Instigator / Pro
Points: 14

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

Finished

The voting period has ended

After 2 votes the winner is ...
Mhykiel
Debate details
Publication date
Last update
Category
Science
Time for argument
Three days
Voting system
Open voting
Voting period
Two weeks
Point system
Four points
Rating mode
Rated
Characters per argument
30,000
Contender / Con
Points: 8
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.
Round 1
Published:
Hello, I appreciate my challenger taking me up this debate.  Good luck to you Sir/Ma'am

I'll begin first by stating that in Math if the inputs are the same, the operators the same, then the outcome must be identical. This is what establishes Math as a repeatable operation for everyone. If I add 1 plus 1 and get 2.  So will anyone properly performing the same equation.

Take for instance the addition of one third plus two thirds:

1/3 + 2/3 = 1.

A third plus two thirds equals a whole, ie 1.

I can also treat the fractions as division sub problems.

1/3 = .333... Repeating
2/3 = .666... Repeating

Adding them together we arrive at .999 repeating.

The equal signs are the correct signs for the conclusion of the division of the 1 divided by 3 and it's compliment.  Because they are equal we can use the substitutive property to rewrite the first equation. This would be substituting 1/3 with .333r and 2/3 with .666r. Thereby giving us the equation written as:

.333r + .666r = 1

Which we established in the second equation also equaling .999 repeating. Affirming 1 equals .999r

Published:
1/3 + 2/3 = 3/3

3/3 = 1 (you will never ever conclude that 3/3 = 0.9 recurring without the context that Pro is tricking you into taking as valid reasoning).

1/3 when rounded to the nearest thousandth is 0.333
2/3 when rounded to the nearest thousandth is 0.667
So if we do actually round correctly, the answer even with decimals replacing fractions is 0.333 + 0.667 = 1.000 =/= 0.999

What Pro is arguing is that if you never ever rounded the 2/3 to end up with a 7 at the end of it (since you round up 0.6 recurring to end with a 7 no matter what) then you'd never end up with the answer of 1 as opposed to 0,9 recurring as the result of 1/3 + 2/3. This 'I'm so smart' quip made by people who think they are math geniuses fails to admit that if a number is recurring, you never ever could finish writing or delivering the answer in any way at all. What I mean by this is that the millisecond you stop typing '6' and '9' you're betraying yourself as you're rounding the answer and if you round the answer you never ever, ever, ever will get anything but 1.000000 (to whatever decimal point you rounded to).

Now let me give you actual 'I am smart and good at math' sums that make the resolution impossible.

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.

Therefore if we are ever tricked by the formatting of sum to conclude that 1/3 + 2/3 = 3/3, we must remind the one tricking us to remember the '7' that never can end up being a '10' so as to make this answer true.
Round 2
Published:

However your argument is mute in that you changed the terms. At no point did I round 1 divided by 3. Nor asserted that a rounded number was the same as it's original number.

You performed a change of the number 1/3 or .333 repeating when you rounded it. It became a different quantity.

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.

When you rounded .333 repeating to .333 and we subtract them we get .00099999.. repeating 9's. That is not zero so your rounded number is not the same as the actual number 1/3 or .333 repeating.

However..

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.

It's important to note that 2.0000 equals 2.00 is not rounding. It's dropping a place holder (which is a semantic use of "0") not the alteration of the value "two".

So just as 2.0000 equals 2.
And
2.0000 infinite 0's equal 2
Then so does
0.00000 infinite 0's equals 0

Therefore, there is no quantity between 1 and .999 repeating. The difference between them is "zero". Because of this the identity of addition implies that they are the same number, or the same quantity, value and exactly equal to each other. Confirmed again that 1 equals .999 repeating.

Published:
If you really didn't round any number involved in 1/3 + 2/3 then your answer could never ever be typed as a decimal given that you'd have to endlessly type '3333333' infinitely but even more so because you'd be lying to type '666666' and not end it with a 7 since you round the last 6 up to 7 given rules of rounding:

When rounding a number, you first need to ask: what are you rounding it to? Numbers can be rounded to the nearest ten, the nearest hundred, the nearest thousand, and so on.

Consider the number 4,827.

4,827 rounded to the nearest ten is 4,830
4,827 rounded to the nearest hundred is 4,800
4,827 rounded to the nearest thousand is 5,000
All the numbers to the right of the place you are rounding to become zeros. Here are some more examples:

34 rounded to the nearest ten is 30
6,809 rounded to the nearest hundred is 6,800
1,951 rounded to the nearest thousand is 2,000
You're completely deceiving the reader when you imply that 1/3+2/3 = 0.9999(recurring)

The reason is that the answer cannot possible be the same as the answer of (1+2)/3 given that 3/3 is 1.0000(recurring obviously) and that the answer of 0.999recurring when multiplied by 3 has to have a 3 added to it at some point because it will always end with a '97' and that 7 can never be the 10 it needs to be without ignoring it.

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.
Round 3
Published:
I'm not rounding.

1/3 is equal to .3 with the 3 repeating ad nauseam to infinity.

And 2/3 is equal to .6 with an infinite series of 6's to infinity.

I can add them together easily. because the addition never carries to another place. Meaning I can say it equals .9 repeating to infinity.

To say they equal anything else would be mathematically wrong.

The reason the numbers repeat is a side effect of the decimal system. That ten is not divisible by 3. There's no digit that ends the math without remainder.

But just because the numbers are written differently does not mean they are different quantities or different numbers.

16 is equal to 4^2 (4 squared).  Those are 2 different ways of writing the same exact number. They are equal in all senses and completely interchangeable.

In the same token, I've shown that without changing the numbers with rounding (as you have), that there is no difference between .999 repeating and "one". The subtraction of the 2 is equal to "zero".

The two  equations:

1/3 + 2/3
and
.333r + .666r

Are the same quantities added together. Because 1/3 is the equal to .333 repeating. They are 2 different ways of writing the same number.
As is 2/3 equals .666 repeating.

Because they are the same quantities with the same operation "addition" the result must be the same. Just like adding 4 squared to 4 makes 20. or adding 16 to 4 makes 20. Because 4 squared and 16 are the same number.

Therefore 1 and .999 repeating are the same number. Equal and interchangeable as any 2 identical quantities are.
Published:
Pro concedes that without rounding, 0.9 recurring never ever is equal to 1.

Pro further concedes by not proving me wrong that 3/3 can never ever equal 0.9 recurring given that there will be a 3 which the final 97 needs in order to have 0.9recurring*3 equal 3.
--> @Juubi_Wolf
*******************************************************************
>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. .
************************************************************************
--> @whiteflame
*******************************************************************
>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.
************************************************************************
--> @3RU7AL
*******************************************************************
>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."
*********************************************************************
--> @3RU7AL
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?
Instigator
#87
--> @Mhykiel, @drafterman
New arguments on this topic here https://www.debateart.com/debates/146
--> @Mhykiel
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.
--> @3RU7AL
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.
--> @drafterman, @Mhykiel
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.
--> @3RU7AL
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.
Instigator
#82
--> @3RU7AL
"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.
--> @Mhykiel
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"?
--> @drafterman
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.
--> @3RU7AL
.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.
Instigator
#78
--> @3RU7AL
"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.
--> @drafterman
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.
Criterion Pro Tie Con Points
Better arguments 3 points
Better sources 2 points
Better spelling and grammar 1 point
Better conduct 1 point
Reason:
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.
Criterion Pro Tie Con Points
Better arguments 3 points
Better sources 2 points
Better spelling and grammar 1 point
Better conduct 1 point
Reason:
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).