Poll: 0.999... = 1

[grammarye]

You missed the point...regardless of how many 3s you write (be it infinite, aleph one or more) there will always be a difference to 1/3 and that is precisely the reason why these numbers need infinity to be represented.

On the contrary, it is well accepted that 0.333... = 1/3

Yes...it is accepted, but that doesn't mean it is true...

0.3 =/= 1/3 correct?

let's continue...

0.33 =/= 1/3

0.333 =/= 1/3

0.3333 =/= 1/3

[.....]

0.3333.... =/= 1/3

There is still a difference between any decimal representation and 1/3, that is why it needs infinity to be represented and that's the reason why it is shady. Anything that needs infinity to be represented cannot be right.

No, no there isn't. You are superimposing your viewpoint of a decimal (that it must at some point finish) over the concept that this is a decimal that never finishes and thus cannot ever be precisely defined as a decimal. It's infinite, much as the Universe is and I don't see the Universe ceasing to exist because it's 'not right'.

I get the struggle - really I do - the human mind does not like to encompass concepts that are without limit - but the case remains that as you divide 1 by 3, you will never ever ever reach a point where it is 'finished' and the same holds true for the reverse operation. It must. Where exactly is the leftover bit going to go? Where is it going to hang around? Has 1 mysteriously lost a bit because the mathematical knife cutting the 1 cake has got a bit of sticky edge on it (I hate it when people do that)?

 

[Athinira]

If you want to allow infinity, you also have to allow infinitesimal values, since that is nothing other than 1/infinity.

There is a difference between using numbers with infinite lengths (eg. infinite decimals) and infinite values.

The Real Number system allows the former but not the latter.

Sorry, but when you use numbers with infinite lengths you need infinitesimal values to differentiate between them...

Wrong. Thats the thing about the Real Number system. You can't differentiate because the numbers aren't different in that system and you never need to.

Allow me to explain: Pi is an example of a number that has an infinite amount of decimals (that aren't repeating btw. The sequence is always unique). But Pi can still be measured and expressed, and formulas exist to calculate Pi, even though it's an infinitely long number.

There doesn't exist any way, however, that in the Real Number system allows you to express or formulate the difference between 0.999... and 1 (for the very simple reason that the difference doesn't exist).

Of all the mathematical proofs shown in this thread, someone posted this one which is the most effective counter to your post:
If the two numbers 0.999... and 1 are different, then you should be able to find the average of the two numbers with the following formula.
Average(A,B) = (A+B)/2
In this case: (0.999... + 1) / 2.

Problem is that you can't find an average between two numbers if they represent the same value. The average of 2 and 2 = 2 and the average of 0.999... and 1 is 0.999... and 1.

 

[Piflik]

Sorry, but when you use numbers with infinite lengths you need infinitesimal values to differentiate between them...

No you don't. Select any two Real numbers, infinite length or otherwise. There will be an uncountable infinity of Real numbers between them. In fact, you can make a bijection of the set of Real numbers to any continuous open subset of the Real numbers, like (-inf, inf) (0.9999, 0.99991).

The only infinite-length numbers that look like they need special treatment are of the form x.y(9), as there are no numbers between x.y(9) and x.[y+1].

Spoiler: And guess how that works out.

x.y(9) = x.[y+1]
For example, 0.(9) = 1

Of course you need them...the difference between 0.99999.... and 1 is infinitesimal...and both are real numbers different from each other. That's the whole deal of this thread.

 

[Lyx]

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.

 

[Piflik]

If you want to allow infinity, you also have to allow infinitesimal values, since that is nothing other than 1/infinity.

There is a difference between using numbers with infinite lengths (eg. infinite decimals) and infinite values.

The Real Number system allows the former but not the latter.

Sorry, but when you use numbers with infinite lengths you need infinitesimal values to differentiate between them...

Wrong. Thats the thing about the Real Number system. You can't differentiate because the numbers aren't different in that system and you never need to.

Allow me to explain: Pi is an example of a number that has an infinite amount of decimals (that aren't repeating btw. The sequence is always unique). But Pi can still be measured and expressed, and formulas exist to calculate Pi, even though it's an infinitely long number.

There doesn't exist any way, however, that in the Real Number system allows you to express or formulate the difference between 0.999... and 1 (for the very simple reason that the difference doesn't exist).

Of all the mathematical proofs shown in this thread, someone posted this one which is the most effective counter to your post: If the two numbers 0.999... and 1 are different, then you should be able to find the average of the two numbers with the following formula: Average(A,B) = (A+B)/2 or (0.999... + 1) / 2. Problem is that you can't.

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

 

[imnot]

Numbers scare me
*curls into a ball*
1243445234398400020040932395382023030

 

[Chrinik]

Yeah, numbers with an infinate amount of dezimal places are always equal to a finite number...
Dudes, the problem here is the infinity of the dezimal places, and abusing math logic to state untruth as truth, which has been pointed out already.
Same with the 1/3=0.333... which is false in it´s concept. One cannot be devided by 3 dezimally, because the number of dezimals would be infinate, and the concept of "1/3rd" is not infinate, 1 does not equal 0.999...It merely states that 3/3rds equal 0.999 if 1/3rd is 0.333...note that 1/3rd can be ANY NUMBER, since 1/3rd of 9 is 3...this does not make 3=1...

Also, for your sakes, the revelation of what infinity is has driven people insane.

 

[lvl9000_woot]

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...

This. /thread

 

[Houmand]

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

 

[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.

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

"Infinitely close to" doesn't exist in the Real Number system.

 

[Piflik]

You missed the point...regardless of how many 3s you write (be it infinite, aleph one or more) there will always be a difference to 1/3 and that is precisely the reason why these numbers need infinity to be represented.

On the contrary, it is well accepted that 0.333... = 1/3

Yes...it is accepted, but that doesn't mean it is true...

0.3 =/= 1/3 correct?

let's continue...

0.33 =/= 1/3

0.333 =/= 1/3

0.3333 =/= 1/3

[.....]

0.3333.... =/= 1/3

There is still a difference between any decimal representation and 1/3, that is why it needs infinity to be represented and that's the reason why it is shady. Anything that needs infinity to be represented cannot be right.

No, no there isn't. You are superimposing your viewpoint of a decimal (that it must at some point finish) over the concept that this is a decimal that never finishes and thus cannot ever be precisely defined as a decimal. It's infinite, much as the Universe is and I don't see the Universe ceasing to exist because it's 'not right'.

I get the struggle - really I do - the human mind does not like to encompass concepts that are without limit - but the case remains that as you divide 1 by 3, you will never ever ever reach a point where it is 'finished' and the same holds true for the reverse operation. It must. Where exactly is the leftover bit going to go? Where is it going to hang around? Has 1 mysteriously lost a bit because the mathematical knife cutting the 1 cake has got a bit of sticky edge on it (I hate it when people do that)?

The universe is not infinite. It is the 'surface' of a 4-dimensional sphere...fly in one direction long enough and you will arrive where you started (quite a similar concept to the surface of a 3-dimensional sphere like our planet...the surface seems endless, if you look at it from a 2-dimensional view, but looked at in 3 dimensions you see that it is confined to a certain volume)

 

[Vanaron]

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.

 

[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

 

[grammarye]

They're not limited. That is the entire point of the notion of recurring decimal places. This is an abstract concept, not what you can fit into a computer's buffer or something.

Not just not in a computer buffer. Actually NOWHERE. It is as you say purely a ruleset for manipulating symbols - there is no representation of it anywhere at all. And with this i do have a problem - there is a big difference between how normal people APPLY infinity, and what you're doing in maths - same words perhaps, but very different meaning.

And you know? Ever since i stopped being a science-fanboy by analyzing the theories i previously so blindly supported, and noticing one conceptual error after another.... i do not take unobservable unprovable axioms for granted anymore. Especially not when those very axioms are used to claim the biggest idiocies in modern science. It's one thing to create an axiomatic rule - its another thing to stop being aware that it is just that, and then infecting others aggressively with such falacies, via appeals to authority (when in fact, "the authority" doesn't understand its own basics).

With respect, when you're quite finished 'sticking it to the man', 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. Gross generalisations tend to not apply very well. 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. I could equally have a fascinating debate about the so-called scientific & mathematical 'facts' that are in fact just theories, but, aside from it being off-topic, you know what? I'm with xkcd on this one - 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? Stop equating science & the pursuit of it with the results - which are usually entirely independent of science & driven by other factors.

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.

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

 

[Jonci]

In programming, we just call this a floating point error.

 

[grammarye]

The universe is not infinite. It is the 'surface' of a 4-dimensional sphere...fly in one direction long enough and you will arrive where you started (quite a similar concept to the surface of a 3-dimensional sphere like our planet...the surface seems endless, if you look at it from a 2-dimensional view, but looked at in 3 dimensions you see that it is confined to a certain volume)

Ah, we're into the phase of 'if I can't find a problem with the opposing debate that is on-topic, we'll highlight the lack of knowledge in another entirely unrelated area in an attempt to make them look foolish'.

Interesting to know, though.

 

[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 infinity, but 1/0 is not defined. So there has to be a difference between the limit and the value.

 

[Max01234]

That is ridiculous. You can not say that 0.999... is equal to 1, has they both have separate values.

 

[Coldie]

Of course you need them...the difference between 0.99999.... and 1 is infinitesimal...and both are real numbers different from each other. That's the whole deal of this thread.

If two Real numbers have different values, then there must be an infinite amount of Real numbers between them. However, there cannot be any numbers between the numbers in question, as the difference is, as you say, 'infinitesimal'. It's not possible to map the Continuum to (0.999..., 1) - if you map 0.(9) to -inf, 1 to +inf, you'd want to map 0 to their average... and oops, there isn't one. They are infinitely close.

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

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?

 

[Vanaron]

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.