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:
Thank you Mad for the quick reply.

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.

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.
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.
Added:
--> @3RU7AL
"Therefore, "1 and .999 repeating are the same quantity. Exactly equal." is false."
Prove it.
Added:
--> @Logical-Master
"...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
Added:
--> @drafterman
Therefore, "1 and .999 repeating are the same quantity. Exactly equal." is false.
Added:
--> @Death23
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.
Added:
--> @3RU7AL
Stick to the debate resolution, please.
Added:
--> @drafterman
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)
Added:
--> @Mhykiel
This is literally rounding 0.0000(r)0001 to 0.0000(r). PRO stakes their case on rounding. This defeats the debate resolution.
Added:
--> @Logical-Master
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.
Added:
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
Added:
--> @3RU7AL
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.
Instigator
#51
Added:
--> @3RU7AL
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.
Added:
--> @3RU7AL
Stick to the debate resolution, please.
Added:
--> @Death23
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).
Added:
--> @Death23
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.
Added:
--> @3RU7AL
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."
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).