[Perm Unresolved] 0.999... = 1?

[capsulecorpjx]

Is 0.999 (repeating) = 1?

I think not. See my sigs.

However I'd like to hear some other opinions, especially if you have a degree in Math.

[kfcSmitty]

0.9999 rounded is 1, otherwise it is 0.99999

[yrwyddfa]

Is 0.999 (repeating) = 1?

I think not. See my sigs.

However I'd like to hear some other opinions, especially if you have a degree in Math.

Why did you ever think that it was?

[capsulecorpjx]

Why did you ever think that it was?

Bunch of gamers argued this on a forum (Blizzard gamers specifically).

[NoteMe]

Bunch of gamers argued this on a forum (Blizzard gamers specifically).

On a computer it might be the same because of the rounding in the IEEE standard. But in real life it is not the same....

[sciguyryan]

On a computer it might be the same because of the rounding in the IEEE standard. But in real life it is not the same....

Exactly.

And it depends on your point of view I suppose but unrounded and not on a computer system 0.99999 is not 1

Cheers,

RyanJ

[zaza]

Hi,

There is no difference between a and b in your sig. Both are 1/0 which is infinite. "Undefined" means nothing.

This, of course, assumes that by 0.999 (repeating) you mean 0.9 recurring, i.e. with an infinite number of 9's on the end.


Here is a proof for you:

Let x = 0.999999999....

Then 10x = 9.99999999999......

Subtracting, 9x = 9

hence x = 0.9999999999....=1


Satisfied?

zaza


OK, Undefined doesn't really mean nothing. That's what a computer will give you. But to all intents and purposes it is infinite. Undefined could, in fact, mean the same thing. But it depends how far into "infinities" you want to go...

[NoteMe]

Then 10x = 9.99999999999......

Subtracting, 9x = 9

How did you do this step?

You can't just subtract. And you don't even subrtract the same amount on the same side????


You can always multiply or devide though. To get 9x on the left side you can always multiply with (9/10) then you will have:

9x = 8.99999999....


Can you please elaborate on how you did that?


- ØØ -

[zaza]

Hi,

10x is simply achieved by moving the decimal point one space to the right. That's what multiplying by 10 does. Don't forget, the 9's go on FOREVER, so there's always another 9 on the end. Hence by subtracting x from 10x, all you are doing is removing the part after the decimal point and so you are left with 9x = 9.

HTH

zaza

[NoteMe]

But you can't simply do that...you are chaning the value of X....

Let x = 0.999999999.... here x = 0.999999999999

Then 10x = 9.99999999999...... here x = 0.999999999999999

Subtracting, 9x = 9 here x is suddenly 1.0000000000, that is impossible, you did something illigal....

hence x = 0.9999999999....=1

[capsulecorpjx]

You're wrong, Undefined <> Infinite.
Sophmore math teaches you that.

Just graph the equations:
y = 1/x

y is undefined if x = 0
y is infinite as x APPROACHES 0

The graph shows the difference.

Hi,

There is no difference between a and b in your sig. Both are 1/0 which is infinite. "Undefined" means nothing.

This, of course, assumes that by 0.999 (repeating) you mean 0.9 recurring, i.e. with an infinite number of 9's on the end.


Here is a proof for you:

Let x = 0.999999999....

Then 10x = 9.99999999999......

Subtracting, 9x = 9

hence x = 0.9999999999....=1


Satisfied?

zaza


OK, Undefined doesn't really mean nothing. That's what a computer will give you. But to all intents and purposes it is infinite. Undefined could, in fact, mean the same thing. But it depends how far into "infinities" you want to go...

[moeur]

To clarify zaza's point
x = 0.999999...
10*x = 9.9999999.....
subtract x from each side
10x - x = 9.99999... - 0.9999....
or
9x = 9
hence
x=1.0

and 1/0 is infinity there is no difference in the graphs

[capsulecorpjx]

a) 1/(1-0.999...) = postive infinity.
b) 1/(1-1) = undefined.

b) 1/(1-1) = n
1/0 = n
1 = 0*n
n being infinity does not solve the equation so n <> infinity.

At the very least, n (or b) can be pos inf, neg inf or 0.
However a) can only be postive infinity.

Which means a <> b.

http://en.wikipedia.org/wiki/Division_by_zero

[moeur]

There is nothing wrong with zaza's logic.
as far as 1/0, you're right, it is undefined.
Therefore we cannot use it in your proof.
zaza did not divide by zero to reach his conclusion therefore performed no undefined procedure.

You did in your attempt so your result is not valid.

[capsulecorpjx]

There is nothing wrong with zaza's logic.
as far as 1/0, you're right, it is undefined.
Therefore we cannot use it in your proof.
zaza did not divide by zero to reach his conclusion therefore performed no undefined procedure.

You did in your attempt so your result is not valid.

All I'm saying is that to say 1 = 0.999..., you must say undefined = infinity.

Which I contend is not true.

You can have undefined in your proof if you intended to use it.

Anyway, thats my proof.

A math prof has another proof that supports my conclusions 0.9999... < 1 he does this using a new way of representing numbers.
http://www.maths.nott.ac.uk/personal/anw/Research/Hack/

[Something Else]

How about this:

How small is the difference between 1 and .999...?

[sql_lall]

sorry, but 0.9999.... = 1
you claim that 1 / (1 - 0.999...) is infinity, but give no proof.
why is this wrong? let x = 1 - 0.99999....
let y be any real number > 0, therefore y > x (as 1 - y < 0.99999....)
so, x must = 0.

another reason:
1 + x + x^2 + x^3 + ... = 1 / (1 - x)
=> 9 + 9x + 9x^2 + 9x^3 + .. = 9 / (1 - x)
let x = 0.1 => 9 + 0.9 + 0.09 + 0.009 + ... = 9.9999... = 9 / (9/10) = 90/9 = 10

another reason (similar to above):
1/9 = 0.111111....
=> 9/9 = 0.99999..... = 1
Note that the 'postponement' multiplication argument that site mentions is not really relevant, as we *know* that all the digits are going to be '1', so it's not a problem of 'postponing' any decision.

The funny thing about that site is that it seems to think recurring decimals in base 10 are nasty, imprecise limits, but recurring decimals in base 2 (which is what hackenstrings effectively are) allow exact representation...

That article is about enumerating games, and the theory can't be applied simply. For one, game enumeration allows concepts such as 'infinity plus one', 'infinity plus two' etc... which are always 'greater than' infinity. In fact, it's not seen as infinity, but is denoted by 'omega'. Real numbers do not have this concept.

Disclaimer: I am *not* a gamer, I am a programmer from a strong maths background.
And don't just take it from me:
http://www.cs.uu.nl/wais/html/na-dir.../0.999999.html
http://forums.topcoder.com/?module=T...start=0&mc=142

[NoteMe]

How about this:

How small is the difference between 1 and .999...?

This is actualy a very interesting question. Since it is here the kernel of the dispute is.......one side here means that it can become ZERO/NIL/NOTHING because they are to lazy to write an infinte amount of 9999999999, the other side says that it goes towards 0, but never reaches it since they are still sitting on the 9 key on their keybarod, and never reach 1.0, at least not before they are out of memory on their computer...


- ØØ -

[moeur]

You can have undefined in your proof if you intended to use it.

I guess you can, but you can't operate on it; therefore you can't do
0*1/0 = 1
by defenition 0*1/0 is undefined.

[capsulecorpjx]

This is actualy a very interesting question. Since it is here the kernel of the dispute is.......one side here means that it can become ZERO/NIL/NOTHING because they are to lazy to write an infinte amount of 9999999999, the other side says that it goes towards 0, but never reaches it since they are still sitting on the 9 key on their keybarod, and never reach 1.0, at least not before they are out of memory on their computer...


- ØØ -

You guys don't understand limits.
0.999... is not a number more than a limit.

And approaching a number IS NOT that number.

Anyway, my proof seems logical to me, but maybe it has some flaws.

What are you thoughts about the proof provided at this university site?
http://www.maths.nott.ac.uk/personal/anw/Research/Hack/

[kfcSmitty]

Okay, I don't get this..maybe I am an idiot...but I believe what I see

0.9999(repeated) doesnt even involve the number 1...how the heck can you say it = 1? Unless you change the number..it will always be 0.9999(repeated)....

[capsulecorpjx]

How about this:

How small is the difference between 1 and .999...?

The difference is
lim x
x-> 0

In other word, the difference is x as x approaches 0.

[szlamany]

Okay, I don't get this..maybe I am an idiot...but I believe what I see

0.9999(repeated) doesnt even involve the number 1...how the heck can you say it = 1? Unless you change the number..it will always be 0.9999(repeated)....

Repeated means forever means infinity - we are talking about a pretty abstract concept here.

Note the only math term used here was "infinity"

In floating point (since this is a computer language forum - at it's roots) even 4.9 = 5 - that's just the way the value gets stored.

I've not yet figured out if this thread is about floating point representations of values or about what appears to be a math-class teaser question.

[NoteMe]

You guys don't understand limits.
0.999... is not a number more than a limit.

Ehhhh...did I ever say that it was?

[moeur]

capsulecorpjx,

You are arguing two points.

my proof seems logical to me, but maybe it has some flaws.

Your argument is flawed so abandon it.

How small is the difference between 1 and .999...?

This is a better way to approach the problem.

What you'll find out is the whole argument depends on how you define the difference: is it zero or infinitesimal; a value arbitrarily close to but greater than zero.

And so the question, "does .999... = 1", cannot be proved but depends on how you define that difference. Those that subscribe to the Arcimedean Axiom say there is no such thing as an infinitesimal number so .999… = 1. Others that accept the idea of an infinitesimal will say that .999 < 1.

I’m not a mathematician, but a Physicist; I work with real world things. Another definition of infinitesimal is: immeasurably or incalculably small. So if it is immeasurable it is indistinguishable from 0.

Hence in the world where we most work, as well as the mathematician's world where the Arcimedean Axiom is accepted; .999… = 1.

[szlamany]

This is actualy a very interesting question. Since it is here the kernel of the dispute is.......one side here means that it can become ZERO/NIL/NOTHING because they are to lazy to write an infinte amount of 9999999999, the other side says that it goes towards 0, but never reaches it since they are still sitting on the 9 key on their keybarod, and never reach 1.0, at least not before they are out of memory on their computer...


- ØØ -

I was trying very hard to keep from getting in this debate (help me - please - don't let me hit "Submit Reply"!

NoteMe - you know very well that it's not "possible" to write an infinite amount of 999999999's.

You might be able to spin them around very fast and crash them into other particles

[NoteMe]

NoteMe - you know very well that it's not "possible" to write an infinite amount of 999999999's.

Yeah I know.......and your point is....



And now you are in...


- ØØ -


EDit:

I’m not a mathematician, but a Physicist; I work with real world things.

you should come to CERN...

[szlamany]

EDit:

Originally Posted by moeur
I’m not a mathematician, but a Physicist; I work with real world things.

you should come to CERN...

Are you saying those particles are real?

[moeur]

you should come to CERN...

I'd love to.

[Something Else]

capsulecorpjx,

Your argument is flawed so abandon it.

Hear, Hear!!!

This is a better way to approach the problem.

What you'll find out is the whole argument depends on how you define the difference: is it zero or infinitesimal; a value arbitrarily close to but greater than zero.

And so the question, "does .999... = 1", cannot be proved but depends on how you define that difference. Those that subscribe to the Arcimedean Axiom say there is no such thing as an infinitesimal number so .999… = 1. Others that accept the idea of an infinitesimal will say that .999[edit]...[/edit] < 1.

I’m not a mathematician, but a Physicist; I work with real world things. Another definition of infinitesimal is: immeasurably or incalculably small. So if it is immeasurable it is indistinguishable from 0.

And, also, if it is incalculably small, it also is indistinquishable from zero

Hence in the world where we most work, as well as the mathematician's world where the Arcimedean Axiom is accepted; .999… = 1.

a difference that makes no difference is no difference.

.00000000000000000000000000...=0

[capsulecorpjx]

Really just a question of semantics. I say 0.999... approaches 1, not 1 itself.
You're saying 0.999... = 1.

Its a different story if you say something that approaches a number is that number, that is wrong.

Hear, Hear!!!


And, also, if it is incalculably small, it also is indistinquishable from zero



a difference that makes no difference is no difference.

.00000000000000000000000000...=0

[zaza]

Hi,

I think you seem to have led yourself up the garden path with your signature. You have assumed that a <> b, and then "discovered" that in fact they are not the same! I would say that your a and b options are in fact exactly the same, by dint of the fact that 0.9 recurring is indeed 1. Thus both a and b are what you term "undefined". You've just decided to call one of them "infinity" and the other "undefined". The semantics, I'm afraid, are in your argument. The lesson, I'm afraid, is that dividing by zero is never a good thing to do.

zaza


Incidentally [carefully puts cat amongst pigeons], what do you call 1/infinity?

[kfcSmitty]

i still dont see how 0.99999 = 1

its like that einstein thing...if you move towards something, moving half the distance of the previous each time, you will never reach your destination...same goes for this

[szlamany]

i still dont see how 0.99999 = 1

its like that einstein thing...if you move towards something, moving half the distance of the previous each time, you will never reach your destination...same goes for this

Yeah - Einstein had a lot of those things - saw a great exhibit at the NY Museam of Natural History on Einstein a couple of years ago...

The keyword you used is never - kind of like the repeated used in many of the posts here...

Those abstract facts (is that an oxymoron?)

[kfcSmitty]

those abstract facts...what? It kinda seems like you stopped in the middle of the sentence there

[szlamany]

I thought to myself how funny the "phrase" abstract fact is - that seems like an oxymoron - in that those two words are polar opposites.

I thought that was enough said..

1 is a number.

.99999 (repeating) is a limit - or so people have said in this thread.

Apples and oranges?

Seems like a game of semantics at this point.

I've seen posts here using incalcuably small to describe the difference - but that's not true, since the .9999 repeats for infinity, then in reality this is not a number - so subtracting it from 1 (a solid good old number) against something that is not a number (we obviosly cannot have a number with an infinite amount of digits - infinity is like a NULL VALUE in a database - rather a lonely thing) makes no sense.

But, then again maybe it's about thinking outside the box here...

Since we seem to have taken a physics turn...

You know how a gas is a loosely organized pile of molecules - a solid is a bit more organized. That desk your PC on is really rather porous (much more vacuum space then "molecule" space) - but gives the illusion of being a flat hard solid object. Those darn sub-nuclear forces at work giving it the appearance of something we cannot pass through. Maybe it's that our own particles interfere/interact with those desk particles.

Maybe if this poor girl was in the "proper frame of mind" she would have avoided that "crash"...

http://www.vbforums.com/showthread.php?t=351029

[zaza]

Which is closer to 1:

0.9 or 0.99?

What about:

0.999 or 0.9999?

How do you make 0.9 closer to 1? Add a 9 on the end = 0.99.
What about 0.99? Add a 9 on the end = 0.999.

Now imagine 256 9's on the end. How do you make it closer to 1? Add another 9 on the end, so you have 257.
Every time you put a 9 on the end, you get closer to 1. However, with any finite number of 9's you would never be exactly at 1, even if you had a hundred million billion of them. There would always be an end point for you to add another 9 onto. But an infinite number of 9's, there is no end point, therefore it is impossible to make it any closer to 1 by adding a 9 on the end. Hence if it can't be any closer to 1, it must BE 1.

BTW, when we say 0.999..., those dots actually mean "going on forever". Nobody is trying to claim that 0.999 is, in fact, 1.


zaza

[szlamany]

Now imagine 256 9's on the end. How do you make it closer to 1? Add another 9 on the end, so you have 257.
Every time you put a 9 on the end, you get closer to 1. However, with any finite number of 9's you would never be exactly at 1, even if you had a hundred million billion of them. There would always be an end point for you to add another 9 onto. But an infinite number of 9's, there is no end point, therefore it is impossible to make it any closer to 1 by adding a 9 on the end. Hence if it can't be any closer to 1, it must BE 1.

Great argument - and I have no problem thinking both ways in this thread...

But you cannot say a hundred million billion - give a fact...

Then say infinite number of 9's and give a fact...

As soon as you say infinite you are on a totally new playing field.

My opinion anyway

[capsulecorpjx]

All you're saying is that it's infintely close to 1.

You're not saying it is 1.

There are many limits and graphs in math that are infinitely close to something, but never quite there.

Which is closer to 1:

0.9 or 0.99?

What about:

0.999 or 0.9999?

How do you make 0.9 closer to 1? Add a 9 on the end = 0.99.
What about 0.99? Add a 9 on the end = 0.999.

Now imagine 256 9's on the end. How do you make it closer to 1? Add another 9 on the end, so you have 257.
Every time you put a 9 on the end, you get closer to 1. However, with any finite number of 9's you would never be exactly at 1, even if you had a hundred million billion of them. There would always be an end point for you to add another 9 onto. But an infinite number of 9's, there is no end point, therefore it is impossible to make it any closer to 1 by adding a 9 on the end. Hence if it can't be any closer to 1, it must BE 1.

BTW, when we say 0.999..., those dots actually mean "going on forever". Nobody is trying to claim that 0.999 is, in fact, 1.


zaza

[zaza]

Hi,

You seem rather fixed on this idea of the difference between limits and actual values. However, you haven't actually said what your limit actually refers to. I would say that it refers to the following series:

0.9 + 0.09 + 0.009 + 0.0009 + ....


This is a geometric progression. The sum to infinity of this is defined (check your maths textbook) to be:

S = lim Sn = 0.9 / 1-(1/10) = 0.9 / 0.9 = 1
n->inf.

Just because there is a limit does not mean that it must always just APPROACH a particular value although, as you say, just because there is a limit does not mean that the limit IS that particular value.

In this case, however, it is.

zaza