Poll: 0.999... = 1

[gl1koz3]

It's not one, it's infinitely close to 1. Christ.

So, on a scale, how would you draw a line that is infinitely close to some other line? Assuming the measure has no width (as the numbers also don't), it would be on that other line. No magic necessary.

 

[Piflik]

If two Real numbers have no other Real numbers between them, they must have the same value.

Every single real number has at least two neighbors that have an infinitesimal distance from it (since the real numbers are a continuum)...with your reasoning this number and his neighbors would have the same value and thus are the same number...continue this and you will end up in a world where every number is equal to any other...

Therefore, 1 and 0.(9) are different notations of the same value. Which has been said countless times, even by you. Why are you still going around in circles?

Stop twisting my words...I never said that 0.999999... and 1 are the same value, I said that there is a difference between 0.9999... and every possible representation of that value with limited amounts of decimals, like 0.9999 or 1. Note the important part,it is bold, just for you.

 

[Piflik]

Ok, one more proof to those still not convinced:

using the a_n = 0.9(n times) notation (a_1 = 0.9, a_2 = 0.99, ...) we have that:

1 - a_n = 1/(10^n).

1 - a_1 = 1/(10^1) = 0.1
1 - a_2 = 1/(10^2) = 0.01
so on and so forth.

so when n->infinity we have:

1 - 0.999... = limit of 1/(10^n) with n->infinity.

And as anyone who took basic calculus can tell you that limit is 0.

therefore

1 - 0.999... = 0


Seriously, it's the same number, different notations... And anyone who tells you different has no business hanging around math.

You cannot interchange a limit with a real value. Yes...the limit for n approaching infinity is 0, but 1/10^n will never reach 0, no matter how far you go...that's why it is called limit or asymptote.

I'm not, the limit IS the real value.

Note that 0.999... will never really be a part of the (a_1...a_n) unless n->infinity. The point being if you don't work with limits 0.999... doesn't make sense.

Sorry, but that is simply not true the limit is not the real value...the limit of 1/10^n for n-> infinity is 0, but 1/10^n will still never reach it.

Other example the limit of 1/n for n -> 0 is infinite, but 1/0 is not defined.

Ok, try again, this time try to read.

0.999... is the limit of a_n when n->infinity and its value is 1, simple as that.

Can you try and understand that limits are not values? If you argue with that one, you would have to define the division by 0...

 

[Athinira]

Is that the proof that I invalidated, because it involves circular reasoning?

Maybe. Invalidate it again then (or link your post where you invalidated it).

Or rather, attempted to invalidate it. The proof can't be invalidated in the real number system, because in the real number system, there is always an average number between two other numbers unless those two numbers are the same. No exceptions.

Here it is: http://www.escapistmagazine.com/forums/read/18.252127-Poll-0-999-1?page=13#9365581

And like i said: You didn't invalidate it.

You invalidated his post, yes, but what you needed to invalidate was a rule in the real number system, more specifically, the rule that states that in the real number system, any number or value must be expressable, either as a value or as a formula.

That is why it's called the Real Number System, because it deals with real numbers. Infinitesimal values are conceptual numbers, that while useable in certain branches of mathematics, isn't allowed in the real number system.

Like i said earlier, thats why the real number system doesn't allow infinite values either (while numbers with infinite decimals still being a different case). An infinite value isn't a Real Number and isn't allowed in the system, and neither is infinitesimal numbers.

Even if his demonstration involves circular reasoning, the formula...
M = Average(A,B) = (A + B) / 2
...must still hold up. The average of two real numbers always produce a real number and can never produce an infinitesimal.

 

[Athinira]

I suck at maths but technically wouldn't 10x (if x is .9~) - x = 8.9999999~?

If 10x = 9.999... and 9x = 8.999... then x = 1 since the difference between 8.9... and 9.9... is is 1, which still proves that 0.999... = 1

 

[Piflik]

Is that the proof that I invalidated, because it involves circular reasoning?

Maybe. Invalidate it again then (or link your post where you invalidated it).

Or rather, attempted to invalidate it. The proof can't be invalidated in the real number system, because in the real number system, there is always an average number between two other numbers unless those two numbers are the same. No exceptions.

Here it is: http://www.escapistmagazine.com/forums/read/18.252127-Poll-0-999-1?page=13#9365581

And like i said: You didn't invalidate it.

You invalidated his post, yes, but what you needed to invalidate was a rule in the real number system, more specifically, the rule that states that in the real number system, any number or value must be expressable, either as a value or as a formula.

That is why it's called the Real Number System, because it deals with real numbers. Infinitesimal values are conceptual numbers, that while useable in certain branches of mathematics, isn't allowed in the real number system.

Like i said earlier, thats why the real number system doesn't allow infinite values either (while numbers with infinite decimals still being a different case). An infinite value isn't a Real Number and isn't allowed in the system, and neither is infinitesimal numbers.

Even if his demonstration involves circular reasoning, the formula...
M = Average(A,B) = (A + B) / 2
...must still hold up. The average of two real numbers always produce a real number and can never produce an infinitesimal.

Real numbers are a continuum, and therefore the difference between two 'neighboring', distinct real numbers is infinitesimal.

 

[grammarye]

Here's the key to the whole thread:

Some people want to define infinity as "infinity plus rounding at the end towards an arbitrarily choosen reference" (how does the number know? Must be the mathematician)

Other people instead think, that infinity means just infinity, and that if one wants to do something on top of it, one needs to do something on top of it.

I have to disagree about this. Only the simple approach of asking this one question is doing that. What is critical to the discussion is the meaning one takes away from the choice you are suggesting.

If this were a simple choice, mathematicians around the world could just pick one, everyone uses it, and everyone speaks the same language. It's not a simple choice.

For two people to communicate effectively they must use the same language & meaning throughout.

So, here's the difficulty:

1/3 done as division, decimal by decimal leads you into an infinite loop of getting 0.3, then 0.33, and so on. Unless you're going to allege that at some point, if we did it enough, we'd get something other than 3, that must hold true.

If that is the case, and we accept that that division is what mathematicians everywhere will call division, and we also have the entirety of algebra agreed upon, such as 2a = b, then:

1 = 0.33... + 0.33... + 0.33...

It has to. You can't have a lossy process in your basic numeric operations. Division of one object into three must be reversed by taking those three objects and merging them back together. 'Something on top' is what exactly?

If that is the case, we must accept that 1/3 = 0.33.. or we haven't even got working division of natural numbers. Back to ye olde drawing board for the entirety of maths. I'm going to naively assume people accept this to be the truth, because otherwise they're effectively saying 'all maths is wrong' - in which case we're no longer speaking the same language and the communication has broken down.

Now lets take addition.

If you take 0.33... and add to it 0.33..., strangely enough you keep getting 0.66... repeating over and over and over.

That inevitably leads to 0.99... = 0.33... + 0.33... + 0.3...

This is, on the face of it, inconsistent. There is only one way to resolve this. 0.99... = 1. Without that, you are essentially saying that the basics of decimal addition & division are incorrect. Mathematics is fundamentally unsound, the entire system needs redoing from scratch.

So, that leaves us with three choices: abandon the decimal system entirely, accept that 0.99... = 1, or invent something better.

I invite you to do the latter, because I doubt people will accept doing the first one.

 

[Coldie]

Every single real number has at least two neighbors that have an infinitesimal distance from it (since the real numbers are a continuum)...with your reasoning this number and his neighbors would have the same value and thus are the same number...continue this and you will end up in a world where every number is equal to any other...

Every pair of Real numbers with different values will have an uncountable, infinite number of Real numbers between them. Sure, the "absolute" distance between 10[sup]-googol[/sup] and 10[sup]-googolplex[/sup] is tiny, but there is still a whole whopping Continuum between them.

How is infinity infinitesimal?

 

[Lyx]

I think you're making something of a leap between people rightly defending basic mathematics, such as what happens when you divide 1 by 3, to science all over the world, as applied by many people, sometimes quite incorrectly or for political reasons.

Quite a leap indeed, though not unfounded (i did not provide the arguments why, and i have no interest to do so, because my intention was simply to state why i am so pissed off about this stuff - not to discuss it.

Gross generalisations tend to not apply very well.

If you read the paragraph again, you will notice that there are almost no quantifiers. Your statement is untrue.

After all, I could get all offended by your implication that I'm a 'science fanboy' - nice assumption there that it's a male-only profession if science were a single profession which it isn't - but I won't.

Notice the phrasing "implication". I never actually said that. And actually, i also never meant it. Your statement is untrue.

I don't care if the theories are quite right - if they produce useful tangible results like mobile phones, why on earth does it matter?

This is where i differ. I seek other things besides of "we can control stuff without understanding why". You know, things like... umm, understanding why. Explanation. A consistent worldview. And finally, sustainability (you can't do shit without concepts and understanding. So if the concepts and understanding doesn't matter, then this is like saying "Who cares about tomorrow? For now, the skyscraper hasn't come down yet."

Stop equating science & the pursuit of it with the results - which are usually entirely independent of science & driven by other factors.

I do not subscribe to the moral, that action and responsibility are isolated from each other. Among other things, it is logically false and untrue. A causal break to be precise. But it certainly is comfortable and popular

Did you know that we still don't have a working model of radio wave propagation that explains how Marconi transmitted across the Atlantic? Should we stop all radio usage until we do? Research & implementation/policy are independent.

Now THAT is quite a leap you're making. Suddenly, that one can do stuff one doesn't understand yet, justifies that one doesn't need to understand (huh!) and justifies lack of responsibility (doublehuh?).

Also I never used the word rule - I believe concept was what I used.

Doesn't change the meaning in the context of what i wrote. If you call a structure of how an operation should be executed a "rule" or a "concept".... who cares? According to current usage of those words, its both anyways.

 

[Piflik]

Here's the key to the whole thread:

Some people want to define infinity as "infinity plus rounding at the end towards an arbitrarily choosen reference" (how does the number know? Must be the mathematician)

Other people instead think, that infinity means just infinity, and that if one wants to do something on top of it, one needs to do something on top of it.

I have to disagree about this. Only the simple approach of asking this one question is doing that. What is critical to the discussion is the meaning one takes away from the choice you are suggesting.

If this were a simple choice, mathematicians around the world could just pick one, everyone uses it, and everyone speaks the same language. It's not a simple choice.

For two people to communicate effectively they must use the same language & meaning throughout.

So, here's the difficulty:

1/3 done as division, decimal by decimal leads you into an infinite loop of getting 0.3, then 0.33, and so on. Unless you're going to allege that at some point, if we did it enough, we'd get something other than 3, that must hold true.

If that is the case, and we accept that that division is what mathematicians everywhere will call division, and we also have the entirety of algebra agreed upon, such as 2a = b, then:

1 = 0.33... + 0.33... + 0.33...

It has to. You can't have a lossy process in your basic numeric operations. Division of one object into three must be reversed by taking those three objects and merging them back together. 'Something on top' is what exactly?

If that is the case, we must accept that 1/3 = 0.33.. or we haven't even got working division of natural numbers. Back to ye olde drawing board for the entirety of maths. I'm going to naively assume people accept this to be the truth, because otherwise they're effectively saying 'all maths is wrong' - in which case we're no longer speaking the same language and the communication has broken down.

Now lets take addition.

If you take 0.33... and add to it 0.33..., strangely enough you keep getting 0.66... repeating over and over and over.

That inevitably leads to 0.99... = 0.33... + 0.33... + 0.3...

This is, on the face of it, inconsistent. There is only one way to resolve this. 0.99... = 1. Without that, you are essentially saying that the basics of decimal addition & division are incorrect. Mathematics is fundamentally unsound, the entire system needs redoing from scratch.

So, that leaves us with three choices: abandon the decimal system entirely, accept that 0.99... = 1, or invent something better.

I invite you to do the latter, because I doubt people will accept doing the first one.

0.333... * 3 =/= 1, since 0.33333... =/= 1/3

0.333... is a flawed representation of 1/3. No matter how far you go, will always be an infinitesimal difference between 1/3 and 0.333...

 

[Vanaron]

snip

They are values, they may not be the values of the function at the point but they are values, that's why we use limits to work with non-continuous functions because they give us an insight of how the function might work on a point where it doesn't exist.

The fact is that infinity is not a number as you may believe it is so there is no such a thing as infinitesimal difference or an infinite number of x, unless we're talking limits.

Like I said: If you actually think 0.999... != 1 you have no business hanging around math.

 

[someonehairy-ish]

If x = 0.999999...
Then 10x = 9.9999...
Therefore, 10x - x = 9
Which implies 9x = 9
Thus, x = 1
x also = 0.99999...

In conclusion, I have just proven 1 = 0.9999...

Doesn't that imply 9x=10/x?

Nope. Where do you get the divide sign from? It shows that 9x = 9 so x = 1

 

[Athinira]

Real numbers are a continuum, and therefore the difference between two 'neighboring', distinct real numbers is infinitesimal.

Now you are the one using circular reasoning. Linear Continuum states that for two values x and y, where x < y, there exists a z such that x < z < y (basically what i said in my last post), and your claim here relies on the up-front assumption that x and y (0.999... and 1) are two different numbers, without having proven it.

In order to prove that there exists an infinitesimal number between 0.999... and 1 you first have to prove that those two are, in fact, different numbers since continuum requires that x < y. You haven't proven that yet, while there has been plenty of proof that shows that the two numbers are the same.

 

[Piflik]

Every single real number has at least two neighbors that have an infinitesimal distance from it (since the real numbers are a continuum)...with your reasoning this number and his neighbors would have the same value and thus are the same number...continue this and you will end up in a world where every number is equal to any other...

Every pair of Real numbers with different values will have an uncountable, infinite number of Real numbers between them. Sure, the "absolute" distance between 10[sup]-googol[/sup] and 10[sup]-googolplex[/sup] is tiny, but there is still a whole whopping Continuum between them.

How is infinity infinitesimal?

10[sup]-googol[/sup] and 10[sup]-googolplex[/sup] are not neighbors...the neighbor of 10[sup]-googol[/sup] would be 10[sup]-googol+10[sup]-googol+10[sup]-googol+10[sup]-googol[/sup][/sup][/sup][/sup]...continue as you see fit, but again you would need infinity to represent it. Since it is a continuum, the difference to the neighbor has to be infinitesimal.

 

[grammarye]

0.333... * 3 =/= 1, since 0.33333... =/= 1/3

0.333... is a flawed representation of 1/3. No matter how far you go, will always be an infinitesimal difference between 1/3 and 0.333...

So I was correct - you are essentially stating that current working knowledge of maths, how we teach division & addition, and all decimal systems are incorrect, flawed, and we should acknowledge this.

That's fine - you can take that standpoint, but lets be clear what it is you're saying.

 

[Staskala]

It is generally accepted among mathematicians that 0.999... = 1.
One "rebellious" college students and people who don't know anything about math in the first place question this FACT.

But hey, how about another question:

0/0 =
0, 1 or &#8734;?

 

[Piflik]

You haven't proven that yet, while there has been plenty of proof that shows that the two numbers are the same.

None of these proofs are acceptable, since they either rely on flawed representation of infinity or circular reasoning.

If you actually think 0.999... != 1 you have no business hanging around math.

Anyone who thinks there is no difference between 0.999... and 1 has no business hanging around math...the thought alone is all kinds of stupid...

 

[Vanaron]

It is generally accepted among mathematicians that 0.999... = 1.
One "rebellious" college students and people who don't know anything about math in the first place question this FACT.

But hey, how about another question:

0/0 =
0, 1 or &#8734;?

You can't divide by 0... not even 0. So that doesn't make sense.

 

[Piflik]

0.333... * 3 =/= 1, since 0.33333... =/= 1/3

0.333... is a flawed representation of 1/3. No matter how far you go, will always be an infinitesimal difference between 1/3 and 0.333...

So I was correct - you are essentially stating that current working knowledge of maths, how we teach division & addition, and all decimal systems are incorrect, flawed, and we should acknowledge this.

That's fine - you can take that standpoint, but lets be clear what it is you're saying.

I say it is flawed and incorrect when we work with infinity (either as a number or as a value), since when we approach the infinitely small or big there happens tons of stuff that basic calculus cannot deal with. The result is the common misconception (I'd almost call it delusion) that 0.99999... = 1.

Also, ever heard of Gödel? Every numerical system is either inconsistent or incomplete (paraphrasing here...), so I'm not the only one arguing that our math is faulty...and in fact will always be.

 

[Coldie]

10[sup]-googol[/sup] and 10[sup]-googolplex[/sup] are not neighbors...the neighbor of 10[sup]-googol[/sup] would be 10[sup]-googol+10[sup]-googol+10[sup]-googol+10[sup]-googol[/sup][/sup][/sup][/sup]...continue as you see fit, but again you would need infinity to represent it. Since it is a continuum, the difference to the neighbor has to be infinitesimal.

Pick two numbers, any two numbers, no matter how close or far they are. If they are different, there's an infinite set of Real numbers between them. There's no "infinitesimally close neighbor" for any number that isn't an infinite set of numbers away (or equal to it anyway).

You either have no idea what you're talking about or you're trolling. And not very good trolling at that.

Anyone who thinks there is no difference between 0.999... and 1 has no business hanging around math...the thought alone is all kinds of stupid...

Indeed you are.