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# 0.999 repeating equals 1

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After not so many votes...

It's a tie!
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Round 1
Pro
Resolution -

0.999 repeating equals 1

Definition -

0.999 repeating means a zero followed by an infinite number of nines after the decimal point.

For purposes of this debate I will express 0.999 repeating simply as 0.9r (the little r means repeating infinitely). This assertion as to the definition isn't prescriptive or a rule of this debate and this definition is itself subject to debate, but I doubt that there's any dispute on this point.

Arguments offered in favor of the resolution -

1. An algebraic argument -

x = 0.9r
10x  = 9.9r       mutiplying both sides by 10
10x = 9 + 0.9r  separating 0.9r and 9
10x = 9 + x       substituting 0.9r with x
9x = 9               by subtracting x from both sides
x = 1                 by dividing both sides by 9

2. Another algebraic argument -

1 - 0.9r = 0.0r = 0 ergo 1 = 0.9r

3. An intuitive argument -

There's no value so close to 1 such that you can't fit another value half way between it and 1. For example, suppose we thought 0.999 was the closest number to 1. We would have thought wrong as 0.9995 is halfway between 0.999 and 1.

If 0.9r were less than 1 then you could fit another value halfway in between 0.9r and 1. For example, 0.9r5. But this can't be done because the 9's go on infinitely. You would have to get past the 9's to fit the 5. You can't get past the 9's.

Con
I notice that Pro forgot to define 'equals' so I will provide that definition by making a hybrid definition based on 2 links.

Being identical with what is about to be or has just been mentioned in quantity, size, degree, or value.

Let's debunk Pro's case now.

x = 0.9r
10x  = 9.9r       mutiplying both sides by 10
10x = 9 + 0.9r  separating 0.9r and 9
10x = 9 + x       substituting 0.9r with x
9x = 9               by subtracting x from both sides
x = 1                 by dividing both sides by 9
Alright, so here is where we will turn Pro's disproof of me against Pro himself/herself.

Pro is denying that the difference between 1 and 0.9r is existent at all. The difference is 0.0r1 (the 1 is going to happen after an infinite number of 0's).

This value: 0.0r1 is argued by people on the side of Pro as being too irrational or far from real consideration because the 1 follows a series of infinite 0's which thus are never going to actually become the 1.

So when Pro multiplies 0.9r by 10 in order to ascertain what 10x (x=0.9r) is, Pro defeats their aforementioned attack on Con and here is why:

To multiply by 10, there must be a 0 placeholding the final number of the number being multiplied by that 10. In this case, that final number is 9. In reality 0.9r is actually able to be written as 0.9r9 because it is a series of infinite 9's with a number 9 at the end just the same in significance as the 1 at the end of 0.0r1 which is the difference between 0.9r and 1 and which Pro is denying is to be considered existent at all. Thus, if we are to multiply 0.9r by 10 either we can't do that since it will actually be 9.9r0 and the 0 will never ever come to be just as the 1 would never come to be in 0.0r1 or we admit that 0.0r1 matters as an actual value and then we concede that the resolution is false.

1 - 0.9r = 0.0r = 0 ergo 1 = 0.9r
This is a lie.

1 - 0.9r = 0.0r1 =/= 0.0r

(=/= means 'is not equal to' and/or 'doesn't equal')

There's no value so close to 1 such that you can't fit another value half way between it and 1. For example, suppose we thought 0.999 was the closest number to 1. We would have thought wrong as 0.9995 is halfway between 0.999 and 1.

If 0.9r were less than 1 then you could fit another value halfway in between 0.9r and 1. For example, 0.9r5. But this can't be done because the 9's go on infinitely. You would have to get past the 9's to fit the 5. You can't get past the 9's.
0.9r5 actually is half way between 0.9r9 and 1. You are forgetting that the 9 at the end of the series is just as impossible to reach as the 5 at the end of those 9's. Either you are denying 0.9r9 ever reaches the last 9 and thus are claiming it doesn't exist and thus can't be equal to 1 which does exist as a number or you admit that 0.9r5 is indeed what you just said we would have thought.

Round 2
Pro

I notice that Pro forgot to define 'equals' so I will provide that definition by making a hybrid definition based on 2 links.

Being identical with what is about to be or has just been mentioned in quantity, size, degree, or value.

I agree with Con's purported definition of "equals" with the reservation that as applied here the appropriate reference would be to quantity or value.

Let's debunk Pro's case now.

x = 0.9r
10x  = 9.9r       mutiplying both sides by 10
10x = 9 + 0.9r  separating 0.9r and 9
10x = 9 + x       substituting 0.9r with x
9x = 9               by subtracting x from both sides
x = 1                 by dividing both sides by 9
Alright, so here is where we will turn Pro's disproof of me against Pro himself/herself.

Pro is denying that the difference between 1 and 0.9r is existent at all. The difference is 0.0r1 (the 1 is going to happen after an infinite number of 0's).

This value: 0.0r1 is argued by people on the side of Pro as being too irrational or far from real consideration because the 1 follows a series of infinite 0's which thus are never going to actually become the 1.

So when Pro multiplies 0.9r by 10 in order to ascertain what 10x (x=0.9r) is, Pro defeats their aforementioned attack on Con and here is why:

To multiply by 10, there must be a 0 placeholding the final number of the number being multiplied by that 10. In this case, that final number is 9. In reality 0.9r is actually able to be written as 0.9r9 because it is a series of infinite 9's with a number 9 at the end just the same in significance as the 1 at the end of 0.0r1 which is the difference between 0.9r and 1 and which Pro is denying is to be considered existent at all. Thus, if we are to multiply 0.9r by 10 either we can't do that since it will actually be 9.9r0 and the 0 will never ever come to be just as the 1 would never come to be in 0.0r1 or we admit that 0.0r1 matters as an actual value and then we concede that the resolution is false.
Con's alleged debunking may be soundly rejected as it's not consistent with the idea of infinity. Infinity isn't a number. It's an idea and by its definition it has no end - And consequently no "after". For that reason, Con's arguments are fatally flawed.

With the first algebraic argument Con claims that multiplying 0.9r by 10 results in 9.9r0 rather than 9.9r and that the algebraic reasoning is flawed as a result. ("there must be a 0 placeholding the final number") This is incorrect as it's inconsistent with the idea of infinity. There is no 0 after 9.9r because there is no "after" in the case of an infinitely long sequence. There is no "final number". Infinity goes on forever. The hypothetical placeholder 0 supposed by Con can't and can't and doesn't happen as it's logically impossible.

1 - 0.9r = 0.0r = 0 ergo 1 = 0.9r
This is a lie.

1 - 0.9r = 0.0r1 =/= 0.0r

(=/= means 'is not equal to' and/or 'doesn't equal')
Con's claim that I lied is baseless. Deception isn't evident here. In fact, there is evidence that I actually do believe what I'm saying. I'll show you - Consider the comments I made in debate 130 - https://www.debateart.com/debates/130 - Those comments predate this debate and indicate a belief that I personally agree with this resolution and specifically my second algebraic argument. In those comments I stated that "1 - 0.999(r) = 0.000(r) = 0 ergo 1 = 0.999(r)". I had no reason to lie there and there's no indication that I've changed my mind in the interim.

There's no value so close to 1 such that you can't fit another value half way between it and 1. For example, suppose we thought 0.999 was the closest number to 1. We would have thought wrong as 0.9995 is halfway between 0.999 and 1.
If 0.9r were less than 1 then you could fit another value halfway in between 0.9r and 1. For example, 0.9r5. But this can't be done because the 9's go on infinitely. You would have to get past the 9's to fit the 5. You can't get past the 9's.
0.9r5 actually is half way between 0.9r9 and 1. You are forgetting that the 9 at the end of the series is just as impossible to reach as the 5 at the end of those 9's. Either you are denying 0.9r9 ever reaches the last 9 and thus are claiming it doesn't exist and thus can't be equal to 1 which does exist as a number or you admit that 0.9r5 is indeed what you just said we would have thought.
Con's response to my intuitive argument similarly fails. Con claims that "0.9r5 actually is half way between [0.9r] and 1." This number supposed by Con - 0.9r5 - is a logical impossibility and can't exist. 0.9r5 requires there to be an "after" with an infinitely long sequence of 9's. There is no after with infinity. Con's claim of equivalence between my arguments and his (e.g. "the 9 at the end of the series is just as impossible to reach as the 5") are false. I never claimed to reach the 9 at the end of the series. There is no end to the series. The end of Con's final argument is a lengthy false dilemma. This form of argument is known to be fallacious. https://en.wikipedia.org/wiki/False_dilemma
Con
Correction: 0.9r05 (not 0.9r5) is the actual midpoint between 0.9r and 1 (I said this in comments section long before Pro posted their Round, check timestamps, this was a blitz-debate where Pro replied in only 2 hours so I replied as quick as I could in the comments about the clarification?). This changes NOTHING about the point I was making but it's wrong to call 0.9r5 the midpoint, I admit that.

0.9r9 is as irrational a number as 0.0r1. Just because the string of infinite 9s ends in a 9 doesn't at all make it less irrational. The number being the same as those preceding it doesn't make it any less attainable, this is an illusion... A Parlour/Parlor Trick if you will.

Either Pro admits 0.9r(ending in a 9) is impossible to be a number and therefore can't equal 1 or Pro concedes that the 0.0r(ending in a 1) is an actual number and therefore the difference between 0,9r and 1 is real.