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

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Mhykiel
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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
Pro
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

Con
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
Pro

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.

Con
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
Pro
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.
Con
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.