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Topic

#130
# 1 and .999 repeating are the same quantity. Exactly equal.

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Finished

The debate is finished. The distribution of the voting points and the winner are presented below.

Winner & statistics

After 2 votes and with 6 points ahead, the winner is...

Mhykiel

Parameters

- Publication date
- Last updated date
- Type
- Standard
- Number of rounds
- 3
- Time for argument
- Three days
- Max argument characters
- 30,000
- Voting period
- Two weeks
- Point system
- Multiple criterions
- Voting system
- Open

1702

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Description

I argue that .999r IS NOT approaching the number 1. Does NOT estimate or round to the number 1. But is in fact the same as the number 1.

By round or estimate to the number one I do not mean the syntactic changing of one number to another. And that any rounding that may occur is no different than rounding 2.000 to 2. They are the same number.interchangeable.

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>Reported Vote: Juubi_Wolf// Mod action: Removed

>Points Awarded:7 points to con

>Reason for Decision: 1=1.000000000

1 does not equal .9999999

>Reason for Mod Action: The voter does not explain any of the points that we awarded. .

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>Reported Vote: Whiteflame // Mod action: Not Removed

>Points Awarded: 3 points for argument

>Reason for Decision: Pro sets up a rather clear equation on which to base his comparison, explaining that by turning each of the fractions he's presented into a decimal, you can find that adding them together leads to a number that is not 1, in spite of the fact that adding those two fractions together does result in 1. The difference is infinitesimally small, but it does exist. He's essentially stating that the number 0.000r is equivalent to 0 for the same reason. While I understand Con's responses regarding the need to round in order to get a real number, I don't think that's necessary when you're comparing what is, effectively, an unmeasurable quantity. That's what Pro is doing with his argument, and while I think he could have defended it better, I don't think just railing against the lack of rounding suffices as a reason for me to vote Con. I do think there are ways to challenge this that involve more complex math, but those aren't presented, leaving me with little choice but to vote Pro.

>Reason for Mod Action: The voter surveys the main arguments and counterarguments, assess the strength of these arguments, and weighs them to produce a result. This meets the basic standard of sufficiency for argument points.

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>Reported Vote: 3RU7AL // Mod action: Removed

>Points Awarded: 1 point to Pro for Conduct

>Reason for Decision: I would call this a tie on arguments, because CON fails to make a clear case, but I'd like to make a case to award points to PRO for conduct.

Round one PRO - "Hello, I appreciate my challenger taking me up this debate. Good luck to you Sir/Ma'am" which is polite.

Round one CON - "...Pro is tricking you..." which is a negative characterization ad hominem strongly suggesting that PRO is intentionally deceptive.

Round one CON - "'I am smart and good at math'" which is not only a bald assertion but also an indirect ad hominem directed at PRO.

Round two PRO - "Thank you Mad for the quick reply." which is polite.

Round two CON - "You're completely deceiving the reader..." which is a negative characterization ad hominem strongly suggesting that PRO is intentionally deceptive.

Round two CON - "Checkmate." which is a rush-to-declare-victory fallacy.

Round three PRO - No positive or negative comments, just arguments.

Round three CON - "Pro concedes..." and "Pro further concedes..." which is another rush-to-declare-victory and by using the term "concedes" falsely suggests that PRO actually conceded the debate.

>Reason for Mod Action: In order to award conduct points, the voter must show that one debater was "excessively rude, profane, or unfair, or broke the debate rules, or forfeited one or more rounds in the debate without reasonable and given cause." The voter listed as evidence of misconduct statements which are not conduct violations. Rush-to-judgement fallacies are faults of logic, not of conduct. Similarly, boasting about one's own abilities is not itself misconduct, unless the voter can contextualize it. The remaining two or three acts of misconduct the voter cites do not rise to the level of "excessive."

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What is so hard in understanding that an infinite amount of zero's equals zero.

1 = 1.0 = 1.00000 = 1.000r

Do you Maths at all?

New arguments on this topic here https://www.debateart.com/debates/146

Ok, here you go -

So, if you take the remainder of 1 - 0.99999(r), lets call it 1/infinityith and you then multiply 1/infinityith by infinity then you end up with a 1 with infinite zeroes behind it. By contrast, if you multiply 0 by infinity you always end up with 0. That's a pretty big difference.

Therefore, 1 =/= 0.99999(r) (without rounding) because 1/infinityith does not equal zero.

I'm not interested in a semantics debate. The statement is an inherently mathematical one. If you are not interested in addressing it within that context, then the conversation is over. Let me know what the case is.

First of all, the debate resolution, which is, once again, "1 and .999 repeating are the same quantity. Exactly equal." says absolutely nothing about mathematics or rules or axioms or authority or popularity. So technically, and I do like to get technical, "1" is only one character and ".999 repeating" is fourteen characters including the space, therefore they are not the same quantity of characters and the resolution is technically defeated.

Second of all, for Mhykiel, if you wanted to use google info, you should have made those references explicit within the actual debate. Appealing to a third party will get you nowhere at this point, either make your own case with logic or I remain unconvinced.

And to drafterman, when you say, "All numbers are "finite" because "infinity" isn't a number. The second number would require an [][]infinite number[][] of digits to represent in written form, but we needn't worry about that because we have the [][]appropriate symbols[][] (r) to account for those infinite digits."

How is that substantively different from the idea that "There is an infinitesimal difference of 0.0000(r)1, which is a non-zero value."?

If we have, as you say, "...the appropriate symbols (r) to account for those infinite digits." it does not follow that the symbol referenced "(r)" could not have a number after it or that we couldn't use some other form of notation to identify the infinitesimal.

I'm also not sure how you can say, "infinite number" right after you say, "infinity is not a number".

"Them's the rules" is missing from the debate resolution.

I would cordially like to invite each of you to present your preferred definitions of "infinitesimal" for further examination.

No the 3 and 6 repeat an infinite amount of times.

Have you googled the resolution? Because I you could benefit from a more solid understanding of fundemental mathimatical concepts such as the difference between rational and irrational numbers. And how number notation define quantities.

"8/9 is finite."

"0.8888(r) is hypothetically infinite."

All numbers are "finite" because "infinity" isn't a number. The second number would require an infinite number of digits to represent in written form, but we needn't worry about that because we have the appropriate symbols (r) to account for those infinite digits.

"While I am willing to grant you they are practically identical, even functionally identical, but they are not perfectly identical."

Incorrect.

"There is an infinitesimal difference of 0.0000(r)1, which is a non-zero value."

It isn't a value. It isn't a number.

"You are making an axiomatic equivocation, which is fine, but logically, this is the same as a bald assertion or an appeal to dogma."

Whatever you call it is part of mathematics and that is the domain in which this resolution is being stated. If you dismiss it, you aren't talking about math anymore and you are off topic.

So when you say in round 1,

"1/3 = .333... Repeating

2/3 = .666... Repeating"

You don't mean implicitly "repeating an infinite number of times"?

Do you in fact more precisely mean "repeating an unknown yet finite number of times"?

8/9 is finite.

0.8888(r) is hypothetically infinite.

The difference is "infinite".

While I am willing to grant you they are practically identical, even functionally identical, but they are not perfectly identical.

There is an infinitesimal difference of 0.0000(r)1, which is a non-zero value.

You are making an axiomatic equivocation, which is fine, but logically, this is the same as a bald assertion or an appeal to dogma.

.888r is not infinite.

It can written as the fraction 8/9

Being writeable as a fraction is one quality that means .888r is not an irrational number.

What is infinite is the representation of the number in decimal form. That's not to say the quantity is infinite.

"While 0.8888(r) may be a very very very very very very close approximation of 8/9, it is not identical."

Then this is the problem. Mathematically 0.8888(r) isn't an approximation of 8/9, it is equal to 0.8888(r) exactly. They are different written representations of the exact same number. If you disagree, then I wonder what you say the answer to 0.8888(r) + 1/9 equals.

Now, I'm not saying you can't construct a form of mathematics that denies this, but it isn't the same mathematics that is in use today. This isn't a "precision" problem or a "flaw" of the decimal system.

Mathematically. 8/9 = 0.8888(r). Literally. Exactly. Precisely.

8/9 is finite.

0.8888(r) is hypothetically infinite.

While 0.8888(r) may be a very very very very very very close approximation of 8/9, it is not identical.

It is virtually identical and in practice, practically identical but not perfectly identical.

This is a limitation (flaw) of the decimal system, not a limitation of common fractions.

If the debate resolution was, "9/9 = 1" then I'm absolutely certain there would be no dispute whatsoever.

Do you assert that 8/9 =/= 0.8888(r) and that 1/9 =/= 0.1111(r)?

8/9 + 1/9 = 1

This was never in dispute when written precisely in this format.

0.8888888888(r) + 0.1111111111111(r) =/= 8/9 + 1/9

The problem identified by PRO (highlighted by the resolution itself) is a limitation (flaw) of the decimal system, not a limitation of common fractions.

This is fundamentally a precision problem.

And what does 8/9 + 1/9 equal?

"...0.8888888888(r) + 0.1111111111111(r) = 0.99999999999999(r)..."

Ok. I'll bite.

What does 0.8888888888(r) + 0.1111111111111(r) equal?

My stated supporting argument directly addresses the debate resolution.

RationalMadman seemed to think that, "Your resolution is impossible because: EITHER We round and get 1 and/or 3. OR We don't round and admit that '7' which is missing in that 0.9recurring * 3 so it can never truly be 3 and thus 3/3 can't be 0.9recurring. Checkmate." closed their case.

However, RationalMadman's argument does not directly address the debate resolution.

The following steel man actually does directly address the debate resolution.

(IFF) 0.8888888888(r) + 0.1111111111111(r) = 0.99999999999999(r) (THEN) 0.99999999999999(r) =/= 1.0000000000(r)

0.99999999999999(r) =/= 1.0000000000(r)

Therefore, "1 and .999 repeating are the same quantity. Exactly equal." is false.

I have no idea why you found my previous comments "annoying and non-productive" since both of these are negative characterizations (ad hominem) and not strictly legitimate objections to my logic.

I really and truly would like to clear up any questions or misunderstandings you or anyone else might have.

Destroying your opponents examples (supporting arguments) alone does not make your case.

That would be like merely blocking your opponent from making a goal.

You actually have to make a goal yourself in order to get a point.

In your own words:

"Focus on the actual debate resolution and not the supporting arguments." Also, I find it funny you accuse me of being a broken record when I hadn't even said it as many times as you had yet.

If you agree that this response is annoying and non-productive, then perhaps we can proceed with an actual discussion of the topic. If you disagree, then I'm afraid I'm going to have to ask you to stick to the debate resolution, please.

It's called a supporting argument. Supporting arguments are actually integral to the very concept of a debate.

(IFF) 0.8888888888(r) + 0.1111111111111(r) = 0.99999999999999(r) (THEN) 0.99999999999999(r) =/= 1.0000000000(r)

0.99999999999999(r) =/= 1.0000000000(r)

Therefore, "1 and .999 repeating are the same quantity. Exactly equal." is false.

Your question is off-topic. You seem to have abandoned the debate resolution. Stick to the debate resolution, please. The debate resolution doesn't say anything about 0.8888888888(r) or 0.1111111111111(r).

0.8888888888(r) + 0.1111111111111(r) = 0.99999999999999(r) =/= 1.0000000000(r)

Therefore, "1 and .999 repeating are the same quantity. Exactly equal." is false.

If you disagree, please present your argument that disproves my argument or in some other way affirms the debate resolution of "1 and .999 repeating are the same quantity. Exactly equal."

Do you for instance, honestly believe that 0.8888888888888(r) + 0.1111111111111(r) = 1.000000000000(r) ?????????????????(r)

I really and truly would like to clear up any questions or misunderstandings you might have, but at this point you sound like a broken record.

Stick to the debate resolution, please.

0.8888888888(r) + 0.1111111111111(r) = 0.99999999999999(r) =/= 1.0000000000(r)

Therefore, "1 and .999 repeating are the same quantity. Exactly equal." is false.

Stick to the debate resolution, please.

0.8888888888(r) + 0.1111111111111(r) =/= 1.0000000000(r)

Therefore, "1 and .999 repeating are the same quantity. Exactly equal." is false.

@drafterman

https://www.debateart.com/debates/146

"Therefore, "1 and .999 repeating are the same quantity. Exactly equal." is false."

Prove it.

"...could one infinitely increase the height of the second rock without ever making the second rock 5 feet tall?"

The realistic and very practical answer is no.

There is a practical limit on how many decimal places you can actually type.

Reality is not infinitely divisible. There is a smallest possible unit of space-time.

This would be the Planck length. - https://duckduckgo.com/?q=Planck+length&atb=v79-2&ia=web

Therefore, "1 and .999 repeating are the same quantity. Exactly equal." is false.

When you say, (IFF) "He's not saying that there's a 1 at the "end". He's saying that it's a mistake to think that there's a 1 at the end." (THEN) how do the "9's" ever change? Why don't they stay exactly 0.9999999999999999(r) if there is no way to convert them into anything else?

When I first read debate, I thought of the classic A =/= A argument. However this seems to be a case of A = A.00000000000(r)0000000000001 which is quantifiably false.

Stick to the debate resolution, please.

Therefore, by PRO's own logic in quote number one, since an infinite set of zeros with a 1 at the end is not exactly and precisely the same as zero, they magically round up to the target value of "1". Because if 1.0 and 0.999(r) were eXactly the same, then the result would obviously be actual zero.

Therefore, "1 and .999 repeating are the same quantity. Exactly equal." is false.

Not to mention 0.8888888888(r) + 0.1111111111111(r) =/= 1.0000000000(r)

This is literally rounding 0.0000(r)0001 to 0.0000(r). PRO stakes their case on rounding. This defeats the debate resolution.

Yes, provided that each incremental increase wasn't equal. In the beginning you may increase the shorter rock's height by 0.0000009 feet to make 4.99999909 feet tall, and next you may increase it by 0.00000009 to make it 4.999999099 feet, and so on. If you do this an infinite number of times you will end up with a rock that is 4.9999990999999999(repeating) feet tall or, expressed differently, as 4.9999991 feet tall.

Question.

Suppose we have two rocks. One rock is 5 feet tall and the other rock is 4.999999 feet tall. If one were to keep increasing the height of the second rock by small enough increments, could one infinitely increase the height of the second rock without ever making the second rock 5 feet tall? :P

The difference between 1 and .999r is not .111r

It's .000... infinity. Never a difference. 2 things with no difference are ergo the same thing.

The round 2 quote you reference - this part - "we are quick to say the answer is an infinite set of zeros then a 1. ie .00000..infinity..somehow ends in a 1. But those zeroes go on for infinity. That singular "1" never appears." - He's not saying that there's a 1 at the "end". He's saying that it's a mistake to think that there's a 1 at the end.

Stick to the debate resolution, please.

And even if I quote from your previous comment, "Pro's case was that 0.999r must be 1 because 1/3 = 0.333r and 2/3 = .666r and 1/3 + 2/3 = 3/3 = 1. It's a pretty weak case but it doesn't have anything to do with rounding." this is also provably false because 0.333(r) and 0.666(r) approach 1/3 and 2/3 respectively but they require rounding.

Since you can never type out infinite anything, anything ending in an infinite sequence must (EITHER) be typed out explicitly (OR) rounded (like pi).

What PRO actually argued was what PRO said.

And I quote from round 2, (1)"We can confirm this because of the additive identity. identity property of addition, which simply states that when you add zero to any number, it equals the number itself. So if the difference between 2 numbers is not zero. They are not equal."

And I quote again from round 2, (2)"If we take 1 and subtract .999 repeating we are quick to say the answer is an infinite set of zeros then a 1. ie .00000..infinity..somehow ends in a 1. But those zeroes go on for infinity. That singular "1" never appears. Making the answer to what is "1" minus ".999 repeating" equal to an infinite set of zeroes."

Therefore, by PRO's own logic in quote number one, since an infinite set of zeros with a 1 at the end is not exactly and precisely the same as zero, they magically round up to the target value of "1". Because if 1.0 and 0.999(r) were eXactly the same, then the result would obviously be actual zero.

This is literally rounding 0.0000(r)0001 to 0.0000(r). PRO stakes their case on rounding. This defeats the debate resolution.

HOwever, I did not give the "arguments" points to CON because CON failed to clearly identify this error in logic.

You also make a good parallel case, and I applaud your excellent steel man, but I am not fabricating a new argument, I have based my critique on CON's actual statements.

When you said, and I quote from your RFD, "Pro correctly pointed out that no rounding was being supposed.", this statement is provably false.

What Pro actually argued was what Pro said. Pro's case was that 0.999r must be 1 because 1/3 = 0.333r and 2/3 = .666r and 1/3 + 2/3 = 3/3 = 1. It's a pretty weak case but it doesn't have anything to do with rounding. Con's attack on it was an obvious strawman. A better attack would have been something like "Pro's argument is based on the premises that 1/3 = 0.333r and 2/3 = 0.666r. These premises haven't been supported by Pro and I challenge them as unsubstantiated. The burden of proof is on Pro to show that these premises are true. If Pro has not met his burden, then Pro's argument fails."

When Death23 makes the bare assertion, "Pro correctly pointed out that no rounding was being supposed." this is provably false.

What PRO actually argues is that the difference between 1.00000(r) and 0.999999(r) is 0.00000(any number)00000010000(r) which is according to PRO "impossible" and therefore virtually equivalent to zero. However, by notating 0.00000(any number)00000010000(r) it would seem to be no more "impossible" than 0.999999(r) or any other "infinite" sequence (like pi).

0.8888888888(r) + 0.1111111111(r) =/= 1.00000(r)

I would call this a tie on arguments, because CON fails to make a clear case, but I'd consider making a case to award points to PRO for conduct.

Round one PRO - "Hello, I appreciate my challenger taking me up this debate. Good luck to you Sir/Ma'am" which is polite.

Round one CON - "...Pro is tricking you..." which is a negative characterization ad hominem strongly suggesting that PRO is intentionally deceptive.

Round one CON - "'I am smart and good at math'" which is not only a bald assertion but also an indirect ad hominem directed at PRO.

Round two PRO - "Thank you Mad for the quick reply." which is polite.

Round two CON - "You're completely deceiving the reader..." which is a negative characterization ad hominem strongly suggesting that PRO is intentionally deceptive.

Round two CON - "Checkmate." which is a rush-to-declare-victory fallacy.

Round three PRO - No positive or negative comments, just arguments.

Round three CON - "Pro concedes..." and "Pro further concedes..." which is another rush-to-declare-victory and by using the term "concedes" falsely suggests that PRO actually conceded the debate.

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>Reported vote: Death23 // Moderator action: NOT removed<

3 points to Pro (arguments). Reasons for voting decision: The instigator's position is a truism. There is no serious debate in mathematics as to whether or not 0.999r = 1. The reason people have a difficult time grasping how 0.999r = 1 has to do with difficulty grasping an infinite series. We are accustomed to the finite and the infinite is something that simply isn't part of our everyday experience, and this is where Pro's position is weakest. Pro merely posits that 1/3 = 0.333r and that 2/3 = 0.666r. This is a bare assertion, but is later supported rather weakly. This assertion is attacked by Con with his rounding argument. However, Con's attack fails because Pro correctly pointed out that no rounding was being supposed. Con's other attack with the 2.9999(r)7 also fails because Pro correctly pointed out that this was a change in terms (i.e. off topic, and this is true - The topic is 0.9999(r).

[*Reason for non-removal*] While the voter does violate the principle of tabula rasa judging by intervening in the debate with their own opinion, they do reference specific arguments in the debate and explain why Pro's counterarguments to Con's case were sufficient to grant Pro the victory in terms of arguments. Thus, the vote is sufficient.

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1 - 0.999(r) = 0.000(r) = 0 ergo 1 = 0.999(r)

There is no 0.000(r) and then a 1 after because there is no "after" with infinity. It doesn't work that way.

therefore an infinite string of 9's can't ever be said to approach 1

Infinity isn't a number. It's an idea. By definition, infinity has no end. Your arguments about having to round (e.g. "0.9 *3 = 2.7

0.99 *3 = 2.97 So if one is to ever conclude that 3/3 = 0.9 recurring there is at some point a '3' that they are ignoring needs to be added on to the '7' in order to ever make this true. ") can be soundly rejected because they're not consistent with the idea of infinity. There's no "at some point". At what point? That point doesn't happen with an infinitely long sequence.