The Instigator
Logic_on_rails
Pro (for)
Winning
62 Points
The Contender
DrStrangeLuv
Con (against)
Losing
52 Points

0.9999 recurring is equal to 1

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Post Voting Period
The voting period for this debate has ended.
after 27 votes the winner is...
Logic_on_rails
Voting Style: Open Point System: 7 Point
Started: 1/19/2011 Category: Education
Updated: 6 years ago Status: Post Voting Period
Viewed: 6,377 times Debate No: 14439
Debate Rounds (3)
Comments (49)
Votes (27)

 

Logic_on_rails

Pro

Intro

The 0.999 recurring is to mean a continuous stream of nines, ever ending. I assume people know what I mean here. The Pro's burden is simple, to prove that 0.99 recurring is indeed equal to 1. I'm looking to see if there's any challengers out there. And before anyone gets at all tricky (sneaky, making new rules), don't get ridiculously tricky, it's also of course in base 10 that we're talking (no base 16 etc.) . Finally not every dash (-) or such is a minus sign or such.

Case

Let's say we have 2 numbers, A and B

Firstly we times both by 10 - we get 10A and 10B
Then we take 1 of the pronumeral from both 10A and 10B - 9A and 9B

Now, if 9A = 9B then it is fair to simplify and say A=B.

Essentially:

A*10 = 10A
B*10 = 10B
10A - A = 9A
10B - B = 9B

If 9A=9B, A=B

Keep this in mind. Now let's do it with the numbers in question.

1*10=10 (A*10)
0.9999... *10 = 9.999... (recurring) (B*10)
10-1=9 (10A-A)
9.999... - 0.999... = 9 (10B-B)

9=9 (9A=9B)

Conclusion - A=B or in this case 0.99999999... (recurring) =1

Is there any person on the website who doesn't believe?


DrStrangeLuv

Con

.999... is not a whole number, it is infinite. You cannot subtract two infinite numbers and get a whole number, which is what you are doing.

Saying [9.999... - .999 = 9] is flawed logic. You are at some point rounding .999... to 1.

In calculus we are taught that an infinite number minus an infinite number will equal an infinite number.

[9.999... - .999] is the same as saying [Infinity - Infinity]
Debate Round No. 1
Logic_on_rails

Pro

Intro

It probably doesn't help that I haven't done calculus, but I can still spy multiple flaws in my opponents logic. Furthermore my bottom argument conforms to the top argument, so by not attacking the top argument (proof if anyone wishes) , my opponent concedes that point, and the bottom is merely an application of the top, rendering it correct. Nevertheless...

My case

1. If we take the opposite method in proving this case 1-0.9999... we see that there is a difference of 0.000... we will never see the one, resulting in a difference of 0.
2. On refuting my opponent the idea of 9.99... - 0.999.. being infinity - infinity is quite wrong. There is clearly a difference of 9. Indeed we times one number by 10 to get the second no.
3. How about adding a negative infinity to positive infinity, that should be 0.
There's a list of other points, but you never need to round - 0.99... - 0.99... is 0, the are the same number.
4. Of course 0.999... isn't a whole number in that form, whole numbers are essentially integers (not quite accurate of course)

What my opponent probably means is something like 0.888... - 0.777... = 0.111... and being unable to simplify 0.111...

Con's case is refuted. Vote Pro.
DrStrangeLuv

Con

If 1 = 0.999... then since 1 - 1/infinity = 0.999..., 1/infinity must be proven to equal zero. Zero, when multiplied by anything, must equal itself (that is, zero). though 22 multiplied by 1/infinity equals 1/infinity (because 22/infinity equals 1/infinity), infinity multiplied by 1/infinity yields one. Zero multiplied by infinity yields zero, not one, so 0 does not equal 1/infinity so 1 does not equal 0.999...
Debate Round No. 2
Logic_on_rails

Pro

Intro

Con still hasn't attacked my main proof, granting it valid. And since the second try is based on the first, attacking the second is pointless. Also, I'm not sure if layout comes under any category of voting, but I'd like voters to realise the incoherence of my opponent's arguments. Without ado, let's proceed.

My Case

1. The point is still unaddressed. To ever assume a difference is to assume a finite number of 9s, but there are an infinite number of 9s.
2-4 - Not countered, not will to waste time on them.

New points:

1. If two numbers are different it should be possible to find another number in between the 2 numbers, but as I'm sure voters can deduce, this is not the case.
2. The number 0.9999.... can be expanded as:

0.9999... = 0.9+0.09....
ie. = 9/10 + 9/100....
= 9/10 + 9/10(1/10)....
=9/10 + 9/10(1/10)^1 +9/10(1/10)^2....

In other words:

0.9999... = (9/10)(1/(1-1/10))
ie. = (9/10)(1/(9/10))
= (9/10)(10/9) =1

Voters, the resolution has been adequately proven in multiple ways and through multiple proofs.

I strongly urge a Pro vote.
DrStrangeLuv

Con

I apologize for the vague second round, I wasn't near a computer for a while so I rushed something out when I saw I was near the time limit.

.999... is an infinitesimally small number. This is part of the reason of why limits exist. If you were to draw this on a graph, the .999... would approach the number 1, but never intersect it. It would just get infinitely closer to it.

The proofs that Pro has provided are correct under his interpretation of infinity. But to me 9.999... -.999... cannot compute, its like trying to divide an apple by an orange. To me they are just words now, since infinity cannot be expressed by any simple numerical means like this. It almost seems too easy, if you catch my drift.

For example, if I wanted to subtract .999... from 9.999... I would have to perform subtraction for every number in 9.999... This process would continue into infinity, since the amount of 9's I would have to take into consideration are infinite. For Pro's proof to work means you have to stop subtracting at some point to get an answer of 1, which to me is not possible.

What this debate has come down to is just an interpretation of infinity, which if I am correct, makes Pro's evidence and proofs null and void.
Debate Round No. 3
49 comments have been posted on this debate. Showing 1 through 10 records.
Posted by john_gabriel 4 years ago
john_gabriel
0.999... interpreted as a number is an ill-defined concept.

http://thenewcalculus.weebly.com...
Posted by XimenBao 6 years ago
XimenBao
argument by calculator limitations? really?
Posted by resolutionsmasher 6 years ago
resolutionsmasher
I agree cons argument was simple and short, but that is all that was required to prove his point.
And when debating mathamatics, by giving your own standpoint you are in the same breath refuting the other's case
i.e. - opponent says 2+2=5 so i say 2+2=4, i don't have to add on that 2+2 doesn't = 5, because my preface automatically does that. a simple 2+2=4 will suffice, even if my opponent takes a page and a half to say 2+2=5, my 2+2=4 will still suffice
Posted by resolutionsmasher 6 years ago
resolutionsmasher
but because the winner is somehow voted the loser and the loser the winner, I will throw my vote all one way to bring justice about
Posted by resolutionsmasher 6 years ago
resolutionsmasher
pro... you need to study up on your rational and irrational numbers and how they relate to each other. an irrational number (goes on continually) can only be logically computed in an equation if it can be turned into a rational fraction i.e. - .333333.... becomes 1/3
I cannot find anywhere a rational fraction that when pluged into a scientific calculator and told not round. gives us .99999999.... or 9.999999999999. thus it debunks your argument
Posted by I-am-a-panda 6 years ago
I-am-a-panda
RFD:

Arguments - Pros argument had a more solid basis in Mathematical logic than Con. Pros assertion were not adequately refuted by Con and Con did not present an adequate argument of his own.
Posted by Grape 6 years ago
Grape
It could equal one or the other depending on the domain. If you are using real numbers only then it's 0.
Posted by bodhiBit 6 years ago
bodhiBit
Could someone please explain whether 1 - 0.999... equals zero or infinitesimal..? And why..?

http://en.wikipedia.org...
Posted by Logic_on_rails 6 years ago
Logic_on_rails
Zabcheckmate, I understand that this was already proven. I wanted to test out my ability to format, lay out a case and also see if my opponent would use semantics. But, although I would agree with my stance, clearly there seems to be great confusion over the issue for some. My next debate will definitely be more controversial.
Posted by Zabcheckmate 6 years ago
Zabcheckmate
Wow... http://en.wikipedia.org......

Please don't run tautological topics. Some things are objective true and debating them is absolutely pointless.

RFD: I want to punish Logic for running dumb things... but conversely want to punish Strangelove for accepting this debate. I couldn't decide who deserved to lose more, so I gave a straight ballot.
27 votes have been placed for this debate. Showing 1 through 10 records.
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