

I thank my contendor PowerPikachu21 for accepting this debate, and look forward to hearing his points!
I will boil my explanation for the two numbers 0.999... and 1 being equivalent down into three components; Intuition, Philosophy, and Mathematical Proof. If at any point I use faulty mathematics, I strongly encourage my opponent to call me out on it. Point 1: Intuition
Arguably the least intensive approach in understanding why these two numbers have to have the same value is through pure mathematical intuition. I`m unsure how easy it is to express numerical equations on here, but I will do my best. We will start by taking the decimal representation of the number 1/3 (onethird).
In other words, if you take three thirds of something, you have one whole. You could construct the exact same argument by taking nine copies of 1/9 with (obviously) the same result that 0.999... = 1.
Point 2: Philosophy
By definition, in order for two numbers to have different value, there must exist an infinite amount of numbers between them. For example, 1 and 2 are not equivalent since 1.5 exists between them. For the same reason, 1 and 1.5 are not equivalent, 1.5 and 2 are not equivalent, etc. However, since there exists no number between 0.999... and 1, it must logically follow that they are the same number.
Numbers which have the same value can be represented in a literallyinfinite different number of ways. 2 is the same as 8/4, which is the same as 1+53, which is the same as the infinite sum ,etc. 0.999... and 1 are simply two different ways to express the same value.
Part 3: Mathematical Proof I will use the two most popular proofs in explaining why 0.999... and 1 must be equivalent. Proof #1:
Proof #2: 0.999... can be expressed as the infinite sum
This is a geometric sum which can be simplified using the equation . In other words, we get . This simplifies to 1, again proving the two values` equivalency.
Sources:
Even though I'm taking a tough stance, I won't forfeit. I have an idea that will help me in this argument. Rebuttal: Intuition: In short, 1/3 (0.333...) x 3 = 1 (Or 0.999...). Let's check a bit closer. 1/3 x 3 = 3/3. This is 3 thirds, or in other words, 1. 0.333... x 3 = 0.999... Let's cross these equations and see if they make sense: If 1/3 = 0.333..., then: 0.333... x 3 = 1. 0.333... + 0.333... + 0.333... = 0.999... Not exactly 1. Let's try the other combination and see if it makes sense. If 0.333... = 1/3, then: 1/3 x 3 = 0.999... We got 3 whole thirds, so they should equal 1, not 0.999... We have conflicting equations. There are 2 ways to explain this: Either 1/3 is not 0.333... or 1 = 0.999. My opponent then states the same thing with 1/9. Well, I've tried to convert 1/18 and 1/36 into decimals to attempt to refute this point, they're infinitely repeating. Philosophy: Since there's no number between 0.999... and 1, they must be the same. I can't really refute that with a counter example... Math: "10xx = 9.999...  0.999... (substitution and subtraction postulates)" I see. Subtracting one tenth. I actually agree with this method... Argument: *sigh* I'm forced to concede on a lot of Pro's points. However, I've come up with my own argument showing that 1 isn't exactly 0.999... My method is math, of course. 1 x 2 = 2 0.999... x 2 = 1.999999999....8 The "8" is not a typo. Multiplying 0.9999... x 2 would (eventually...) have an 8 at the end. No matter how many 9's we write down, it would come out as 1.9999999....8. When multiplying, you start at the end of the number (Example: 17 x 3, we'd first do 3 x 7, than add that to 3 x 10), or at least it's easiest like that. When multiplying 9 x 2, we get 18. Taking that 1 from the tens place, we move on to the next 9, and get 18, adding the forementioned 1, creating 19. Taking the 1 from the tens place, we move on to the next 9, get 19, and repeat the process. Since the number is 1.99999....8, there's a number between it, and 2; 1.99999... Under the philosophical law, no number must exist between two numbers in order for them to be the same. Since 1 x 2 = 2, and 0.999... x 2 = 1.99999....8, and 1.9999... exists between 2 and 1.99999....8, 2 and 1.999...8 aren't the same number. To assume the factors (1 and 0.999...) are the same, despite having different multiples, wouldn't make much sense. I admit I'm unable to refute my opponent's arguments. However, my own argument has a point. So what say you, Pro? Are you going to refute my arument? 

I thank my opponent for his timely and wellrounded response, albet with some mathematical inconsistensies. I`m having a blast discussing this topic.
The crux of my counterargument for Round 2 in refuting Con`s points will be the irrefutable fact that if you take 1 away from infinity, you still have infinity.
Looking back at my Proof #1 under Part 3: Mathematical Proof, we notice under step 2 that when multiplying 0.999... by 10, we are obviously left with 9.999... . In other words, taking a single 9 away from an infinite amount of them still leaves you with an infinite amount of nines. I will reference this later.
Now, on to the rebuttal!
"If 0.333... = 1/3, then: 1/3 x 3 = 0.999... We got 3 whole thirds, so they should equal 1, not 0.999..."
There is no reason to believe this disproves the notion that 0.999... and 1 are the same number. If they were (and are), then there would be nothing wrong with these simple arithmetic points. Saying "I believe they aren`t the same because they look different" isn`t sound reasoning in math.
"Since there's no number between 0.999... and 1, they must be the same. I can't really refute that with a counter example..."
I agree.
"1 x 2 = 2"
Yes.
"0.999... x 2 = 1.999999999....8" I can see why you would think this. I will try my best to explain why this is also false.
An infinitelyrepeating, neverending sequence of numbers is just that, neverending. It is nonterminating, meaning there is no "last" nine. The number 1.999...998 has a terminating number of nines if you can slap an 8 at the end of it and be done. If you have a "..." between two numbers, then that ellipsis cannot represent a nonterminating sequence because it is bound by two other numbers. If can only represent infinity when it is not followed by anything else. When you multiply 2 by 0.999..., you actually get 1.999... . The nines still never end. Another way to think of this is by using another nonterminating rational number.
2/3 = 0.666...
(2/3) x 3 = 1.999... , or in other words, 2.
HOWEVER, when you look at the terminating decimal 0.666 for example, and multiply it by 3, you do in fact get 1.998. That`s because in that case the sixes terminate, or they end. 2/3 in decimal form does not.
The problem with a lot of people`s conception about infinity is they treat it like a number. If you do this, however, mathematics starts to break apart. For example:
∞ + 1 = ∞
Therefore, by subtracting ∞ from both sides, 1 = 0.
Infinity is a concept, it is not a number.
People also view a sequence like 0.999... by looking at each individual number being written out one at a time, and so it must have a finishing point. This is also obviously not a good way to look at the number, or to look at the concept of infinity. There is not enough space in the universe to contain all these flipping nines, there`s literally infinitely many of them. There is no last 9, ergo, there is no 8 to terminate the sequence when doubling the number.
My final rebuttal is to look at one other point you made.
"Since the number is 1.99999....8, there's a number between it, and 2; 1.99999... Under the philosophical law, no number must exist between two numbers in order for them to be the same."
What happens when you take 1.999... and divide it by 2, then? Obviously, you get 0.999... , which is the same as 1.
I hope I was able to convince my opponent that these two number are in fact equivalent.
Please vote Pro. Thank you.
Yeah, you got me with the infinity part. I honestly don't think there's any possible way to argue out of this. I concede, vote Pro. (Though, you can't really use math with infinity.) 

I thank my contender for debating me on this topic. I`m glad we were able to reach an agreement. I would like to make one final point that you can absolutely use infinity in mathematics, you just can`t treat it as a number. Infinity is used all the time in equations involving limits and sums. As my opponent stated, vote Pro. 
sboss18  PowerPikachu21  Tied  

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sboss18  PowerPikachu21  Tied  

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sboss18  PowerPikachu21  Tied  

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sboss18  PowerPikachu21  Tied  

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