Instigator / Pro
4
1702
rating
574
debates
67.86%
won
Topic
#157

# 1/3 is not actually 0.3r and also 1 doesn't equal 0.9r, the reason the misconception of 1/3=0.3r is accepted by mainstream math is due to a flaw in the decimal number system.

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Finished

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

Winner & statistics
Better arguments
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3
Better sources
2
2
Better legibility
1
1
Better conduct
1
1

After 1 vote and with 3 points ahead, the winner is...

Death23
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Standard
Number of rounds
5
Time for argument
Three days
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30,000
Voting period
One month
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Contender / Con
7
1553
rating
24
debates
56.25%
won
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Round 1
Pro
#1
r=recurring

1/3 is actually unanswerable. If you really are going to be honest about mathematics, you need to admit there are limitations to it. It is literally 100% identical in the impossibility to answer as getting the even-numbered-root of any negative value. What's the square-root or power-of-four-root of a negative number? Exactly.

There are things in math that result in no answer at all. The reason is that if we were to use binary numbers we'd clearer see how '1' can never ever be broken into 3 parts. '101' would repeat infinitely, yes it's the same thing but what happens is the following:

(1/3)*y SHOULD EQUAL y*1/3

So multiply 0.3r by 30 (not 3, 30). Now multiply 1 by 30 and divide it by 3.

When we make the first number a whole number it becomes quite blatant that 9.Something (in this case 9r) is not equal to 10.Something (in this case 0)

The argument that if you have an infinite series of decimal places that happen to be the same number again and again that it's in any way more real or actualised as a number than a series that alters itself after an infinite number of digits is infantile to say the least and here is why:

If you were a child or someone generally immature when it comes to wisdom and logic, you would maybe thing 'ah so because the final number of .3r is the same as the rest it's somehow not equally impossible to ever get as the last number of pi or the last '1' in the answer to 1-0.9r (it is 0.0r1, the 1 being after a series of infinite 0's).

All irrational numbers (that's what 1/3 is) are actually placeholders for a value that never ever could be real. It doesn't matter how many threes you had, it never ever would be a third of 1, it would ALWAYS be less.
Con
#2
In this round I will argue only my case. I will not argue against Pro's case. I will argue against Pro's case in round 2.

The resolution is as follows:

1/3 is not actually 0.3r and also 1 doesn't equal 0.9r, the reason the misconception of 1/3=0.3r is accepted by mainstream math is due to a flaw in the decimal number system.

Little "r" simply means repeating. For example, 0.99999 repeating could be expressed as 0.9r. The resolution contains three separate claims:

A) 1/3 does not equal 0.3r
B) 1 does not equal 0.9r
C) The reason that mainstream mathematicians accept that 0.3r = 1 is due to a flaw in the decimal number system.

The resolution claims A, B and C. In order for me to show that the resolution is false, it isn't necessary for me to show that A, B and C are all false. Rather, showing that just one of those claims - A, B or C - is false is sufficient to disprove the resolution.

My argument is as follows: The resolution is false because claim B is false. (Claim B is that 1 does not equal 0.9r)

Claim B is false because 1, in fact, does equal 0.9r. I offer the following arguments in support of this assertion:

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.

Round 2
Pro
#3
Pro is copying himself so I will do it to myself too. Credit to myself in this debate: https://www.debateart.com/debates/146

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.
0.9r05 (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.

Con
#4
In this round  I will argue only against Pro's round 1 arguments.

In general, Pro argues that the resolution is true because it's paradoxical for the sum of an infinite series number to result in a finite number. Pro supports this argument with the fact that it's not possible to get to infinity and contrasts this with the finite nature of natural numbers.

The paradox presented by Pro is resolvable. It isn't necessary for an infinite series to "get to infinity"; The challenge isn't to "get to infinity" but merely to answer this question - "What would the value of an infinite series be supposing it somehow got to infinity?" The answer to that is calculable in many cases as I have shown in my round 1 arguments. The sum of an infinite series may result in a finite value. This may happen when, for example, the number keeps getting larger but each time it gets larger the additional value is reduced (e.g. 0.9 + 0.09 + 0.009...). In a sense, there is the infinitely getting bigger component but there is also the infinitely getting smaller part. The two infinities work against one another, somewhat canceling each other out and the result is a finite convergence.

Aside from the general argument, Pro makes many baseless claims in his round 1 argument which should be rejected, e.g. -

A) The value of 0.3r is "100% identical in the impossibility to answer as getting the even-numbered-root of any negative value."

B) 1 can never ever be broken into 3 parts

C) it is "quite blatant that [9.9r] ... is not equal to [10.0r]"

D) the contention that there's a true sum for an infinite series of decimals is infantile

E) Straw man argument. "you would maybe thing 'ah so because the final number of .3r is the same as the rest it's somehow not equally impossible to ever get as the last number of pi or the last '1' in the answer to 1-0.9r (it is 0.0r1, the 1 being after a series of infinite 0's)."

I challenge all of these claims and other similar round 1 claims as unsubstantiated.

Round 3
Pro
#5
Con states the following:
It isn't necessary for an infinite series to "get to infinity"; The challenge isn't to "get to infinity" but merely to answer this question - "What would the value of an infinite series be supposing it somehow got to infinity?"
This concedes that the 0.0r1 difference between 0.9r and 1 is a valid value. It also then proves that since 0.3r multiplied by 3 is not 3/3 but rather (2.9r7)/3, that it cannot possibly be a third of 1.0r
Con
#6
In this round I will respond only to Pro's round 2 arguments. In the prior debate, Pro was Con. In the quoted text below, Pro/Con references may be switched, but I have made my responses here consistent with Pro/Con in this debate.

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 Pro'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.
Pro'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, Pro's arguments are fatally flawed.

With the first algebraic argument Pro 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 logically inconsistent with the idea of infinity. By definition, infinity has no end and consequently there is no "after" with something infinite. 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 Pro can't and doesn't happen as it's a logical impossibility.

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')
Pro'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.

Pro's argument here is that 0.9r = 0.0r1. This is not true. This is incorrect as it's logically inconsistent with the idea of infinity. By definition, infinity has no end and consequently there is no "after" with something infinite. There is no 1 after 0.0r because there is no "after" in the case of an infinitely long sequence. There is no "final number". Infinity goes on forever. The hypothetical 1 occurring after 0.0r supposed by Pro can't and doesn't happen as it's a logical impossibility.

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.
Pro's response to my intuitive argument similarly fails. Pro claims that "0.9r5 actually is half way between [0.9r] and 1." This number supposed by Pro - 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. Pro'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 Pro's final argument is a lengthy false dilemma. This form of argument is known to be fallacious. https://en.wikipedia.org/wiki/False_dilemma

Round 4
Pro
#7
In this debate I never ever said that 0.9r5 was the midpoint, fail copy pasting. I correct myself before Con pasted the 'error' because I knew he'd paste it.

0.9r05 is the midpoint as the (05) becomes the '10' to then be the '1' in 0.0r1 which is the difference between 0.9r and 1.0r

Anyway, let's stick to the topic.

The classic argument people make that seems irrefutable isn't the 10x one that Con did but rather the trickier illusion of:

1.3 = 0.3r (which mainstream math, especially that taught at a high-school level considers to be correct)

Based on this lie, it then says that since 0,3r*3 = 0.9r and 1/3*3 = 3/3 that there's a connection to be drawn to make 0.9r = 1.

THIS IS A LIE!!!!!!!

Not the connection, no. I will show the lie:

1/3 is an impossible value. This is something we have to ACCEPT. 1/3 does not exist beyond an idea or symbol we could invent for it like we did for pi. The actual number can never ever be written in full because you'll never get past the 3's to read the 'third of 1' to add onto them.

1 doesn't split into 3 parts because 10 doesn't and to make the '1' you need a split. This is why you can find the midpoint between 0,9r and 1 since 05*2 = 10 to become the 1 difference to turn it into 1 but why you can never ever make a third of one be an actual number. Yes, even a calculator will give you 0.3r as the answer because a calculator is PROGRAMMED BY the people who have this delusion and rely on long-division as their way of approaching a division sum when in fact the way to 'divide' is simply to ask 'is there a number that multiplied by the divisor (in this case 3) gets us to the result of '1'?' THE ANSWER IS NO!!!!!!

1/3 doesn't exist in the real-value number system and Con has refused to address this despite explicitly promising to in an earlier Round.

Con says the following:
It isn't necessary for an infinite series to "get to infinity"; The challenge isn't to "get to infinity" but merely to answer this question - "What would the value of an infinite series be supposing it somehow got to infinity?"
But then Con concedes the following:
This concedes that the 0.0r1 difference between 0.9r and 1 is a valid value. It also then proves that since 0.3r multiplied by 3 is not 3/3 but rather (2.9r7)/3, that it cannot possibly be a third of 1.0r
Because of the following:
The argument that if you have an infinite series of decimal places that happen to be the same number again and again that it's in any way more real or actualised as a number than a series that alters itself after an infinite number of digits is infantile to say the least and here is why:

If you were a child or someone generally immature when it comes to wisdom and logic, you would maybe thing 'ah so because the final number of .3r is the same as the rest it's somehow not equally impossible to ever get as the last number of pi or the last '1' in the answer to 1-0.9r (it is 0.0r1, the 1 being after a series of infinite 0's).

All irrational numbers (that's what 1/3 is) are actually placeholders for a value that never ever could be real. It doesn't matter how many threes you had, it never ever would be a third of 1, it would ALWAYS be less.

Con
#8
In this debate I never ever said that 0.9r5 was the midpoint, fail copy pasting. I correct myself before Con pasted the 'error' because I knew he'd paste it.

0.9r05 is the midpoint as the (05) becomes the '10' to then be the '1' in 0.0r1 which is the difference between 0.9r and 1.0r

Anyway, let's stick to the topic.

The classic argument people make that seems irrefutable isn't the 10x one that Con did but rather the trickier illusion of:

[1/3] = 0.3r (which mainstream math, especially that taught at a high-school level considers to be correct)

Based on this lie, it then says that since 0,3r*3 = 0.9r and 1/3*3 = 3/3 that there's a connection to be drawn to make 0.9r = 1.

THIS IS A LIE!!!!!!!

Not the connection, no. I will show the lie:

1/3 is an impossible value. This is something we have to ACCEPT. 1/3 does not exist beyond an idea or symbol we could invent for it like we did for pi. The actual number can never ever be written in full because you'll never get past the 3's to read the 'third of 1' to add onto them.

1 doesn't split into 3 parts because 10 doesn't and to make the '1' you need a split. This is why you can find the midpoint between 0,9r and 1 since 05*2 = 10 to become the 1 difference to turn it into 1 but why you can never ever make a third of one be an actual number. Yes, even a calculator will give you 0.3r as the answer because a calculator is PROGRAMMED BY the people who have this delusion and rely on long-division as their way of approaching a division sum when in fact the way to 'divide' is simply to ask 'is there a number that multiplied by the divisor (in this case 3) gets us to the result of '1'?' THE ANSWER IS NO!!!!!!

1/3 doesn't exist in the real-value number system and Con has refused to address this despite explicitly promising to in an earlier Round.
Pro is overly emphasizing the importance of 1/3. My case against the resolution has nothing to do with 1/3 and everything to do with 1 = 0.9r. Even if I drop the argument, it doesn't matter. What ultimately matters is that Pro must prove the entirety of the resolution to be true in order to win, not just the 1/3 issue.

Nonetheless, I will address these points. Pro would have us believe that 1/3 is an impossible value, but it isn't. 1/3 can be written in full, easily if you use a different base number system. First, writing 1/3 as 1/3 is writing it in full. Second, supposing you had a base 3 number system (i.e. count like 0 1 2 10 11 12 20 21 22 100...) then you could write 1/3 as 0.1. Alternatively, you could use a base 9 number system (i.e. count like 012345678-10), then 1/3 could be written as 0.3.

Con says the following:
It isn't necessary for an infinite series to "get to infinity"; The challenge isn't to "get to infinity" but merely to answer this question - "What would the value of an infinite series be supposing it somehow got to infinity?"

But then Con concedes the following:
This concedes that the 0.0r1 difference between 0.9r and 1 is a valid value. It also then proves that since 0.3r multiplied by 3 is not 3/3 but rather (2.9r7)/3, that it cannot possibly be a third of 1.0r
Pro states that there was a concession. There wasn't, and no argument was dropped. In round 3, I stated specifically that "In this round I will respond only to Pro's round 2 arguments."

Because of the following:

The argument that if you have an infinite series of decimal places that happen to be the same number again and again that it's in any way more real or actualised as a number than a series that alters itself after an infinite number of digits is infantile to say the least and here is why:

If you were a child or someone generally immature when it comes to wisdom and logic, you would maybe thing 'ah so because the final number of .3r is the same as the rest it's somehow not equally impossible to ever get as the last number of pi or the last '1' in the answer to 1-0.9r (it is 0.0r1, the 1 being after a series of infinite 0's).

All irrational numbers (that's what 1/3 is) are actually placeholders for a value that never ever could be real. It doesn't matter how many threes you had, it never ever would be a third of 1, it would ALWAYS be less.

This is largely a straw man argument, and 1/3 is not an irrational number. 1/3 is a rational number - a real number. The mathematical definition of a rational number is "any number that can be expressed as the quotient or fraction p/q of two integers, a numerator p and a non-zero denominator q." Real numbers are the numbers that can represent distance on a number line. Irrational numbers are all the real numbers which are not rational numbers (e.g. pi). https://en.wikipedia.org/wiki/Rational_number
Round 5
Pro
#9
I said all that had to be said
Con
#10
Pro has dropped the arguments.