The Instigator
Masterful
Pro (for)
Tied
0 Points
The Contender
MagicAintReal
Con (against)
Tied
0 Points

0.999...Does not equal 1

Do you like this debate?NoYes+0
Add this debate to Google Add this debate to Delicious Add this debate to FaceBook Add this debate to Digg  
Vote Here
Pro Tied Con
Who did you agree with before the debate?
Who did you agree with after the debate?
Who had better conduct?
Who had better spelling and grammar?
Who made more convincing arguments?
Who used the most reliable sources?
Reasons for your voting decision - Required
1,000 Characters Remaining
The Voting Period Ends In
09days21hours36minutes36seconds
Voting Style: Open Point System: 7 Point
Started: 6/14/2018 Category: Science
Updated: 2 hours ago Status: Voting Period
Viewed: 336 times Debate No: 115545
Debate Rounds (4)
Comments (40)
Votes (0)

 

Masterful

Pro

Prove that an undefined value does equal a value.
MagicAintReal

Con

Thanks for instigating the debate, Pro.
I accept.
I also noticed a lack of definitions, so I'll supply them to guide the debate.
I request voters use the Opt-In voting standards and moderators remove crappy votes.

*The Resolution Is Not True*

I reject the resolution that 0.999 does not equal 1, because of the algebraic arguments in mathematics to the contrary.


*Algebraic Argument #1*

For this argument x = 0.999...

If we multiply x times 10, we then get 9.999...
So, we can say 10x = 9.999...
Nothing crazy so far, right?

If 10x = 9.999, then 10x is also equal to 9 + 0.999...
Well, 9 + 0.999... is just 9 + x, because x = 0.999...
So, 10x = 9 + x.
Right?

By using algebra and subtracting the x from the right side of the equation, 10x = 9 + x becomes 9x = 9.
From 9x = 9, one divides both sides by 9 and x = 1.

Not bad for a science guy, huh?


*Algebraic Argument #2*

For this argument x = 0.999...

If you take the number 1 and divide it by 3, you then get 0.333...
When you multiply 0.333... times 3 you get 0.999...
Therefore, x/3 = 1/3.
x = 1.


*Conclusion*

It makes sense to reject this resolution, because the claim is that 0.999... does not equal 1 and it demonstrably does.
One can even go to Google's calculator and type in:
0.999999999999 X 2 and the calculator will yield 2.
https://www.google.com...


Definitions

0.999... - denotes the repeating decimal consisting of infinitely many 9s after the decimal point (and one 0 before it) that represents the smallest number no less than all decimal numbers 0.9, 0.99, 0.999, etc. shown to equal 1.
https://en.wikipedia.org......

equal - be the same as in number or amount.
https://en.oxforddictionaries.com...

1 (one) - the first number of the infinite sequence of natural numbers.
https://en.wikipedia.org...
Debate Round No. 1
Masterful

Pro

0.999.. does not equal 1.

The most compelling reason you give is when you divide 1 by 3 to get 0.333..
If you type 0.33333333333333333333 x 3 into googles calculator, you in fact get the answer 1 and not 0.999.. Try it.
This is because Googles calculator rounds up the answer it is given due to 0.999.. not being a correct value. In fact, 0.333.. is not a correct value either, the reason 0.333.. resorts to infinity, is because you can not divide 1 by 3.

Your example of multiplying 0.999.. by 10 to get 9.999.. is a trick. During this trick your sum is 9.999.. - 0.999.. = 9
The problem is, you're taking infinity away from infinity and suggesting it equals 0. This is false, infinity - infinity = undefined. Therefore the answer to 9.999.. - 0.999.. = 9.0/0 [A]

Not only do you misunderstand the current concept of infinity, but you misunderstand that infinity has no place in the current mathematical model.
All mathematics should be able to represent the real world in some aspect. However, we have no examples in the universe of infinity, we only have examples of infinite potential.


[A Infinity minus infinity] https://www.philforhumanity.com...
MagicAintReal

Con

Thanks for that last round Pro.
Pro seems to think that 1/3 isn't a "correct" value.
Oh well, at least Pro had no problems with the definitions.
Therefore, the definitions from my 1st round stand.

*Responding to Pro*

Pro points out:
"If you type 0.33333333333333333333 x 3 into googles calculator, you in fact get the answer 1 and not 0.999.. Try it."

My response:
Yes, this speaks to the fact that 1/3 times 3 gives you both the fraction 1/1 and the decimal 0.999...
Hence why 0.999... in fact equals 1.


Pro asserts:
"In fact, 0.333.. is not a correct value either, the reason 0.333.. resorts to infinity, is because you can not divide 1 by 3."

My response:
Ok, why is the fraction 1/3 not a correct value and why can it be added to, multiplied by, subtracted from, and divided by other numbers even yielding natural numbers?


Pro goes on an infinity rant:
"multiplying 0.999... by 10 to get 9.999... is a trick...you're taking infinity away from infinity and suggesting it equals 0. This is false, infinity - infinity = undefined...Not only do you misunderstand the current concept of infinity, but you misunderstand that infinity has no place in the current mathematical model."

My response:
Let's make some things clear.
First, the Mathematics noun INFINITY.

infinity - a number greater than any assignable quantity or countable number.
https://en.oxforddictionaries.com...

While I agree that infinity is neither a natural or real number, it is treated as a number in mathematics, particularly calculus, and given these definitions, 0.999... is not infinity, because 0.999... is both a real and rational number and is not greater than any countable number.

While I also agree that numbers like 0.333... and 0.999... repeat *infinitely*, they are not the same thing as INFINITY (greater than any countable number) because numbers like 2 and 3 are bigger than both 0.333... and 0.999...

Either way, one can subtract 0.999... from 9 just like one can subtract 0.333... from 3.
What Pro is saying about infinite decimals' inability to be added and subtracted is complete bunk.


*Questions for Pro*

Both of my algebraic arguments stand and if Pro cannot answer all of these following questions, Pro must concede:

1. What's the difference between the product of 3 times 1/3 and the product of 3 times 0.333...?

2. Why is it that we can use arithmetic with a number like 1/3 even though it's an infinitely repeating decimal and, as you call it not a "correct" value?

3. Can you ever divide by 3 without multiplying by 0.333..., why/why not?
(Try dividing 1/3 on a calculator and using that answer the calculator gives, 0.333..., to multiply by any number)

4. Can you see how 3 times 1/3 must necessarily involve an infinite decimal in both the factor and the product?

Pro?
Debate Round No. 2
Masterful

Pro

The basics

I would like to begin this round by going over various assumptions that my opponent has made, I would then like to refute and point out con's poor understanding of mathematics and the concept of infinity.
Let us start by understanding the difference between rational and irrational decimals. Rational decimals repeat the same digit, thus 0.999.. is an rational decimal. Irrational decimals do not repeat the same digit, pi is an example of an irrational decimal (3.14159..)

My opponent has stopped speaking in decimal terms and begun favoring fractions, he fails to realize that irrational numbers, including pi cannot be represented in fractions. This is important to note as this suggests the limitations of fractions to represent an infinite sequence of numbers, which is important to this debate. [A]




0.333.. is defined as a string of 3s that increase in value by one tenth the value of the previous 3, this suggests an increase in value that is by definition infinite.
1/3 does not increase in value and could only be an accurate representation of splitting an object into 3 equal parts if we could split an object into 3 equal parts without losing any value. 0.333.. is different to 1/3 as it suggests we lose 0.00..1 of value in total. [1] [2]
I therefore urge readers to reject my opponents assumptions.


Refutation

"While I agree that infinity is neither a natural or real number, it is treated as a number in mathematics, particularly calculus, and given these definitions, 0.999... is not infinity, because 0.999... is both a real and rational number and is not greater than any countable number."

My opponent here makes numerous contradictions, he even goes so far to suggest that 0.999.. is not an example of infinity as it is not greater than every number. I urge con to prove this by counting from 0.9 to 0.999..

Aside from the abundant contradictions. Con believes infinity is only infinity, because it's bigger than any other countable number. Laughable.
Infinity is not larger than infinity +1 Therefore it's reasonable to assert an infinity can exist in a value lesser than 1, in this instance , in the form of 0.999.. [P1]

It is not possible to subtract specifically a rational infinite decimal from anything, as the rule for rational infinite decimals involves an increase in value by one tenth of the previous digit, the answer to the subtraction would therefore have to be a decrease in value by one tenth of the previous digit, giving us a number that is an irrational number. Thus one could not assert 0.999..-0.333..=0.666..
This is the fatal flaw in my opponents argument. The value of infinite = undefined. [3]


Answers to my opponents question:

1: Has been marked as [1]
2: 1/3 is a fraction and has no properties that repeat. This has been marked under [2] for a full break down.
3: Has been marked under [3] Calculators have been programmed to round up or round down. Therefore your argument here is weak.
4: You're using fractions here, which as I've explained are different to using the base 10 decimal system. The problem is, the decimal system is known as base 10, which limits certain mathematical equations to represent values through 0 to 9. Using fractions gives us more consistent values. To get an infinite value is essentially to get an error, the error is a tiny one that doesn't ruin the base 10 system entirely, but shows it's not perfect. This doesn't mean we should assume 0.999..=1.


If con cannot answer these questions, con must concede.
I have very clearly answered all my opponents questions, shown his ignorance on the concept of infinity and I now present him with questions of my own that he must answer.

1: 0.999.. follows a pattern, whereby we begin with 0.9, does that equal 1? Does 0.9999 equal 1? Where exactly along the infinite string of 9s do we find the value equal to 1?

If you cannot identify that point, then you can't ever truly prove your claim to be true, unless you use tricks that rely on misconceptions of how the properties of infinity work.

2: When you consider the diminishing value of the 9s within 0.999.. becoming so infinity small, can you tell me what's the difference between a value being so infinity small and simply not existing?

If you can't tell me the difference, then we can assert that infinities have no place in mathematics.

3: As we both agree that infinity is not a number, how is it that in one of your algebraic equations, you minus a number with an infinite string from another number of an infinite string to get something other than undefined?

I fully expect your only counter argument to involve a semantics based argument, whereby you attempt to separate the word infinity from infinite. I've already disproven this. Please refer to my argument labeled [P1]

Con?

[A] https://www.mathsisfun.com...




Both sides have agreed that Djksp cannot vote on this debate.
MagicAintReal

Con

Though not very direct, thanks for your last round, Pro.
Pro either dodged or spun answers to my questions.
Figures.
Well, let's remedy all of that garbage.

*Questions From Round 2*

1. I had asked Pro what the difference was between the product of 3 times 1/3 and the product of 3 times 0.333...

Pro responds:
"0.333.. is defined as a string of 3s that increase in value...1/3 does not increase in value...0.333...is different to 1/3 as it suggests we lose 0.00..1 of value in total."

My response:
Opt-in voters, pay attention.
I had asked about the difference between the PRODUCTS of 3 times 1/3 and 3 times 0.333... and Pro responds with how 1/3 and 0.333... have different values (take any calculator and divide 1 by 3 and you will ALWAYS get 0.333... they are the same) without ever mentioning the products of the two sets of factors I asked about.

By not directly responding to the question posed in the 2nd round, Pro concedes that the products of those two sets of factors are in fact equal thereby equating 1/3 and 0.333...


2. I had also asked Pro WHY we can use arithmetic with infinitely repeating decimals, like 0.333... (1/3).

Pro responds:
"1/3 is a fraction and has no properties that repeat."

My response:
Wow!
1/3 means "one divided by three."
You cannot divide 1 by 3 without "properties that repeat."
You do know that a fraction is just the numerator divided by the denominator, right?
Therefore, 1/3 = 0.333...
Also, this "response" does not address why it is that we can use infinite decimals in arithmetic.
Boo Pro.


3. I had asked Pro if you can ever DIVIDE by 3 without MULTIPLYING by 0.333...

Pro responds:
"It is not possible to subtract specifically a rational infinite decimal from anything."

My response:
Readers, I had asked about DIVISION and MULTIPLICATION, not subtraction, obvious dodge by Pro, and though irrelevant to my question, Pro's wrong here too, infinite decimals can be subtracted from numbers.
https://www.dpmms.cam.ac.uk...

Pro adds:
"Calculators have been programmed to round up or round down."

My response:
Red herring.
Readers, take a calculator and divide 1 by 3.
Take THAT answer and multiply it by 10.

Now, take 10 and divide by 3.
The answers are identical AND not rounded!

Boom.
Multiplying by 0.333... is the same as multiplying by 1/3.


4. I directly asked Pro if he could see how 3 times 1/3 must necessarily involve an infinite decimal in both the factor and the product.

Pro responds:
"You're using fractions here, which as I've explained are different to using the base 10 decimal system."

My response:
All fractions equal a rational decimal, like 0.333... or 0.999... and "decimals are rightfully restored as fractions of a special kind..."
https://math.berkeley.edu...


Pro asks:
"Where exactly along the infinite string of 9s do we find the value equal to 1?"

My response:
When you subtract a number from itself, the result is zero.
For instance, 4 " 4 = 0.
So what is the result when you subtract 0.999... from 1? Without an infinite decimal, you get:

1.0 - 0.9 = 0.1

1.00 - 0.99 = 0.01

1.000 - 0.999 = 0.001

1.0000 - 0.9999 = 0.0001

1.00000 - 0.99999 = 0.00001

Then what about 1.000... " 0.999...?
You'll get an infinite string of zeroes.
What about that '1' at the end?
Ah, 0.999... is an infinite decimal; there is no "end", and thus there is no "1 at the end".
The zeroes go on forever. And 0.000... = 0.
http://www.purplemath.com...


*Conclusion*

Man that was a thorough refute.
How's that for only 30 minutes?

Voters, 0.999... = 1 and Pro has done nothing to show otherwise...just a lot of dodging facts.
Debate Round No. 3
Masterful

Pro

My contender and his flawed mathematics.

This section regards cons slippery strategy, where he focuses on an equation that is simply untrue.

Con responds to my question 1.
"When you subtract a number from itself, the result is zero."

This claim of his is relevant to his single equation regarding 9.999..-0.999..=9 I shall prove this equation to be untrue.

We've already agreed that infinity is not a number, so when 0.999.. has a string of 9s that go on for infinity (without end, to be infinite) how can we then, be sure of the value of this 'number' we can't.
Therefore 9.999...-0.999.. cannot equal 9. This sum relies on taking away a number with a decimal value of infinity from a number with a decimal value of infinity. The actual answer to this equation is, 9.0/0 (Nine Undefined) I will now prove this mathematically.



Let's prove that infinity - infinity is not 0 as con has assumed.

First, we assume the claim to be true,

∞-∞=0

Next I will add a 1 to both sides of the equation.

∞-∞+1=0+1

Since ∞+1=∞ and 0+1=1 we will simplify both sides of the equation.

We get ∞-∞=1

Following my opponents logic, I am able to make infinity-infinity= any number I desire, thus proving his equation to be flawed and nothing more than a trick. To further solidify my argument I will prove this using a second mathematical equation.

Let's assume ∞-∞=0

Since we know that ∞ = ∞ + ∞, then if we substitute this equation into the first infinity in the equation above, we get:

(∞+∞)-∞=0 which is the same as ∞+∞-∞=0

Since we already assumed ∞ - ∞ = 0, then we can substitute to this:

∞+0=0 This simplifies to ∞=0

This is clearly incorrect and proves that ∞=Undefined.

I urge my opponent to attempt to prove otherwise, or else he cannot win this debate.

Con does not respond to my question 2

Con does not respond to my question 3

In response to cons misunderstandings.

It seems my opponents arguments weighs heavily on calculators agreeing with him.
Being able to prove that a calculator doesn't round up, is something my opponent is unable to do, therefore perusing this argument is pointless and would only show the limitations of calculators, rather than, the limitation of his own calcualtions.

It's clear that con has misunderstood that there is a need for infinite decimals in the base 10 system, due to the base 10 system not being a perfect system. Mathematics is a reflection of the real world, the real world is not a reflection of mathematics, so just because we haven't devised a perfect mathematical system yet, without the need for infinite decimals to cover its flaws, we shouldn't assume such axioms to be true.

Despite the fact that I've answered all of cons questions in my previous rounds, con has failed to answer 2 key questions I had for him. I've also proven his equations to be false and he failed to show any understanding of the concept of infinity, he has in fact, resorted to using semantics in order to tackle the various issues infinity presents. Therefore con has lost this debate.

I urge voters to not lose track of the important arguments made, and overlook the fallacious arguments by con has presented.
I would like to thank con for participating and for remaining stubborn in the face of overwhelming odds. May his final argument be better than his last.

Djksp cannot vote on this debate due to been a vote spammer.
MagicAintReal

Con

Well, this has been fun, hasn't it?
Pro either can't or purposefully won't answer my questions.
That's fine.
Opt-in voters will take note and vote Con.

*Review*

Aside from clearly showing that 9 + 0.999... = 10 (0.999...) thereby making 0.999... = 1, I provided a very quick, easy example of this fact with 1/3.

When you divide 1 by 3 you MUST get 0.333...
When you divide 0.999... by 3 you also get 0.333...

When you multiply ANY NUMBER by 1/3 you get the same answer if you multiply that same number by 0.333..., so 3 (1/3) = 3 (0.333...) or another way we could say it is 3 (0.999.../0.333...) = 1.

The number 1/3 basically destroys Pro's case by its mere existence as a rational decimal that repeats infinitely and can be used in standard arithmetic.

I also pointed out that,
1 - 0.9 = 0.1
1 - 0.99 = 0.01
1 - 0.999 - 0.001
but that when you attempt this same subtraction with the infinite decimal 0.999..., there's no LAST 9 and therefore, in the difference, there is no last 1 so 1 - 0.999... = 0.000...
Zeroes to infinity is still equal to zero.


*Responding to Pro*

While I know that Pro cannot respond to this, Pro has said, within this debate, that 1/3 and 0.333... aren't equal or the same thing, so I need to respond to Pro's assaults on the truth.

Pro lies:
"We've already agreed that infinity is not a number."

My response:
No, readers can check the debate, I never once agreed to that and, given the uncontested-therefore-no-reason-to-reject definition of infinity from the 2nd round, INFINITY IS A NUMBER!

infinity - a NUMBER greater than any assignable quantity or countable number.
https://en.oxforddictionaries.com...

Pro didn't question the credibility of the source, Oxford Dictionaries, and made no attempt to demonstrate why/how the definition is wrong...he just asserts it.


Pro conflates:
"when 0.999.. has a string of 9s that go on for infinity."

My response:
Simply because something goes on infinitely does not make it infinity.
If this were the case, then all rational decimals that repeat infinitely would have the same value, infinity, and multiplying by 1/3 (0.333...) would be the same as multiplying by 2/3 (0.666...).
Pro would have you believe that there is no difference between 1/3 and 2/3 because both equal infinity.
What a garbage argument, I mean, even for Pro.


Pro lies again:
"Let's prove that infinity - infinity is not 0 as con has assumed."

My response:
I have not once mentioned this in the debate, so Pro cannot say "Con has assumed," because I never mentioned this at all.
Check the debate.
Pro's following infinity - infinity argument is a straw man argument, an argument that was never constructed by me.
Also, 0.999... ISN'T INFINITY!!!
So Pro's "proof" is really irrelevant.


Pro perceives:
"It seems my opponents arguments weighs heavily on calculators agreeing with him."

My response:
Ugh.
While trusting a calculator over Pro is CERTAINLY a good idea, even if calculators did not exist, 1/3 would be a RATIONAL DECIMAL THAT DOES NOT EQUAL INFINITY and I urge EVERYONE to divide 1 by 3 without a calculator and see if you get to 0.333...
Man, that was another garbage argument from Pro!


Pro errors:
"It's clear that con has misunderstood that there is a need for infinite decimals in the base 10 system."

My response:
I've already said that 0.999... and 0.333... are INFINITE decimals; they're just not INFINITY.
I also understand their role within the decimal system, hence 1/3 = 0.333...


Pro lies:
"Despite the fact that I've answered all of cons questions in my previous rounds."

My response:
Yeah like how Pro answered the question "What's the difference between the product of 3 times 1/3 and the product of 3 times 0.333...?"
Oh wait, he never answered it AND I've already indicated his lack of response to it, so voters should take note...twice.


Pro buries himself:
"I urge voters to not lose track of the important arguments made."

My response:
And that should do it voters!
Do not lose track of all relevant, resolution-impacting arguments made and give a proper opt-in vote.


Thanks for the debate, Pro.
Thanks for the win opt-in voters.
Debate Round No. 4
40 comments have been posted on this debate. Showing 1 through 10 records.
Posted by Masterful 1 day ago
Masterful
It's true
Posted by MagicAintReal 1 day ago
MagicAintReal
hahahah, you're disgraceful.
Posted by Masterful 2 days ago
Masterful
Do you think that using my current profile picture is a good strategy? I think it's perfect.
-Looking like a mathematician gives the voter more confidence in my arguments.
-Being black helps one to gain votes, just ask Obama.
Posted by MagicAintReal 2 days ago
MagicAintReal
"May his final argument be better than his last."
Classic.
Posted by MagicAintReal 5 days ago
MagicAintReal
I guess...
Posted by Masterful 5 days ago
Masterful
Not bad for half and hours worth of work hu?
Posted by MagicAintReal 5 days ago
MagicAintReal
Props to you completely owning your delusion.
Posted by Masterful 6 days ago
Masterful
Quote me.
Posted by MagicAintReal 1 week ago
MagicAintReal
Oh you will see Mr. 1/3 isn't a real number.
Posted by Masterful 1 week ago
Masterful
https://en.oxforddictionaries.com...
"A number greater than any assignable quantity or countable number (symbol W34;)" That's the definition you're playing with. Here is an example of the definition in use.

Take the link then click more sentence examples.
""As classically conceived, a real number can be thought of as an infinite decimal, a completed infinity.""

I think you're trying to suggest that 'infinity' and to be 'infinite' are completely different. The above quote would disagree with you.

Would you kindly like to post your debate round now?
No votes have been placed for this debate.