The Instigator
sboss18
Pro (for)
Winning
12 Points
The Contender
PowerPikachu21
Con (against)
Losing
1 Points

0.999... is equal to 1

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Post Voting Period
The voting period for this debate has ended.
after 4 votes the winner is...
sboss18
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
Debate Rounds (4)
Comments (22)
Votes (4)

 

sboss18

Pro

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!
PowerPikachu21

Con

I'll accept. I hope you stay on the site, since you look like a nice debater.
Debate Round No. 1
sboss18

Pro

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 (one-third).

  • 1/3 has the repeating decimal representation of 0.333...
  • If you add 1/3 to itself three times, you get: (1/3)+(1/3)+(1/3) = 0.333... + 0.333... + 0.333... = 0.999...
  • However, a number added to itself three times is simply three times that number. i.e. 3(1/3) = 3/3 = 1
  • Therefore, it must hold true that 0.999... = 1

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 literally-infinite different number of ways. 2 is the same as 8/4, which is the same as 1+5-3, 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:

  • Let x = 0.999...
  • 10x = 9.999... (multiply both sides by 10)
  • 10x-x = 9.999... - 0.999... (substitution and subtraction postulates)
  • 9x = 9
  • x = 1
  • Therefore, 0.999... = 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:

http://math.fau.edu...

http://www.purplemath.com...

PowerPikachu21

Con

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:

"10x-x = 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?
Debate Round No. 2
sboss18

Pro

I thank my opponent for his timely and well-rounded response, albet with some mathematical inconsistensies. I`m having a blast discussing this topic.
The crux of my counter-argument 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 infinitely-repeating, neverending sequence of numbers is just that, neverending. It is non-terminating, 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 non-terminating 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 non-terminating 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.
PowerPikachu21

Con

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.)
Debate Round No. 3
sboss18

Pro

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.
PowerPikachu21

Con

Then again, I have Pokepuffs. 11 different flavors, including Lime.
Debate Round No. 4
22 comments have been posted on this debate. Showing 1 through 10 records.
Posted by PowerPikachu21 1 month ago
PowerPikachu21
@NDECD1141 You misread the debate. It's "0.999..."; aka 0 with infinite nines following it.
Posted by NDECD1441 1 month ago
NDECD1441
Wow, I couldnt understand those mathamatical equations. Nice debate however, 0.999 is obvi not 1. 1 is 1, 2 is 2, 0.999 is 0.999. Kind of, in a really indirect way, rigged.
Posted by PowerPikachu21 3 months ago
PowerPikachu21
How about you guys debate this in an actual debate? I kind of want to see that, and I'll be glad to vote the winner.
Posted by sboss18 3 months ago
sboss18
I mentioned it in my first comment.
Posted by DrCereal 3 months ago
DrCereal
Though I understand limiting the debate to the real numbers, I fail to see where you wrote that in the debate, and, specifically, you remark in the opening, "You may use any logic or reasoning you wish...".
Posted by sboss18 3 months ago
sboss18
"Hyperreals" use infinitesimals. That's why I said real numbers only, no hyperreals.
Posted by PowerPikachu21 3 months ago
PowerPikachu21
I'm currently in 9th grade, and haven't heard of Surreal Numbers before, let alone how this disproves 0.999... = 1.
Posted by DrCereal 3 months ago
DrCereal
No, there is. Surreal numbers. Epsilon. You can prove that writing 0.999... explicitly implies it is not equal to 1.
Posted by PowerPikachu21 3 months ago
PowerPikachu21
@DrCereal Maybe if you go to 100+ 9's maybe. But there's no real argument to suggest that 0.999999999... is not equal to 1.
Posted by DrCereal 3 months ago
DrCereal
Yes, when only considering the real numbers, the two are equal, but when you extend the numbers to surreal numbers, they suddenly have different values.
4 votes have been placed for this debate. Showing 1 through 4 records.
Vote Placed by fire_wings 10 months ago
fire_wings
sboss18PowerPikachu21Tied
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Total points awarded:30 
Reasons for voting decision: concession from con
Vote Placed by warren42 10 months ago
warren42
sboss18PowerPikachu21Tied
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Reasons for voting decision: Con concedes
Vote Placed by dsjpk5 10 months ago
dsjpk5
sboss18PowerPikachu21Tied
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Reasons for voting decision: Concession.
Vote Placed by ThinkBig 10 months ago
ThinkBig
sboss18PowerPikachu21Tied
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Total points awarded:31 
Reasons for voting decision: Concession. This is the only time I give the conduct point for good conduct.