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I just found this debate, it amuses me, so I thought I'd bring it here and test the intelligence of all you CNCNZers.

The theory: 0.999... = 1

Their proof (people actually believe this) :

• If 1/3 = 0.333..., multiply each side by three. Wouldn't 3/3 = 0.999...? In other words, wouldn't 1 = 0.999...?

• Some believe that if there is no possible number to "be between" another, then they equal each other. In other words, since it is not possible to fit a number between 0.999... and 1, they are technically equivalent.

• Take 1/3. You will get 0.333..., and multiply it by three. It's kinda like the first step, only you are using a formula of repetition. It should equal 0.999..., and since, if you complete a formula and try to cover it up by succeeding with it again, it should be equivalent with the same number as before.

Then one guy wrote this 'proof' which they thought was 'very nice':

let x = 0.999...

so 10x = 9.999...

subtract x from both sides:

9x = 9

divide:

x = 1

Therefore, 0.999... = 1

You could also prove this using subtraction:

1 - 0.999... = 0.000...

because the zeroes are infinitely repeating, there is no place to put the 1 in so therefore, the answer would be 0 (just like how 1 - 1 = 0)

These people are very entertaining because they are so very stupid while appearing to be smart to those even stupider yet and are actually convincing others they are right in their claims. I've never heard this mathematical theory before, but if it was invented by a mathematician, he should be stripped of all respect and burned at the stake.

Anyway, I want to see how fast you people can debunk these horribly faulted assertions. Or if you believe them, I want to laugh at you too. That's horrible... infinite decimals =/= equivilance

Maths is all about being precise.

This is not.

What an argument, it's simple maths!

The problem here though is the decimal place. The whole argument completely ignores it, and that frustrates me. That'd be like saying that (during a marathon) an athlete who collapses from exhaustion (and dies) just meters from the finishing line 2 minutes ahead of everyone else should win because it would be kinda to his living memory. I'd say tough s**t! The dead bloke came last/DNF, he run himself into the ground... end of story.

If such a theory was to exist as viable. Then we as a species should all be pretty much dead from poor calculation.

This mind boggling stuff is so full of BS.

This is one of those things people will discuss when they have simply run out of topics to talk while trying so hard to strike a decent conversation. It'll get entertaining and keep each other occupied for quite a while, after that you just feel its all BS. Meh. good stuff.

You might laugh at me but I seriously don't understand this one. Mind****: I guess it has something to do with the top one being just a tad bigger than the bottom one somehow.'

I can't see where else the space should come from.

EDIT: sorry for going off-topic or whatever, but seriously though, 1 = 1 end of discussion.

Edited by Stevie_K

Does anyone have a proof to invalidate it?

Does anyone have a proof to invalidate it?

What? That 1=0.999 or that 1=/=0.999? I can disprove every single one of those 'proofs' I posted or I wouldn't have posted them. If no one else feels like putting up proofs (or is too ashamed to admit they can't) I'll post them in about 16 hours or so.

Oh yes, as for the triangle puzzle...

I guess it has something to do with the top one being just a tad bigger than the bottom one somehow.'

Yes, it is. The top one's slope bends inward just slightly, while the bottom one bumps outward. It is too small of a difference to be immediately visible, but it is enough to add up to the area of one square. You can see it best on the square two up, six to the right, upper-left corner, compare.

http://qntm.org/?pointnine

I rest my case.

0.999... repeating equals 1. There is no counterproof, and if you have one, it is wrong.

http://qntm.org/?pointnine

I rest my case.

Pretty much the same arguments I already posted.

0.999... repeating equals 1. There is no counterproof, and if you have one, it is wrong.

I didn't expect you to be on that side on this one... that means this will probably turn into a lengthy, unending (irony) debate...  0.999... repeating equals 1. There is no counterproof, and if you have one, it is wrong.

no... EDIT: well I don't know if it's counterproof or not, really...

I didn't expect you to be on that side on this one... that means this will probably turn into a lengthy, unending (irony) debate... no...

This should be a more correct way of saying it anyways.

(image file, ImageShack won't verify my e-mail at the moment)

Edited by Stevie_K

http://qntm.org/?pointnine

I rest my case.

0.999... repeating equals 1. There is no counterproof, and if you have one, it is wrong.

Thanks DD. The counterproof is the fact that 0.9999... etc. will forever continue to get closer to 1 as you read more and more decimal places however it will never actually reach it... kinda like an asymtope... (i think)

How does one number get closer to another number if one number only represents one value?

Edited by sith_wampa

How does one number get closer to another number if one number only represents one value?

as you read more and more decimal places
He was talking about the number as a progression into infinity.

I believe you meant 'asymptote' though, BioBen.

I will explain how all of their proofs are invalid in five hours. Don't worry, not one of my counter-proofs have been brought up either in this discussion or the posted link.

However, even if some of you refuse to accept my proofs (as is inevitable), I am not coming back here again. Although I started it, I do not find mathematics to be an interesting enough topic to debate it, especially pitted against the hot-tempered one.

He was talking about the number as a progression into infinity.

Yes

I believe you meant 'asymptote' though, BioBen.

That too.

One value cannot approach another value. 0.999... is one value. Just because the value has infinitely many decimal places does not mean it approaches another value. Would you also say that 0.111... or 0.222... approaches 1?

0.999... is a number just like any other number, be it whole, real, integer, irrational, whatever. It represents one value, it's value does not increase just because you decide to look further to the right.

First of all, I would like to point out the biggest faulty logic used through most of their ‘proofs’, however, I will address each proof in addition to this.

Their problem is the misunderstanding of infinity. ∞ is not an abstract number, but an absolute, comparable with zero because zero is, in fact, infinite nothingness. Zero, however, is one half of ∞’s absolutes, the other is absolute everything. There is no ½ ∞, there is no 2 ∞. You cannot have multiples of ∞, it can only be multiplied by 1 or 0. Why? Because ∞ either is, or is not. It is absolute.

Them: Yes you can!

Me: No, you can’t.

Them: Yes you can times a hundred!

Me: No you can’t times a million.

Them: Yes you can times infinity!

Me: No you can’t, infinity times infinity.

Them: Hey, you can’t multiply infinity times... oh... wait...

Me: Exactly.

Now then, first ‘proof’:

If 1/3 = 0.333..., multiply each side by three. Wouldn't 3/3 = 0.999...? In other words, wouldn't 1 = 0.999...?

No, you cannot multiply ∞ times 3.

But you can multiply 1/3 times 3 and 1/3 = 0.333…

Yes, but you can’t multiply ∞. I’ll explain, as this is an easy concept to miss as you are looking at it finitely.

0.333 * 0.333 = 0.999

0.333… * 0.333… = Impossible, you cannot multiply ∞, but let’s ignore that for a moment.

0.333… * 0.333… = 1

What the @#&%???

When multiplying 0.333… by 3, you will keep getting repeating 9’s, however, look over to the next decimal, what is that? Why, it’s another 3! The problem can never be completed because there will always be another 3 to be multiplied (the answer will continue to be 9 forever, but is not the final answer because there will always still be one more three to be multiplied).

Looks like 0.999… to me...

That’s because you are stuck looking at ∞ as if it were a finite number, there is no end to the 3’s all equaling 9, which appears to equal 999…, but as long as there are still 3’s to be multiplied, the problem is not solved. You are looking at the infinite answer while ignoring that it is an infinite problem.

Then how do you solve 0.333… * 3?

Like this:

Given: 1/3 = 0.333…

Problem: 0.333… * 3

Solution: 0.333 1/3 * 3 (removed ∞, inserted the accurate amount of 1/3)

Answer: 0.333 1/3 * 3 = 1 (0.333 1/3 * 3 = 0.999 3/3 = 1)

You’re substituting fractions for decimals!!!

Irrelevant. They are equal, we’ve already agreed on that.

But if 1/3 and 0.333… are equal, you are saying your 1/3 can be replaced with 0.333…!

If you do, you will have 0.333… * 3 = 1, which I've already stated. Aside from that, the reason you don't use decimals is because you cannot have multiples of ∞

Second ‘proof’:

Some believe that if there is no possible number to "be between" another, then they equal each other. In other words, since it is not possible to fit a number between 0.999... and 1, they are technically equivalent.

This is made up rule based on cycling logic. The only numbers this rule would apply to would be any ending with 999…. So the rule is verified by the ‘proof’ that 1=0.999…, but then the rule is used to prove that 1=0.999…. I will not debate cycling logic as it is already wrong to begin with.

Third ‘proof’:

Take 1/3. You will get 0.333..., and multiply it by three. It's kinda like the first step, only you are using a formula of repetition. It should equal 0.999..., and since, if you complete a formula and try to cover it up by succeeding with it again, it should be equivalent with the same number as before.

This is using the same faulty solution as the first ‘proof’ and therefore, is incorrect for the same reason.

0.333… * 3 ≠ 0.999…

0.333… * 3 = Invalid, you cannot have multiples of ∞

0.333… * 3 = 1 (pretending you can have multiples of ∞)

Forth ‘proof’:

let x = 0.999…

so 10x = 9.999…

subtract x from both sides:

9x = 9

divide:

x = 1

Therefore, 0.999… = 1

Let’s see you debunk that! Numbers don’t lie!

People do, and this is actually an eye trick from the finite perspective.

x = 0.999… (Ok)

so 10x = 9.999… (Invalid, you cannot have multiples of ∞)

Quit with the “Invalid, you cannot have multiples of ∞” argument already!

Then we must treat ∞ as a number, let’s try that again.

x = 0.999… (Ok)

so 10x = 9.999… (Error, where are all the rest of the ∞s?)

What do you mean “∞s”? There is only one ∞ there!

You mean to say you cannot have multiples of ∞? Where have I heard that before?

Ok, so treat it as a number...

x = 0.999… (Ok)

so 10x = 9.999 ∞∞∞∞∞∞∞∞∞∞ (There are now 10 ∞s)

subtract x from both sides:

9x = 9.999 ∞∞∞∞∞∞∞∞∞ (There are now 9 ∞s)

divide:

x = 0.999 ∞ (There is only 1 ∞ left)

Therefore, 0.999… = 0.999… (Good job! You can match identical numbers!)

Don’t patronize me!

Why not, you disproved yourself?

You changed the numbers around, I know the original still equals 1.

I had to change them, they didn’t make sense, now watch how I prove 256 = 1:

256 ≠ 1

Multiply by 0:

0 = 0

1 = 1, therefore, 256 = 1.

You multiplied by zero... you can’t do that to prove something...

That’s because zero is an absolute, it is …000.000…, it is ∞. As an absolute nothing cannot be multiplied to prove my point, an absolute something cannot be multiplied to prove your point, because... (here it comes) you cannot have multiples of ∞.

Fifth ‘proof’:

1 - 0.999… = 0.000…

because the zeroes are infinitely repeating, there is no place to put the 1 in so therefore, the answer would be 0 (just like how 1 - 1 = 0)

Key words here: there is no place to put the 1. So you acknowledge there is supposed to be a 1?

There is no 1, because there is no end to ∞...

So you acknowledge there is supposed to be a 1?

Whatever, so there’s an imaginary 1, you still can’t put it anywhere.

The 1 is as imaginary as the ‘fact’ that it does not exist. I do not argue the one can go on to the end of infinity, obviously it cannot, but you also cannot ignore its existence. You are rounding 0.000…1 to 0.000… because 0.000…1 isn’t a real number. However, just as true as it is not a real number, as there is no end to infinity, it is also true that 0 ≠ 0.000…1 no matter how you slice it.

Now I'm going to defend BioBen's arguement because he is actually right.

Draw a grid, the bottom value is 0.9, the top value is 1.0. Your starting point is 0.9, then you go up to 0.99, then 0.999.

It is getting pretty small, so we'll make a new grid, 0. followed by 1064 9's is the bottom value, 1.0 is the top value. Your starting point is 0. 1064 9's, then you go up to 0. 1065 9's, then 0. 1066 9's.

It is getting pretty small, so we'll make a new grid, 0. followed by 10512 9's is the bottom value, 1.0 is the top value. Your starting point is 0. 10512 9's, then you go up to 0. 10513 9's, then 0. 10514 9's.

Are we at 1 yet? Let's keep going...

We'll make a new grid, 0. followed by 104096 9's is the bottom value, 1.0 is the top value. Your starting point is 0. 104096 9's, then you go up to 0. 104097 9's, then 0. 104098 9's.

It is getting pretty small, so we'll make a new grid, 0. followed by 101000000000 9's is the bottom value, 1.0 is the top value. Your starting point is 0. 101000000000 9's, then you go up to 0. 101000000001 9's, then 0. 101000000002 9's.

That's odd, we still aren't at one? Can this keep going on forever?

Why, yes, it will, it's called an asymptote. It will never merge with 1 into infinity. Not even when we use 0.999….

Lastly, I'm going to use a quote from the posted link:

"My mate/my dad/my mathematics teacher/Professor Stephen Hawking told me that 0.9999... and 1 were different numbers."

They were wrong. In science, credentials are as worthless as intuition. Proof is everything.

My mate/my dad/my mathematics teacher/Professor Stephen Hawking/Wikipedia told me that 0.999... and 1 were the same numbers.

They were wrong. In science, credentials are as worthless as intuition. Proof is everything.

You don't make sense, Nmenth. There is no counter argument. They are all wrong. Every single one, even yours. If you can't understand the basic principles laid out in the link, then stop now.

0.999... = 1. There is no getting around it. I feel educated... ish.

You don't make sense, Nmenth. There is no counter argument. They are all wrong. Every single one, even yours. If you can't understand the basic principles laid out in the link, then stop now.

0.999... = 1. There is no getting around it.

Your response is not unexpected. That is pretty much exactly what I thought you would come back with and that is why I said I would not participate in a debate over this. You suggest I stop now, and that I will, but not because I don't understand... rather vice versa. It is finished.

I don't have much interest in 1/3 = 0.333... arguments, because the equivalency of this with 0.999... = 1 is so immediate that a failure to understand why the latter is true will surely lead to a dismissal of the former. I address only the following:

Second ‘proof’:

Some believe that if there is no possible number to "be between" another, then they equal each other. In other words, since it is not possible to fit a number between 0.999... and 1, they are technically equivalent.

This is made up rule based on cycling logic. The only numbers this rule would apply to would be any ending with 999…. So the rule is verified by the ‘proof’ that 1=0.999…, but then the rule is used to prove that 1=0.999…. I will not debate cycling logic as it is already wrong to begin with.

I don't understand what you mean by "it applies only to numbers ending in 999...", as it's certainly true for all real numbers. 'Why' it's true is fairly sophisticated, but it should be intuitively obvious that given any two distinct numbers a and b, (a+b )/2 exists and certainly lies between them. Regardless, explicit construction of the real numbers is often ignored (all most people need to know is that it can be done), and instead we simply define them as a system of numbers with certain properties, this being one of them.

Fifth ‘proof’:

1 - 0.999… = 0.000…

because the zeroes are infinitely repeating, there is no place to put the 1 in so therefore, the answer would be 0 (just like how 1 - 1 = 0)

Key words here: there is no place to put the 1. So you acknowledge there is supposed to be a 1?

There is no 1, because there is no end to ∞...

So you acknowledge there is supposed to be a 1?

Whatever, so there’s an imaginary 1, you still can’t put it anywhere.

The 1 is as imaginary as the ‘fact’ that it does not exist. I do not argue the one can go on to the end of infinity, obviously it cannot, but you also cannot ignore its existence. You are rounding 0.000…1 to 0.000… because 0.000…1 isn’t a real number. However, just as true as it is not a real number, as there is no end to infinity, it is also true that 0 ≠ 0.000…1 no matter how you slice it.

The 1, in fact, does not exist. Furthermore, 0.000...1 is not a well-defined number as stands. The only possible definition it could have is the limit point of {0.1, 0.01, 0.001, 0.0001, ... }, but this sequence clearly has 0 as a limit, and limit points are unique, hence 0.000...1 = 0. If you can think of another logical way to define it then by all means go ahead, but "0 followed by an infinite number of 0's, followed by a 1" is, again, not well-defined.

Now I'm going to defend BioBen's arguement because he is actually right.

Draw a grid, the bottom value is 0.9, the top value is 1.0. Your starting point is 0.9, then you go up to 0.99, then 0.999.

It is getting pretty small, so we'll make a new grid, 0. followed by 1064 9's is the bottom value, 1.0 is the top value. Your starting point is 0. 1064 9's, then you go up to 0. 1065 9's, then 0. 1066 9's.

It is getting pretty small, so we'll make a new grid, 0. followed by 10512 9's is the bottom value, 1.0 is the top value. Your starting point is 0. 10512 9's, then you go up to 0. 10513 9's, then 0. 10514 9's.

Are we at 1 yet? Let's keep going...

We'll make a new grid, 0. followed by 104096 9's is the bottom value, 1.0 is the top value. Your starting point is 0. 104096 9's, then you go up to 0. 104097 9's, then 0. 104098 9's.

It is getting pretty small, so we'll make a new grid, 0. followed by 101000000000 9's is the bottom value, 1.0 is the top value. Your starting point is 0. 101000000000 9's, then you go up to 0. 101000000001 9's, then 0. 101000000002 9's.

That's odd, we still aren't at one? Can this keep going on forever?

Why, yes, it will, it's called an asymptote. It will never merge with 1 into infinity. Not even when we use 0.999….

Indeed, you have observed that the limit of {0.9, 0.99, 0.999, .... } is 1. We define 0.999... to be the limit of this sequence. There is no other possible logical definition; it's certainly not greater than its limit, and it cannot be less than it (the important point here), so they must be equal.

I like this new guy.  yeah that's cute.

no way.. I'm with Nmenth on this one.

Now then, first ‘proof’:

If 1/3 = 0.333..., multiply each side by three. Wouldn't 3/3 = 0.999...? In other words, wouldn't 1 = 0.999...?

No, you cannot multiply ∞ times 3.

But you can multiply 1/3 times 3 and 1/3 = 0.333…

Yes, but you can’t multiply ∞. I’ll explain, as this is an easy concept to miss as you are looking at it finitely.

0.333 * 0.333 = 0.999

0.333… * 0.333… = Impossible, you cannot multiply ∞, but let’s ignore that for a moment.

0.333… * 0.333… = 1

An infinite number of digits does not represent an infinite value.

Given: 1/3 = 0.333…

Problem: 0.333… * 3

Solution: 0.333 1/3 * 3 (removed ∞, inserted the accurate amount of 1/3)

Answer: 0.333 1/3 * 3 = 1 (0.333 1/3 * 3 = 0.999 3/3 = 1)

You’re substituting fractions for decimals!!!

Irrelevant. They are equal, we’ve already agreed on that.

But if 1/3 and 0.333… are equal, you are saying your 1/3 can be replaced with 0.333…!

If you do, you will have 0.333… * 3 = 1, which I've already stated. Aside from that, the reason you don't use decimals is because you cannot have multiples of ∞

I'm not sure what your point here is. Nobody is multiplying infinity. Multiplying an infinite amount of digits is far different from multiplying the value of infinity.

Second ‘proof’:

Some believe that if there is no possible number to "be between" another, then they equal each other. In other words, since it is not possible to fit a number between 0.999... and 1, they are technically equivalent.

This is made up rule based on cycling logic. The only numbers this rule would apply to would be any ending with 999…. So the rule is verified by the ‘proof’ that 1=0.999…, but then the rule is used to prove that 1=0.999…. I will not debate cycling logic as it is already wrong to begin with.

And how exactly is it wrong to begin with? Can you mathematically disprove this assumption?

Third ‘proof’:

Take 1/3. You will get 0.333..., and multiply it by three. It's kinda like the first step, only you are using a formula of repetition. It should equal 0.999..., and since, if you complete a formula and try to cover it up by succeeding with it again, it should be equivalent with the same number as before.

This is using the same faulty solution as the first ‘proof’ and therefore, is incorrect for the same reason.

0.333… * 3 ≠ 0.999…

0.333… * 3 = Invalid, you cannot have multiples of ∞

0.333… * 3 = 1 (pretending you can have multiples of ∞)

Again, multiplying a recurring decimal is not the same as multiplying infinity. 0.999... clearly is not equal to infinity.

Forth ‘proof’:

let x = 0.999…

so 10x = 9.999…

subtract x from both sides:

9x = 9

divide:

x = 1

Therefore, 0.999… = 1

Let’s see you debunk that! Numbers don’t lie!

People do, and this is actually an eye trick from the finite perspective.

x = 0.999… (Ok)

so 10x = 9.999… (Invalid, you cannot have multiples of ∞)

Quit with the “Invalid, you cannot have multiples of ∞” argument already!

Then we must treat ∞ as a number, let’s try that again.

x = 0.999… (Ok)

so 10x = 9.999… (Error, where are all the rest of the ∞s?)

What do you mean “∞s”? There is only one ∞ there!

You mean to say you cannot have multiples of ∞? Where have I heard that before?

Ok, so treat it as a number...

x = 0.999… (Ok)

so 10x = 9.999 ∞∞∞∞∞∞∞∞∞∞ (There are now 10 ∞s)

subtract x from both sides:

9x = 9.999 ∞∞∞∞∞∞∞∞∞ (There are now 9 ∞s)

divide:

x = 0.999 ∞ (There is only 1 ∞ left)

Therefore, 0.999… = 0.999… (Good job! You can match identical numbers!)

You divided 9.999... by ten, not nine.

256 ≠ 1

Multiply by 0:

0 = 0

1 = 1, therefore, 256 = 1.

You multiplied by zero... you can’t do that to prove something...

That’s because zero is an absolute, it is …000.000…, it is ∞. As an absolute nothing cannot be multiplied to prove my point, an absolute something cannot be multiplied to prove your point, because... (here it comes) you cannot have multiples of ∞.

I'm not sure what you're saying here either. This makes no sense.

Fifth ‘proof’:

1 - 0.999… = 0.000…

because the zeroes are infinitely repeating, there is no place to put the 1 in so therefore, the answer would be 0 (just like how 1 - 1 = 0)

Key words here: there is no place to put the 1. So you acknowledge there is supposed to be a 1?

There is no 1, because there is no end to ∞...

So you acknowledge there is supposed to be a 1?

Whatever, so there’s an imaginary 1, you still can’t put it anywhere.

The 1 is as imaginary as the ‘fact’ that it does not exist. I do not argue the one can go on to the end of infinity, obviously it cannot, but you also cannot ignore its existence. You are rounding 0.000…1 to 0.000… because 0.000…1 isn’t a real number. However, just as true as it is not a real number, as there is no end to infinity, it is also true that 0 ≠ 0.000…1 no matter how you slice it.

I'm going to say this even though you may not like it. Can you find any value between 0.000...1 and 0? No, because they are the same value.

Now I'm going to defend BioBen's arguement because he is actually right.

Draw a grid, the bottom value is 0.9, the top value is 1.0. Your starting point is 0.9, then you go up to 0.99, then 0.999.

It is getting pretty small, so we'll make a new grid, 0. followed by 1064 9's is the bottom value, 1.0 is the top value. Your starting point is 0. 1064 9's, then you go up to 0. 1065 9's, then 0. 1066 9's.

It is getting pretty small, so we'll make a new grid, 0. followed by 10512 9's is the bottom value, 1.0 is the top value. Your starting point is 0. 10512 9's, then you go up to 0. 10513 9's, then 0. 10514 9's.

Are we at 1 yet? Let's keep going...

We'll make a new grid, 0. followed by 104096 9's is the bottom value, 1.0 is the top value. Your starting point is 0. 104096 9's, then you go up to 0. 104097 9's, then 0. 104098 9's.

It is getting pretty small, so we'll make a new grid, 0. followed by 101000000000 9's is the bottom value, 1.0 is the top value. Your starting point is 0. 101000000000 9's, then you go up to 0. 101000000001 9's, then 0. 101000000002 9's.

That's odd, we still aren't at one? Can this keep going on forever?

Why, yes, it will, it's called an asymptote. It will never merge with 1 into infinity. Not even when we use 0.999….

Yes, we all know that if you keep looking further to the right you will find more digits, forever. But that does not change the value of what you're looking at. Pi is a constant. It does not change in value when we discover more digits. Same idea with 0.999...

Lastly, I'm going to use a quote from the posted link:

My mate/my dad/my mathematics teacher/Professor Stephen Hawking/Wikipedia told me that 0.999... and 1 were the same numbers.

They were wrong. In science, credentials are as worthless as intuition. Proof is everything.

Sorry Nmenth, I believe you're wrong this time. You have not provided any mathematical statements to disprove 0.999... = 1.

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