Do you think 0.9999999~ =1 , that is infinite 9s.

[worzel]

What if we decided to use base 0.999...? :D
Back to counting beans! Are you ducking my question?
No, I didn't see your question when I posted. But I am wondering what the equivalent question would be in base 1 (base 0.999..)

0.000...=1 ? :-?

Guess you can't represent fraction in unary anyway.
No bases. Does 1=.9999....? Simple question, no fluffing. Use the same process as you did with .9999....=1. Don't use any other mathematic formulas, please. YES OR NO.
Say what?

YES. 1=0.999.... because 0.999...=1

Am I missing something?

[Candy]

Be Honest! 8-[

[worzel]

Is that a YES or NO to my question?
Do you want a slap? ;)

[Candy]

YES. 1=0.999.... because 0.999...=1

Am I missing something?
OK, I don't think you understood what I was asking, but I'm sure a math wizard will. :D

[Candy]

1=0.999.... because 0.999...=1

Does this mean...

2=1.999... because
3=2.999... because

Please, explain.

[Bad jcsd]

We don't want just any one-to-one correspondence; we want one that preserves algebraic properties (and so we're back to that...), i.e. an isomorphism.

There is a slight problem though in that the only algebra defined is that defined on the reals (i.e. what kind of algebra is defined on the decimal expansion alone?). Unfortuanelty the only additional structure that decimal expansions have is that which comes from the structure of the reals

There are defintely limittations on the bijections that can be defined between two sets with additonal structure, for example though there obviously do exist bijections between N and Q there are no bijections such that f(a) > f(b) iff a > b or Z and Q are not isomorphic rings, etc.

[VTBoy]

1=0.999.... because 0.999...=1

Does this mean...

2=1.999... because
3=2.999... because

Please, explain.

Yes it does. Just like 0.499999.... = 0.5 = 0.4999999....

[Disinfo Agent]

But in unary every postion is 1^x so 1 in unary could be 10, 100, 1000, etc.
Good point. 8)


No bases.
"1" and "0.999..." are just abbreviations. Some base is always implicit. In most cases, that's base 10, but there are others. The number denoted by "1" is the same as the number denoted by "0.999..." only in base 10. In bases lesser than 10, there is no "0.999..." numeral, and, in bases greater than 10, "0.999..." is strictly less than 1 -- just like "0.333..." is strictly less than 1 in base 10, but equal to 1 in base 4.


Does 1=.9999....? Simple question, no fluffing. Use the same process as you did with .9999....=1.
The former follows from the latter.

[worzel]

1=0.999.... because 0.999...=1

Does this mean...

2=1.999... because
3=2.999... because

Please, explain.
No. if I am a hippopotamus then a hippopotamus am I. A = B means that B = A. You're winding me up aren't you?

[Candy]

You're winding me up aren't you?
Is it working?

I just don't understand how you can take a fraction and make it a whole.

You can keep repeating the same old ways you calculate it, but it doesn't change the way my brain works. :-?

[gzhpcu]

Given the number of proofs that have failed to convince people, I don't think proof of any kind is going to work.

Why don't you accept that 0.999r=1?

Is it because you don't believe a number whos significant digits are all after the decimal point can be equal to 1?

Is it because you don't believe two decimal representations can correspond to the same number?

I'm giving you the benefit of the doubt here and assuming the doubters aren't just trolling/refusing to admit they are wrong.

Ît is because I do not believe that a number with an infinite number of 9's behind the decimal point equals the next higher number.

[Musashi]

Can you make 4/2 into a whole?

[Disinfo Agent]

I just don't understand how you can take a fraction and make it a whole.
2*(1/2)=1
3*(1/3)=1
4*(1/2)=2
...
:)

[Candy]

I just don't understand how you can take a fraction and make it a whole.
2*(1/2)=1
3*(1/3)=1
4*(1/2)=2
...
:)
You know what I mean, Einstein. YOU'RE TAKING A COMPLEX FRACTION AND MAKING IT A WHOLE NUMBER. :D

[Musashi]

What is the fraction?

[Disinfo Agent]

I just don't understand how you can take a fraction and make it a whole.
Candy, you're assuming that 0.999... is a fractional number, as opposed to an integer. But you won't really know unless you prove it. And when you try, you see that it isn't a fractional number.

[worzel]

Ît is because I do not believe that a number with an infinite number of 9's behind the decimal point equals the next higher number.
And you said that our point of view sounds like religion to you.

Other than your belief, have you got any other reason?

[Candy]

I just don't understand how you can take a fraction and make it a whole.
Candy, you're assuming that 0.999... is a fractional number, as opposed to an integer. But you won't really know unless you prove it. And when you try, you see that it isn't a fractional number.
Then what's Pi? An integer, too?

[Musashi]

Can you represent 0.9.... as a fraction for us?

[gzhpcu]

Ît is because I do not believe that a number with an infinite number of 9's behind the decimal point equals the next higher number.
And you said that our point of view sounds like religion to you.

Other than your belief, have you got any other reason?

I only said "believe" because I am being polite. I am convinced 100% of the above statement. Also convinced that 0.999999999... is not an integer.

[Musashi]

Pi is an irrational number.

[Candy]

Can you represent 0.9.... as a fraction for us?
Not the way the math wizard's mean it....

[Disinfo Agent]

I just don't understand how you can take a fraction and make it a whole.
Candy, you're assuming that 0.999... is a fractional number, as opposed to an integer. But you won't really know unless you prove it. And when you try, you see that it isn't a fractional number.
Then what's Pi? An iteger, too?
Now you're assuming that what I wrote applies to all numbers with infinite decimal expansions. Not so. [-X

[Disinfo Agent]

I only said "believe" because I am being polite. I am convinced 100% of the above statement. Also convinced that 0.999999999... is not an integer.
Let's check that. Define "integer" for us.

[Musashi]

How about Pi? Try putting that into a fraction using integers. ;)

[Candy]

I just don't understand how you can take a fraction and make it a whole.
Candy, you're assuming that 0.999... is a fractional number, as opposed to an integer. But you won't really know unless you prove it. And when you try, you see that it isn't a fractional number.
Then what's Pi? An iteger, too?
Now you're assuming that what I wrote applies to all numbers with infinite decimal expansions. Not so. [-X You know what I think, your theory of 0.9999.... means just that....

just like pi equals.... it never equals 3 or 4.

[Musashi]

An integer should never repeat or terminate. So, it seems that .9... would not be an integer.

[Musashi]

What is 0.999... divided by 3?

[01101001]

You have to admit, this is a tough concept, and not easy to wrap your head around. Logic dictates that 0.999...does not equal 1, but logic cannot resolve the number before or after .999...
It's big of you to say so. Thanks for thinking.

I hadn't stopped to think of it, but .999... and 1 are no special case. It's been hinted at here. There are an infinty of numbers whose decinal expansion has at least 2 forms.

We usually think of numbers that have a repeating form and ones that don't, but those that don't can always be considered to be have a repeating form ending in 000... We just use a shorthand to say all those zeros don't really matter, and don't express them.

All those infintely many non-repeating numbers (that really repeat zeros) have at least 2 different decimal expansions. A number that ends in d000... (d not zero) also can be written as ending in (d-1)999...

And for those who like to think in other bases, an alternate form has an ending of (d-1)(r-1)(r-1)(r-1)... where r is the radix.

For instance, 1.25000... can also be expressed at 1.24999...
(and 1.01000... base two can also be written as 1.00111...)

Ooo, almost generated my own handle!

Should it be unsettling to have more than one form of decimal expansion -- that a single real number can have more than one "name", indeed infinitely many names? Only if you live in a world where you think 2, 2.0, 2.00, 02, 02.00 and 01.999... have different values.

Pity poor 0. It's the only zero-repeating number that doesn't also have a 9-repeating form.

[worzel]

Ît is because I do not believe that a number with an infinite number of 9's behind the decimal point equals the next higher number.
And you said that our point of view sounds like religion to you.

Other than your belief, have you got any other reason?

I only said "believe" because I am being polite. I am convinced 100% of the above statement. Also convinced that 0.999999999... is not an integer.

So "polite" means repeatedly stating what you think with no justifiction?

[Disinfo Agent]

Now you're assuming that what I wrote applies to all numbers with infinite decimal expansions. Not so. [-X You know what I think, your theory of 0.9999.... means just that....
It's not my theory (I should be so lucky!), and you're wrong, the fact that 0.999... is a whole number does not make pi a whole number. It doesn't even make 0.333... a whole number. Why should it?
[/url]

[Candy]

An integer should never repeat or terminate. So, it seems that .9... would not be an integer.
According to you, it is a number that never terminates... that fact that it repeats itself is not valid. IT NEVER TERMINATES.

[Musashi]

An integer should never repeat or terminate. So, it seems that .9... would not be an integer.
According to you, it is a number that never terminates... that fact that it repeats itself is not valid. IT NEVER TERMINATES.

What?

A number that EITHER repeats or terminates is not an integer.

0.999... repeats so it cannot be an [edit] irrational.

I'm sorry, it is early for me. I meant irrationals.

[Candy]

A number that EITHER repeats or terminates is not an integer.

0.999... repeats so it cannot be an integer. Then it's a turd, it doesn't terminate, let's call it what it is not one.

[Disinfo Agent]

A number that EITHER repeats or terminates is not an integer.
Except when its fractional part is 0.9999... or 0.000... [-X

[Musashi]

0.999... does not terminate. That is what the ... is for.

An integer is Whole numbers plus negatives. Whole numbers are Natural numbers along with zero. Natural numbers are 1,2,3,4,5, ...

[Disinfo Agent]

0.999... does not terminate.
But it repeats.


An integer is Whole numbers plus negatives. Whole numbers are Natural numbers along with zero. Natural numbers are 1,2,3,4,5, ...
There are other possible ways of writing those numbers, among which 0.999..., 1.999..., 2.999..., 3.999...

[Musashi]

0.999... does not terminate.
But it repeats.


An integer is Whole numbers plus negatives. Whole numbers are Natural numbers along with zero. Natural numbers are 1,2,3,4,5, ...
There are other possible ways of writing those numbers, among which 0.999..., 1.999..., 2.999..., 3.999...

I think we are talking at cross purposes here.

I understand that it repeats. Therefore it is rational. I made a mistake earlier and started talking about rational/irrational instead of integers. I have already tried to clear that up.

[Musashi]

So, Candy, what happens if you divide 0.999... by 3?

[Candy]

So, Candy, what happens if you divide 0.999... by 3?
you get a fraction

[Musashi]

Any particular fraction?

[Candy]

Any particular fraction?
A fraction that doesn't equal 1. Your point.

[Musashi]

WOuld you agree that 0.999..../3 = 1/3?

[Candy]

WOuld you agree that 0.999..../3 = 1/3?
No

[Normandy6644]

WOuld you agree that 0.999..../3 = 1/3?
No

That's kind of the crux of the argument though.

[Musashi]

so, 1/3 does not equal 0.333.....?

[Disinfo Agent]

The division algorithm tells me they are equal... :-?

[Candy]

so, 1/3 does not equal 0.333.....?
Is this a different question?

[Musashi]

Not particularly, but if you wish I can expand it.

However, first, could you tell me if you think that 1/3 = 0.333....?

[Candy]

Not particularly, but if you wish I can expand it.

However, first, could you tell me if you think that 1/3 = 0.333....?
Yes

(Are you going to do the process of elimination equation with me?) :lol:

[kucharek]

If 0.999999... isn't equal to 1, Achilles would never overtake the turtle.
These limits are a concept not easy to understand and they are often counterintuitive. But they were what mathematics brought a tremendous step forward after it was leanrt how to use them. Quantum physics or relativity is also pretty counter-intuitive, but there is overwhelming proof for being correct.

[Musashi]

Not particularly, but if you wish I can expand it.

However, first, could you tell me if you think that 1/3 = 0.333....?
Yes

(Are you going to do the process of elimination equation with me?) :lol:

I am not sure what you mean exactly.


What happens if you multiply 0.333... by 3? Wouldn't you get 0.999...?

Now, working on the other side of the equation, what happens if you multiply 1/3 by 3?

[Candy]

If 0.999999... isn't equal to 1, Achilles would never overtake the turtle.
These limits are a concept not easy to understand and they are often counterintuitive. But they were what mathematics brought a tremendous step forward after it was leanrt how to use them. Quantum physics or relativity is also pretty counter-intuitive, but there is overwhelming proof for being correct. And that is why we are advancing into the new mathematical era. Let's not have close minds.

[Candy]

Not particularly, but if you wish I can expand it.

However, first, could you tell me if you think that 1/3 = 0.333....?
Yes

(Are you going to do the process of elimination equation with me?) :lol:

I am not sure what you mean exactly.


What happens if you multiply 0.333... by 3? Wouldn't you get 0.999...?

Now, working on the other side of the equation, what happens if you multiply 1/3 by 3? You're making math simplified, when it's complicated. I understand your thought process. I just wish you would understand mine. 8-[

[Bad jcsd]

And that is why we are advancing into the new mathematical era. Let's not have close minds.

But the fact that 0.999.. = 1 is trivial and not open for debate, it's either a case of understanding why 0.99.. = 1 or not understanding.

[Musashi]

Then explain it. I do not understand what is complicated about it.

Here is another way I thought of showing it

0.333... = 1/3 +
0.333... = 1/3
------------------
0.666... = 2/3 +
0.333... = 1/3
-------------------
0.999... = 3/3 = 1

Where is the trouble?

[Candy]

Then explain it. I do not understand what is complicated about it.

Here is another way I thought of showing it

0.333... = 1/3 +
0.333... = 1/3
------------------
0.666... = 2/3 +
0.333... = 1/3
-------------------
0.999... = 3/3 = 1

Where is the trouble?
Think simple. PUT YOURSELF IN MY SHOES.

[Grey]

No bases. Does 1=.9999....? Simple question, no fluffing. Use the same process as you did with .9999....=1. Don't use any other mathematic formulas, please. YES OR NO.
Yes, it does.

[Disinfo Agent]

Yes, it does.

[Candy]

Put yourself in my shoes, and think the way I think for a moment.

[Candy]

Just try it. 8-[

[Musashi]

Then explain it. I do not understand what is complicated about it.

Here is another way I thought of showing it

0.333... = 1/3 +
0.333... = 1/3
------------------
0.666... = 2/3 +
0.333... = 1/3
-------------------
0.999... = 3/3 = 1

Where is the trouble?
Think simple. PUT YOURSELF IN MY SHOES.

Explain the problem. It looks like simple arithmetic to me. Since the non-terminating numbers have a 1 to 1 ratio of places, it seems like simple math. We can agree that 0.333.... represents 1/3 and that 0.666... represents 2/3, ands when we ad 3 to 6 we get 9. It seems simple enough to show that 0.999 = 1/3+2/3 or 1/3+1/3+1/3, either of which equal 1. I am not sure if you are saying you don't understand how I got there. If so, is there a particular step in the equation that is bothering you, or is it just the conclusion?

[Disinfo Agent]

Put yourself in my shoes, and think the way I think for a moment.
I have tried it, but your shoes don't fit me. :)

[Candy]

When you can't understand what I think, that tells me you are closed minded.

Try arguing that 0.9999... is not equal to 1.

[Musashi]

I have been asking you to clarify what you meant for several posts. If that is closeminded than I guess I will have to cede the point.

[Disinfo Agent]

When you can't understand what I think, that tells me you are closed minded.
Nancy Lieder will love to hear that.

[Moose]

Here is another way I thought of showing it

0.333... = 1/3 +
0.333... = 1/3
------------------
0.666... = 2/3 +
0.333... = 1/3
-------------------
0.999... = 3/3 = 1



I like this one. Simple, direct, intuitive.

[Candy]

I have been asking you to clarify what you meant for several posts. If that is closeminded than I guess I will have to cede the point.
Yes, this is close minded, because you want me to agree with you.

[worzel]

When you can't understand what I think, that tells me you are closed minded.

Try arguing that 0.9999... is not equal to 1.

What do you call it when one can't explain what they think?

[Musashi]

No, I want you to explain the problem.

[Candy]

When you can't understand what I think, that tells me you are closed minded.
Nancy Lieder will love to hear that.
I'M NANCY NOW! KISS MY A....

[Disinfo Agent]

I'M NANCY NOW!
Obviously not, yet you are arguing like her.

[Candy]

Hey, I'm the dumb one here. Try to convince me.

[Musashi]

Then explain it. I do not understand what is complicated about it.

Here is another way I thought of showing it

0.333... = 1/3 +
0.333... = 1/3
------------------
0.666... = 2/3 +
0.333... = 1/3
-------------------
0.999... = 3/3 = 1

Where is the trouble?
Think simple. PUT YOURSELF IN MY SHOES.

What does this mean? What is not simple about the explanation? I do not want to assume that it is over your head, is it? Is there a step that you think is wrong? Is there some problem with my math? I do not know what you are trying to say when you say "think simple." Does that mean, "abandon math and learning and just agree with me when I say 0.999... does not equal one"? If you could explain what you mean it would be helpful. If you cannot explain, that is fine, I would appreciate if you said so. If you can explain but choose not to, that is just rude.

[Disinfo Agent]

Try arguing that 0.9999... is not equal to 1.
O.K.
A few pages ago, 01101001 tried assuming that 0.9999... was <1 (http://www.badastronomy.com/phpBB/viewtopic.php?p=379193#379193), to see what he got. That led him to a contradiction.
Since I don't expect anyone to argue that 0.9999... is >1, I say case closed: 0.9999... can only be =1.

[Candy]

A Thousand Pardons, you must know what I am saying, cause these boys don't. I go to bed.

[Musashi]

Gee, I guess it sucks we are not mind readers. :roll:

[A Thousand Pardons]

What do you call a thread that never ends and doesn't repeat? :)

[Bad jcsd]

Hey, I'm the dumb one here. Try to convince me.

Okay 0.999... is the the limit of the sequence of partial sums of the series of 9*10^-n, this sequence is a Cauchy sequence (all convergaent sequences are Cauchy sequences). In one construction of the real numbers each real number is regarded as an equivalency class of Cauchy sequence of rationals, the Cauchy sequence above belongs to the equivalency class 1. Therefore it is a simple matter of definition that 0.999... = 1.

[Musashi]

irr.. wait a second!

:D

Actually, I think the thread does repeat, but for the sake of your joke...

[kucharek]

Think simple. PUT YOURSELF IN MY SHOES.

I've 13 1/2. Do I have any chance?

:D

Harald

[Candy]

What do you call a thread that never ends and doesn't repeat? :)
My hero! 8)

[Bad jcsd]

btw I hope everyone's paying attention 'cos this is one of the quetsions St. Peter asks when you get to the pearly gates.

[Disinfo Agent]

I still think worzel was right (http://www.badastronomy.com/phpBB/viewtopic.php?p=379394#379394), and Candy was teasing... :D

[Candy]

btw I hope everyone's paying attention 'cos this is one of the quetsions St. Peter asks when you get to the pearly gates.
If you meet St. Peter, tell him I said HELLO. :wink:

[Candy]

I still think worzel was right (http://www.badastronomy.com/phpBB/viewtopic.php?p=379394#379394), and Candy was teasing... :D
I stuck to my guns, didn't I? :wink:

[Candy]

CAN WE OFFICIALLY CLOSE THIS THREAD?

[Grey]

CAN WE OFFICIALLY CLOSE THIS THREAD?
Does the discussion offend you? Is someone forcing you to read it at gunpoint?
:D

[A Thousand Pardons]

Don't we need two ayes to close this thread

Neeiiigghhh! :)

I went back to all your posts, Candy, and I noticed the early contribution (http://www.badastronomy.com/phpBB/viewtopic.php?p=372152#372152) that seemed you were anticipating the downfall of calculus. Sorry, if you had a bad experience.

Some famous mathematicians fought similar battles over a hundred years ago--they had the same problems with dealing with concepts of infinity, and the implications. Some thought that the methods were too loose, and might be making unwarranted assumptions. Logicians spent whole lifetimes trying to pick up the pieces after the shambles created in those arguments, and putting the pieces back together into a system that was reasonable and satisfactory. We may not have succeeded. "Playing" with infinity still has a nasty feeling for some, so mathematicians (and logicians) have tended to try to build systems that avoid it. Their definitions are mathematical and logical, and have survived decades of attacks by some of the most incredible minds. You're in good company in that regard.

However, the general consensus is still against you. Until some superstar intellect comes up with a counterargument, it will remain a well-established result. There's nothing on the horizon, so you might be doomed to wait for a long time.

Sometimes, it just takes a little getting used to.

Try this. Find a decimal representation of pi on the internet (say, out to a hundred digits) and start dividing it by 3 until you think you have a pretty good idea of what is going on. Now, try dividing 0.999... by 3, and (after your nap :) ) come back and tell us what you've noticed/learned about the process.

[Disinfo Agent]

Maybe posters like mutineer, gzhpcu, and Zjm7891 will come back with answers to the many questions we asked them and they still haven't answered...

[01101001]

CAN WE OFFICIALLY CLOSE THIS THREAD?
Candy, the answer to this and all your other questions can be found in this thread (http://www.badastronomy.com/phpBB/viewtopic.php?t=17953).

Enjoy. Enjoy. Enjoy...

[Careless]

CAN WE OFFICIALLY CLOSE THIS THREAD?
Candy, the answer to this and all your other questions can be found in this thread (http://www.badastronomy.com/phpBB/viewtopic.php?t=17953).

Enjoy. Enjoy. Enjoy...
Are you aiming for a suicide?

[Lycus]

Candy, the answer to this and all your other questions can be found in this thread (http://www.badastronomy.com/phpBB/viewtopic.php?t=17953).

Enjoy. Enjoy. Enjoy...
And once you get to the 24th page of that thread, there's another link there that may help to explain things even better.

[Candy]

Do you think 0.9999999~ infinite 9s is exactly the equal to 1.
Yes it is equal
59% [ 64 ]
No it is not equal
40% [ 44 ]

Total Votes : 108
The percentage doesn't equal 100, or does it?
http://planetsmilies.com/smilies/scared/scared07.gif

[Lycus]

The percentage doesn't equal 100, or does it?
http://planetsmilies.com/smilies/scared/scared07.gif
Apparently, phpBB only rounds down.

[01101001]

The percentage doesn't equal 100, or does it?
http://planetsmilies.com/smilies/scared/scared07.gif
Apparently, phpBB only rounds down.

Of course, if it used infinte precision, the 64/108 + 44/108 result would total 59.2592592...% + 40.7407407...% which of course equals 99.999...%

Nobody can believe this poll is honest! What happened to the remaining vote? Why doesn't it equal 100%?????

I call shenanigans!

[gzhpcu]

Unfortunately, being in a different time zone everytime things start picking up, I miss the action.

What I see all the time is the assertion that 1/3 = .3333333......
which is exactly the same as asserting that 1 = .99999999999....

This type of reasoning will not convince anyone who has the viewpoint that 1 and .999...... are not equal.

Why do you have an infinite number of numbers after the decimal point? Because the division can not be resolved. It goes on and on . You can not get out of the problem by just simply equating it to the next higher number. It is a fundamental problem, and mathematics has conveniently circumvented it. No problem with this. Just that it is a circumvention and not the answer.

[A Thousand Pardons]

Why do you have an infinite number of numbers after the decimal point? Because the division can not be resolved. It goes on and on . You can not get out of the problem by just simply equating it to the next higher number. It is a fundamental problem, and mathematics has conveniently circumvented it. No problem with this. Just that it is a circumvention and not the answer.
The answer to what?

[Candy]

Think simple. PUT YOURSELF IN MY SHOES.

I've 13 1/2. Do I have any chance?

:D

Harald
Heck, I'd marry you. :o 8-[ :P :D

[Normandy6644]

Why do you have an infinite number of numbers after the decimal point? Because the division can not be resolved. It goes on and on . You can not get out of the problem by just simply equating it to the next higher number. It is a fundamental problem, and mathematics has conveniently circumvented it. No problem with this. Just that it is a circumvention and not the answer.
The answer to what?

The only question the internet can't answer (http://www.badastronomy.com/phpBB/viewtopic.php?t=18037), I think. :D

[gzhpcu]

Why do you have an infinite number of numbers after the decimal point? Because the division can not be resolved. It goes on and on . You can not get out of the problem by just simply equating it to the next higher number. It is a fundamental problem, and mathematics has conveniently circumvented it. No problem with this. Just that it is a circumvention and not the answer.
The answer to what?

The result of the division. It can not provide an exact result.

[01101001]

The result of the division. It can not provide an exact result.
I'm having a little problem with your assertion.

Please prove 1/3 is not equal to 0.333... i.e. show

∞
Σ 3/(10^n) ≠ 1/3
n=1

[Candy]

In order to represent 0.333... as a fraction, then it would be 1/3. That doesn't mean its equal.

[01101001]

In order to represent 0.333... as a fraction, then it would be 1/3. That doesn't mean its equal.
I'm slow. Please prove your assertion.

[Maksutov]

In order to represent 0.333... as a fraction, then it would be 1/3. That doesn't mean its equal.
Then here's a test for equivalence. Divide 1 by 3 and tell me when you stop getting repeated "threes" in the series "0.333...".

[Candy]

In order to represent 0.333... as a fraction, then it would be 1/3. That doesn't mean its equal.
I'm slow. Please prove your assertion. I can't. I thought I was being witty. :lol:

[01101001]

In order to represent 0.333... as a fraction, then it would be 1/3. That doesn't mean its equal.
I'm slow. Please prove your assertion. I can't. I thought I was being witty. :lol:
OK.

Candy : wit :: 01101001 : half-wit

[Candy]

In order to represent 0.333... as a fraction, then it would be 1/3. That doesn't mean its equal.
I'm slow. Please prove your assertion. I can't. I thought I was being witty. :lol:
OK.

Candy : wit :: 01101001 : half-wit :P

[Bad jcsd]

The result of the division. It can not provide an exact result.

Everything in pure maths is exact, numerical approximations are of little interest.

[worzel]

What I see all the time is the assertion that 1/3 = .3333333......
which is exactly the same as asserting that 1 = .99999999999....

This type of reasoning will not convince anyone who has the viewpoint that 1 and .999...... are not equal.

The 10x-x=9x proof didn't rely on 1/3=0.33.., nor did the limit proof. But if you don't believe 1/3=0.33.. then what do you think 1/3 does equal?

[gzhpcu]

What I see all the time is the assertion that 1/3 = .3333333......
which is exactly the same as asserting that 1 = .99999999999....

This type of reasoning will not convince anyone who has the viewpoint that 1 and .999...... are not equal.

The 10x-x=9x proof didn't rely on 1/3=0.33.., nor did the limit proof. But if you don't believe 1/3=0.33.. then what do you think 1/3 does equal?

Certainly not 0.333333333....... because you need an infinite number of threes and you are still not there, any more than you are there with an infinite number of 9's to reach 1. Infinitely close, granted, but infinitely close is still not exactly equal.

[gzhpcu]

The result of the division. It can not provide an exact result.
I'm having a little problem with your assertion.

Please prove 1/3 is not equal to 0.333... i.e. show

∞
Σ 3/(10^n) ≠ 1/3
n=1

The proof is in the expression of the number itself. The number of 3's is infinite because it can not resolve itself. It is infinitely close to 1/3 but not accurately identical.

[gzhpcu]

The result of the division. It can not provide an exact result.

Everything in pure maths is exact, numerical approximations are of little interest.
Theoretically yes, in this case no. 0.99999999....... is not exact. It is infinitely close but not precisely exact.

[Candy]

I admit defeat, only under the condition that no one finds a value for a said infinite amount in the next 50 years. :D

Theoretically speaking, of course. :roll:

[Bad jcsd]

Theoretically yes, in this case no. 0.99999999....... is not exact. It is infinitely close but not precisely exact.

Pure maths is entierly theoretical. Where is the ambuigty in 0.9999...? Apart from the faulty logic applied by some here there is no ambugity in the representation "0.999...", it is precise and it is precisely equal to one.

This is very basic maths.

[Donnie B.]

It's been said before in this thread, but I think it bears repeating.

The symbols 0.99... represent a value.

They do not represent an algorithm.

That is to say, it is not necessary to actually do an infinite number of additions. Mathematiics has tools that allow us to understand the exact value that is produced by an infinite sum without actually performing the infinite sum.

And when we apply those tools to the expression 0.99... (that is, the infinite sum 9/10 + 9/100 + 9/1000 + ... ) we determine that its value is exactly 1.

Those proofs have been shown several times already, so I won't repeat them. However, to reject them is to reject a substantial chunk of mathematics. You can do that if you like, of course, but I don't see where it's going to get you (other than to page 26 of this thread).

[gzhpcu]

Theoretically yes, in this case no. 0.99999999....... is not exact. It is infinitely close but not precisely exact.

Pure maths is entierly theoretical. Where is the ambuigty in 0.9999...? Apart from the faulty logic applied by some here there is no ambugity in the representation "0.999...", it is precise and it is precisely equal to one.

This is very basic maths.

Oh yes? Please prove it. :D

Calling it "basic math" is no proof...

[Moose]

Oh yes? Please prove it. :D

You'll find many such proofs here (http://www.badastronomy.com/phpBB/viewtopic.php?t=17953&postdays=0&postorder=asc&sta rt=0).

[gzhpcu]

Oh yes? Please prove it. :D

You'll find many such proofs here (http://www.badastronomy.com/phpBB/viewtopic.php?t=17953&postdays=0&postorder=asc&sta rt=0).

So we are in a loop. If I had found one convincing I would have said so a long time ago. Others as well apparently....

[Bad jcsd]

Theoretically yes, in this case no. 0.99999999....... is not exact. It is infinitely close but not precisely exact.

Pure maths is entierly theoretical. Where is the ambuigty in 0.9999...? Apart from the faulty logic applied by some here there is no ambugity in the representation "0.999...", it is precise and it is precisely equal to one.

This is very basic maths.

Oh yes? Please prove it. :D

Calling it "basic math" is no proof...

This whole thread contains various proofs and arguments, I notice you did not answer the quetsion I posed.

Yes this is basic maths and I advise anyone who thinks that two real numbers can be 'infinitely close' to find out what the real numbers are.

[gzhpcu]

Theoretically yes, in this case no. 0.99999999....... is not exact. It is infinitely close but not precisely exact.

Pure maths is entierly theoretical. Where is the ambuigty in 0.9999...? Apart from the faulty logic applied by some here there is no ambugity in the representation "0.999...", it is precise and it is precisely equal to one.

This is very basic maths.

Oh yes? Please prove it. :D

Calling it "basic math" is no proof...

This whole thread contains various proofs and arguments, I notice you did not answer the quetsion I posed.

Yes this is basic maths and I advise anyone who thinks that two real numbers can be 'infinitely close' to find out what the real numbers are.

Sorry, this thread is long.... which question are you alluding to?

[Normandy6644]

gzhpcu, what would convince you?

[gzhpcu]

First of all, my apologies if I appear to some as unreasoningly stubborn. I am honestly attempting to be open-minded, but I still see the two as non-identical numbers, even if infinitely close. For me, "infinitely close" is not synonymous to identical, and no forumula can convince me, as long as I see 0.9999999999999...........

[Normandy6644]

First of all, my apologies if I appear to some as unreasoningly stubborn. I am honestly attempting to be open-minded, but I still see the two as non-identical numbers, even if infinitely close. For me, "infinitely close" is not synonymous to identical, and no forumula can convince me, as long as I see 0.9999999999999...........

That's kind of contradictory though. You're saying your open minded, yet at the same time no "math" can convince you. So again, what would convince you?

[gzhpcu]

Someone making it clear that "infinitely close" = 0

[Bad jcsd]

First of all you must define 'infinitely close', do you mean that two numbers are 'infintely close' iff (a,b) = {} (i.e. there are no numbers inbetween them)? It follows on directly from the fact that the reals are an ordered field (the fact that the reals are a field guarantees that a + (b-a)/2 is defined for all a, b in R) that if (a,b) = {} then a = b as:

a < a+ (b-a)/2 < b for all a,b in R where a is not equal to b.

[gzhpcu]

First of all you must define 'infinitely close', do you mean that two numbers are 'infintely close' iff (a,b) = {} (i.e. there are no numbers inbetween them)? It follows on directly from the fact that the reals are an ordered field (the fact that the reals are a field guarantees that a + (b-a)/2 is defined for all a, b in R) that if (a,b) = {} then a = b as:

a < a+ (b-a)/2 < b for all a,b in R where a is not equal to b.

Hmmmmmm.... I think you have a point there.... I think you have finally convinced me (with a probablity of 0.999999.....) :D Thanks! If I don't come back within a short time, then the probability becomes 1....

[worzel]

What I see all the time is the assertion that 1/3 = .3333333......
which is exactly the same as asserting that 1 = .99999999999....

This type of reasoning will not convince anyone who has the viewpoint that 1 and .999...... are not equal.

The 10x-x=9x proof didn't rely on 1/3=0.33.., nor did the limit proof. But if you don't believe 1/3=0.33.. then what do you think 1/3 does equal?

Certainly not 0.333333333....... because you need an infinite number of threes and you are still not there, any more than you are there with an infinite number of 9's to reach 1. Infinitely close, granted, but infinitely close is still not exactly equal.

If 0.333... is less than 1/3 then 0.666... must be less than 2/3, and 1 - 0.666... = 0.333... But 1 minus something a bit less than 2/3 shoulld be a bit more than 1/3. So if 0.333... is slightly less than 1/3 then it is also slightly more than 1/3!

EDITED: for clarity

[worzel]

First of all you must define 'infinitely close', do you mean that two numbers are 'infintely close' iff (a,b) = {} (i.e. there are no numbers inbetween them)? It follows on directly from the fact that the reals are an ordered field (the fact that the reals are a field guarantees that a + (b-a)/2 is defined for all a, b in R) that if (a,b) = {} then a = b as:

a < a+ (b-a)/2 < b for all a,b in R where a is not equal to b.

Hmmmmmm.... I think you have a point there.... I think you have finally convinced me (with a probablity of 0.999999.....) :D Thanks! If I don't come back within a short time, then the probability becomes 1....
I'm glad that has almost convinced you, but you know that argument has been put many times in thins thread in other guises. It simply says that if 0.99.. < 1 then there must be (infintely) many numbers in between them.

[01101001]

Hmmmmmm.... I think you have a point there.... I think you have finally convinced me (with a probablity of 0.999999.....) :D Thanks! If I don't come back within a short time, then the probability becomes 1....

That is big of you. Thank you for letting us know.

Hey, a mathy friend suggested yet another approach that I don't recall seeing.

0.999... + 0.999... = 1.999...
Subtract 0.999... from both sides.
0.999... = 1

Of course, that's probably not going to convince anyone that won't accept the more common 10*0.999... = 9.999... but it's kind of elegant.

[gzhpcu]

Thanks. You convinced me and am sure you are right, even though looking at the original equation, it still does not seem right...

[worzel]

Thanks. You convinced me and am sure you are right, even though looking at the original equation, it still does not seem right...
Thanks for arguing your case rather than just accepting the wisdom of the maths wizzes here. I always understand things a whole lot better when someone forces me (and the maths wizzes) to think about it. Playing devil's advocate never seems to work quite as well for me.

[Gmann]

then 0.666... must be less than 2/3

Be careful with the .666... thing. Is it less than 2/3, or something far more sinister? (especially when a devil's Advocate is involved) :D

(runs and hides under the coffee table) 8-[

[Normandy6644]

Hmmmmmm.... I think you have a point there.... I think you have finally convinced me (with a probablity of 0.999999.....) :D Thanks! If I don't come back within a short time, then the probability becomes 1....

That is big of you. Thank you for letting us know.

Hey, a mathy friend suggested yet another approach that I don't recall seeing.

0.999... + 0.999... = 1.999...
Subtract 0.999... from both sides.
0.999... = 1

Of course, that's probably not going to convince anyone that won't accept the more common 10*0.999... = 9.999... but it's kind of elegant.

Yeah, I can't see that convincing many people, but it is cool!

[worzel]

Yeah, I can't see that convincing many people, but it is cool!
How about this? (a reformulation of my earlier post)

1) let d = 1/3 - 0.33..

2) 1/3 = 0.33.. + d

3) 2 /3 = 0.66.. + 2d

4) 1 - 2/3 = 1 - 0.66.. - 2d = 0.33.. - 2d

5) 1 - 2/3 = 1/3

6) 1/3 = 0.33.. - 2d

7) 0.33..+ d = 0.33.. - 2d

8) d = 0, i.e. 1/3 = 0.33..

[Makgraf]

then 0.666... must be less than 2/3

Be careful with the .666... thing. Is it less than 2/3, or something far more sinister? (especially when a devil's Advocate is involved) :D

(runs and hides under the coffee table) 8-[
Now I hate to pedantic (No, that's not right, I love being pedantic) but 666 (the number of the beast) refers to the antichrist. The Devil's Advocate was a position in the Catholic Church who would argue the case against someone being made a saint.

[A Thousand Pardons]

4) 1 - 2/3 = 1 - 0.66.. - 2d = 0.33.. - 2d
This step might not be obvious, where you assume 1 - 0.66.. equals 0.33..

You've already assumed that 1/3 = 0.33.. + d and 2/3 = 0.66.. + 2d, so 1 would equal 0.99.. + 3d, wouldn't it?

[worzel]

4) 1 - 2/3 = 1 - 0.66.. - 2d = 0.33.. - 2d
This step might not be obvious, where you assume 1 - 0.66.. equals 0.33..

You've already assumed that 1/3 = 0.33.. + d and 2/3 = 0.66.. + 2d, so 1 would equal 0.99.. + 3d, wouldn't it?

Yeah, 1 would equal 0.99.. + 3d. I think you're probably right, I hid an assumption that 0.66.. + 0.33.. = 1 which makes it rather circular :( I thought I was on to a good idea before, I'll have a think about it tomorrow.

[Normandy6644]

4) 1 - 2/3 = 1 - 0.66.. - 2d = 0.33.. - 2d
This step might not be obvious, where you assume 1 - 0.66.. equals 0.33..

You've already assumed that 1/3 = 0.33.. + d and 2/3 = 0.66.. + 2d, so 1 would equal 0.99.. + 3d, wouldn't it?

Yeah, 1 would equal 0.99.. + 3d. I think you're probably right, I hid an assumption that 0.66.. + 0.33.. = 1 which makes it rather circular :( I thought I was on to a good idea before, I'll have a think about it tomorrow.

I was wondering about that myself, though I didn't have time to post anything before my physics exam.

[Nicolas]

1 = 0.99... + 3d so 1 = 0.99.... if d=0 which is the same as 1/3 = 0.33... which in fact says that any infinitely correct approximating value is equal to the real value.

Another way in which I view it: take A(ssumption) = V(alue) + E(rror)

so A <> V. If I make E a factor 10 smaller, A gets closer to V.

If I iterate this proces, the error gets infinitely small in the limit (lim(n=>inf) of (1/(10^n)) = 0). In fact, believing this limit is believing 1 = 0.9999999999... It's all about limits.

[worzel]

4) 1 - 2/3 = 1 - 0.66.. - 2d = 0.33.. - 2d
This step might not be obvious, where you assume 1 - 0.66.. equals 0.33..

You've already assumed that 1/3 = 0.33.. + d and 2/3 = 0.66.. + 2d, so 1 would equal 0.99.. + 3d, wouldn't it?
My reasoning was that, like 0.33.. _+ 0.33.. = 0.66.., 1 - 0.66.. is undeniably 0.33.. whem you do the arithemtic and notice the repetition, wahtever you believe 0.33.. and 0.66.. actually represent. So if 0.33.. < 1/3, then 0.66.. < 2/3, and 1 - 0.66.. > 1/3, but this implies 0.33.. > 1/3. A similar contradiction would arise if 0.33.. > 1/3, but no such contradiction arises when 0.33.. = 1/3 (i.e. when d=0). I'm sure you realized that was my intent though. If not wrong, maybe it was just a long winded version of the 1 / 3 = 0.99.. / 3 argument.

[A Thousand Pardons]

My reasoning was that, like 0.33.. _+ 0.33.. = 0.66.., 1 - 0.66.. is undeniably 0.33..
My personal rule of thumb is, whenever a step in a proof was obvious, that was where I could find my error. :)

PS: Wait a minute! I mentioned this to you before (http://www.badastronomy.com/phpBB/viewtopic.php?p=346821&highlight=error+mistake#346 821).

[worzel]

My reasoning was that, like 0.33.. _+ 0.33.. = 0.66.., 1 - 0.66.. is undeniably 0.33..
My personal rule of thumb is, whenever a step in a proof was obvious, that was where I could find my error. :)
But to doubt the process that gives 1-0.66..=0.33.. would be to doubt the process that gives 0.33..+0.33..=0.66.. and 0.33..+0.66..=0.99.., wouldn't it?

[A Thousand Pardons]

My reasoning was that, like 0.33.. _+ 0.33.. = 0.66.., 1 - 0.66.. is undeniably 0.33..
My personal rule of thumb is, whenever a step in a proof was obvious, that was where I could find my error. :)
But to doubt the process that gives 1-0.66..=0.33.. would be to doubt the process that gives 0.33..+0.33..=0.66.. and 0.33..+0.66..=0.99.., wouldn't it?
No, of course not, since with those you can almost immediately derive 1 = 0.99..

IOW, if you are doubting 1 = 0.99.., then you'd surely also doubt that.

[worzel]

My reasoning was that, like 0.33.. _+ 0.33.. = 0.66.., 1 - 0.66.. is undeniably 0.33..
My personal rule of thumb is, whenever a step in a proof was obvious, that was where I could find my error. :)
But to doubt the process that gives 1-0.66..=0.33.. would be to doubt the process that gives 0.33..+0.33..=0.66.. and 0.33..+0.66..=0.99.., wouldn't it?
No, of course not, since with those you can almost immediately derive 1 = 0.99..
Of course, but any of these proofs show that, the point is to try to prove it from only those elements that the inequality believers belive in, surely. The process of addition and subtraction are essentially the same, so if you accept that 0.33.. + 0.33.. = 0.66.. then how can you doubt 1 - 0.66.. = 0.33..?


IOW, if you are doubting 1 = 0.99.., then you'd surely also doubt that.
I disagree. Several people on this thread have not doubted that 0.33.. + 0.33.. = 0.66.. but have doubted that 0.33.. = 1/3 (i.e. that 0.99.. = 1) and have claimed that 0.33.. < 1/3 (i.e. that 0.99.. < 1). Maybe what I was trying to show was trivial, but I was trying to show that by the process that they are happy with, 0.33.. <> 1/3 is contradictory.

EDIT: On second thoughts, doing 1-0.66.. on paper leads to ever more accurate approxminations which decrease rather than increase as in 1/3. Although at infinity they become the same, the argument for 0.33.. < 1/3 would content that 1-0.66.. > 1/3 and while different, are both approximated by 0.33..

My apologies ATP, you're absolutely right, I guess it's difiifcult (for me anyway) when working from an assumption you know to be false not to sneak in the correct assumption when trying to reach a contradiction. Oh well, back to my day job.

[fosley]

My reasoning was that, like 0.33.. _+ 0.33.. = 0.66.., 1 - 0.66.. is undeniably 0.33..
My personal rule of thumb is, whenever a step in a proof was obvious, that was where I could find my error. :)
But to doubt the process that gives 1-0.66..=0.33.. would be to doubt the process that gives 0.33..+0.33..=0.66.. and 0.33..+0.66..=0.99.., wouldn't it?
No, of course not, since with those you can almost immediately derive 1 = 0.99..
Of course, but any of these proofs show that, the point is to try to prove it from only those elements that the inequality believers belive in, surely. The process of addition and subtraction are essentially the same, so if you accept that 0.33.. + 0.33.. = 0.66.. then how can you doubt 1 - 0.66.. = 0.33..?


IOW, if you are doubting 1 = 0.99.., then you'd surely also doubt that.
I disagree. Several people on this thread have not doubted that 0.33.. + 0.33.. = 0.66.. but have doubted that 0.33.. = 1/3 (i.e. that 0.99.. = 1) and have claimed that 0.33.. < 1/3 (i.e. that 0.99.. < 1). Maybe what I was trying to show was trivial, but I was trying to show that by the process that they are happy with, 0.33.. <> 1/3 is contradictory.

EDIT: On second thoughts, doing 1-0.66.. on paper leads to ever more accurate approxminations which decrease rather than increase as in 1/3. Although at infinity they become the same, the argument for 0.33.. < 1/3 would content that 1-0.66.. > 1/3 and while different, are both approximated by 0.33..

My apologies ATP, you're absolutely right, I guess it's difiifcult (for me anyway) when working from an assumption you know to be false not to sneak in the correct assumption when trying to reach a contradiction. Oh well, back to my day job.

The problem with assuming that 1-0.66...=0.33... is evident when you stop at some arbitrary point, say four decimal places:

1.0000
- 0.3333
-----------
0.6667


So:

1.0...00
- 0.3...33
------------
0.6...67

That is, 0x1 + 6x(1/10) + . . . + 6x(1/<infinity-1>) + 7x(1/<infinity>)

It could be looked at as a pattern:

1 - 0.3 = 0.7
1 - 0.33 = 0.67
1 - 0.333 = 0.667
1 - 0.3333 = 0.6667
...
1 - 0.3...33 = 0.6...67

That pattern continues as far as I care to follow it - I see no reason to think it would stop working when you get out to infinity.

Just like the long division of 1/3 is a pattern:
<answer for this iteration><r=remainder><value of current remainder after considering it's place value>
0r0.3
<decimal point>
3r0.03
3r0.003
3r0.0003
...
3r0.0...03

I don't see why that pattern would stop working, either.

1/3 = 0.3...3 + 1/<infinity>/3

So, it really depends on whether 1/<infinity>=0. Now, according to everything I know about such things, it's generally accepted that infinitesimals do exist.

I was going to use something about the space-time continuum here, but, continuum, by definition, means that you can't distinguish between 0 and whatever lay next to it. However, you can obviously distinguish between 0 and 1, so somewhere in between [0 and 1) is a point that can be distinguished from 0. I would guess that the thing that lay two points from 0 can be distinguished from 0.

However, 1/<infinity> can still be divided by something and made smaller, so it doesn't lie next to 0. Assuming that my "guess" is correct, that means 1/<infinity> > 0.

I really need to go to bed though, so I'm not going to try to go any further with this at the moment :D.

[worzel]

The problem with assuming that 1-0.66...=0.33... is evident when you stop at some arbitrary point, say four decimal places:A recurring decimal expansion never stops, ever.

That pattern continues as far as I care to follow it - I see no reason to think it would stop working when you get out to infinity.You can easily prove that the pattern repeats forever.

So, it really depends on whether 1/<infinity>=0. Now, according to everything I know about such things, it's generally accepted that infinitesimals do exist.But an infinitesimal (http://mathworld.wolfram.com/Infinitesimal.html) that is not zero is not (http://mathworld.wolfram.com/NonstandardAnalysis.html) part of the standard real number system.

[kucharek]

I don't know if I'm happy that this thread got bumped... [-X :D

[worzel]

I don't know if I'm happy that this thread got bumped... [-X :D
Cheer up, you could always exercise the right not to click on it anymore :)

[Fortis]

The problem with assuming that 1-0.66...=0.33... is evident when you stop at some arbitrary point, say four decimal places:A recurring decimal expansion never stops, ever.
And neither does a thread about them. ;) :)

Folsley, this has been mentioned earlier in the thread, but ask yourself what number comes between 1 and 0.9999... Then ask yourself what this means for numbers which are point objects (with no finite extent) on the number line. :)

[farmerjumperdon]

Very strange. What if I only knew arithmetic? Wouldn't I be correct in saying .99 repetend 9 does not equal 1?

Why does knowing calculus change that? I guess I have a problem with the use of the term "exactly." Does it really exactly equal 1, or for certain calculations is it close enough to mean the same. Or close enough it doesn't matter.

I understand that the concept of infinity is difficult for a lot of people to get. They think of it quantitatively (as in think of the biggest number you can and then add something) rather than qualitatively (as in an abstract you can not contain).

Which makes me wonder, does infinity really exist, other than in concept? Is there an example of something we can prove is infinite, other than as an idea based on mathematical constructs?

For example, in geometry we are taught there are an infinite number of points on a line segment because between any 2 points there can be another point. But isn't that just by our own definition of terms? Isn't there a certain minimal size for a physical thing, like a singularity? If there were 2 singularities sitting next to and adjoining each other, could anything fit in between? Isn't there a downward limit? Can one singularity be smaller than another? If a singularity represents a limit to smallness, then things can not be infinitely small.

Doesn't infinity only exist as a mathematical construct?

[worzel]

Very strange. What if I only knew arithmetic? Wouldn't I be correct in saying .99 repetend 9 does not equal 1? If you do the arithemtic then when you dicide to stop you'll have to round up because you know that the next digit is a nine. The longer you do it for, the less the error. If do you it for infinity, the error is infinitly small (which is zero).

Why does knowing calculus change that?There are some proofs on this thread that don't rely on calculus. The easiest one is:

1/3 = 0.33...
2/3 = 0.66...
3/3 = 0.99...


I guess I have a problem with the use of the term "exactly." Does it really exactly equal 1, or for certain calculations is it close enough to mean the same. Or close enough it doesn't matter.According to the definitions of real numbers, it exactly equals one.


Which makes me wonder, does infinity really exist, other than in concept? Is there an example of something we can prove is infinite, other than as an idea based on mathematical constructs?As infinity is a mathematical concept I don't think we can prove it exists without maths, anymore than we can prove that the number one exists.


For example, in geometry we are taught there are an infinite number of points on a line segment because between any 2 points there can be another point. But isn't that just by our own definition of terms?Yes. But 2+2 is only 4 because of the way we define '+', and the angles of a triangle only add up to 180 degrees because we assume they do (or something similar that implies they do).


Isn't there a certain minimal size for a physical thing, like a singularity? If there were 2 singularities sitting next to and adjoining each other, could anything fit in between? Isn't there a downward limit? Can one singularity be smaller than another? If a singularity represents a limit to smallness, then things can not be infinitely small.0.99..=1 is a mathematical statement only. Whether a particular mathematical structure accurately models the universe or not is what physics is about, and unlike maths, is never certain. "Does 0.99..=1" is short hand for "Is it implicit in the assumptions of the real number system that 0.99..=1" and the answer is definitely yes, whatever the nature of the universe. The only way you could question it is by questioning logical reasoning itself, but it would then be quite hard to articulate your reasoning :)

[farmerjumperdon]

So if I do not round, does it still exactly equal 1?

I get your points, and appreciate them. I'm definitely in way over my head on the topic. It does occur to me that not only does math have a language all it's own, but a logic all it's own.

If I lived on a street that I was told went on forever, and my house number was .9999999 and I was told the house numbers kept getting bigger, but never reached 1; would a mathemetician still insist that way doen the street somewhere was a house numbered 1?

Reminds me of a great quotation from Lederman's The God Particle: Physics is to math as sex is to masturbation.

[jfribrg]

The thread that never dies. I predict that eventually the number of posts on this thread will exceed the number of 9's in 1.0000.... :lol:

[Severian]

I don't know if I'm happy that this thread got bumped... [-X :D
Yeah, this is my fault. fosley said something similar in another thread and I turned him/her to this one.

[A Thousand Pardons]

Reminds me of a great quotation from Lederman's The God Particle: Physics is to math as sex is to masturbation.
Wasn't that Feynman? but he hated mathematicians :)

[farmerjumperdon]

I think you are right about it being Feynman. I only meant I read it in Lederman's book.

[worzel]

So if I do not round, does it still exactly equal 1?
Yes, if you go on for ever.


If I lived on a street that I was told went on forever, and my house number was .9999999 and I was told the house numbers kept getting bigger, but never reached 1; would a mathemetician still insist that way doen the street somewhere was a house numbered 1?
I don't think house numbering is a helpful analogy, there are uncountably infinite real numbers between any two real numbers (that aren't equal), so labelling them with house numbers would get you nowhere, even if you did it forever.

But if you want to imagine houses, then a better analogy might be to say that even if you thought you lived next door to house number 1, there would still be infintely many housees between you and your neighbour - and between each of these infinte inbetweens there would be infintely more inbetweens etc. etc.

[farmerjumperdon]

So that's how my neighborhood got so crowded. You know, I moved out to the country thinking I could get away from the madding crowd. I knew I should have went beyond algebra and geometry.

[kucharek]

So that's how my neighborhood got so crowded. You know, I moved out to the country thinking I could get away from the madding crowd. I knew I should have went beyond algebra and geometry.
To handle the crowd, you may open a Hilbert hotel. Unfortunately, it is too small for real numbers of people (pun intended).

[worzel]

So that's how my neighborhood got so crowded. You know, I moved out to the country thinking I could get away from the madding crowd. I knew I should have went beyond algebra and geometry.
To handle the crowd, you may open a Hilbert hotel. Unfortunately, it is too small for real numbers of people (pun intended).
But if you opened an infinte chain of Hilbert Hotels you might just squeeze them all in.

EDIT: I think that should that be - open a new hotel for person in your original hotel.

EDIT 2: rats, how do you get infinity to the power of infinity guests?

[Normandy6644]

Wow this thing sure made a comeback. :o

[jfribrg]

A Hilbert Hotel has countably infinite rooms. An infinite number of Hilbert Hotels is countably infinite. Does that mean that the number of rooms in all the infinite number of hotels is also countably infinite? My answer is that it is countable, using the same triangulation technique that is used to prove that the rationals are countable. Is this true of all countable sets of countable sets? Is there a single necessary condition for a set to be uncountable? The proofs that I am familiar with deal with showing that an isomorphic mapping to the natural numbers is impossible. Is there any other way of proving uncountablility?

[Severian]

A Hilbert Hotel has countably infinite rooms. An infinite number of Hilbert Hotels is countably infinite. Does that mean that the number of rooms in all the infinite number of hotels is also countably infinite? My answer is that it is countable, using the same triangulation technique that is used to prove that the rationals are countable. Is this true of all countable sets of countable sets? Is there a single necessary condition for a set to be uncountable? The proofs that I am familiar with deal with showing that an isomorphic mapping to the natural numbers is impossible. Is there any other way of proving uncountablility?
A countably infinite number of Hilbert hotels is countably infinite. It is generally true that a countable union of countables is countable.

The times I've had to show something was uncountable, most of the time I've just shown it either bijects with the reals or contains the reals as a subset. The proof that the set of real numbers is uncountable is pretty neat.

[worzel]

A Hilbert Hotel has countably infinite rooms. An infinite number of Hilbert Hotels is countably infinite. Does that mean that the number of rooms in all the infinite number of hotels is also countably infinite? My answer is that it is countable, using the same triangulation technique that is used to prove that the rationals are countable. Is this true of all countable sets of countable sets?
I guess so. But infinity to the power of inifinity is a countable set of countable sets of countable sets of ... which is uncountable. Have a look here (http://mathworld.wolfram.com/Aleph-0.html) and follow the "See also"s.

[Disinfo Agent]

But an infinitesimal (http://mathworld.wolfram.com/Infinitesimal.html) that is not zero is not (http://mathworld.wolfram.com/NonstandardAnalysis.html) part of the standard real number system.
Those entries discuss two different concepts of infinitesimal. The one you want is in the latter link.


[...] in geometry we are taught there are an infinite number of points on a line segment because between any 2 points there can be another point. But isn't that just by our own definition of terms? Isn't there a certain minimal size for a physical thing, like a singularity?
Mathematics does not have to be constrained by physical limits.
Even if it were, how would we know what those limits are?

Edited.

[Disinfo Agent]

EDIT: I think that should that be - open a new hotel for person in your original hotel.

EDIT 2: rats, how do you get infinity to the power of infinity guests?
Goodness, you don’t need that much! :o :wink:

Invite the guests in the hotel to an infinite number of parties, like this:

To the first party, invite no-one.

To each of the following parties, invite only one guest; do this once for every guest in the hotel.

To the parties after those, invite just two guests, each time a different pair of guests. Include all possible pairs.

To the following parties, invite exactly three guests, each time a different trio of guests. Include all possible trios.

And so on, for every natural number of guests per party.

Next, start inviting an infinite number of guests for each party, but leave out a finite number of them. Do this for every possible combination of guests, and every natural number of guests left out per party.

Then, invite an infinite number of guests for each party, but leave out an infinite number of them, too. Do this for every possible (infinite) set of left out guests.

Finally, invite all guests to one last party, for good measure.

Now, suppose you wanted to use a different ballroom for each of those parties. That would make 2^aleph_0 ballrooms (http://mathworld.wolfram.com/PowerSet.html), which equals the continuum (http://mathworld.wolfram.com/Continuum.html). :)

[fosley]

[Edited a couple times for spelling corrections]

Ok. . . here's the best I've come to so far:

This part is just because everyone seems to think that you cannot go beyond infinity:

If there is such thing as 1/<infinity>, infinity is provably not the largest number possible - that is, it's possible to go past infinity.

There are an infinite number of infinitesimals in the number 1. There are twice as many of these infinite infinitesimals in the number 2:

3/3 = 1
2 * 3/3 = 2 * 1 = 2

<inf>/<inf> = 1
2 * <inf>/<inf> = 2 * 1 = 2

So, I decided to use the word "absolute" to represent the opposite of zero, since that's pretty much what it means in non-mathematical contexts. That is:

0=<nothing>
<abs>=<everything>
<inf>=<abstract set of numbers between 0 and <abs>>

So, <absolute> is <infinity>*<infinity>+<double your infinity>, repeated <forever>*<always>^<even more always>*<nuh-uh, I said "always" first!>. . .

The difference between a number that goes out to infinity, and one that goes out "absolutely":

One number higher than 0.9...9 is 0.9...9+1/<abs>. One number lower than 1 is 1-1/<abs>. The difference between those two numbers is 1/<inf> - 2*1/<abs>.

Also, due to the nature of <abs>, <abs>-1/<abs>!=<abs> where <inf>-1/<inf>=<inf>, because <inf> is not a number in and of itself - it is a collection of numbers that are defined by our inability to count them.

So, let us assume that the argument was about this number right above 0 that I define as 1/<abs>, instead of 1/<inf>:

First, you could then have 0.9.....<to absolute place value>....9, and it wouldn't round up because it is exactly that number.

Now, because in a continuum you can't distinguish two objects that lay next to each other, 1-(1/<abs>) (that is, 0.9...9) is indistinguishable from 1. However, that does not mean they are equal:

If
1-1/<abs>=1
and
(1-1/<abs>)-1/<abs>=1-1/<abs>
then
(1-1/<abs>)-1/<abs>=1

This pattern could be continued from any number to any other number, if it were true. Therefore, according to the logic that 1=0.9...9, 0=1 and 3=3,343,223. Because you can keep applying the logic the entire distance, all numbers would necessarily be equal. Since this is obviously not true, 0.9...9 cannot equal 1, although it can be indistinguishable.

[Chuck]

EDIT: I think that should that be - open a new hotel for person in your original hotel.

EDIT 2: rats, how do you get infinity to the power of infinity guests?
Goodness, you don’t need that much! :o :wink:

Invite the guests in the hotel to an infinite number of parties, like this:

To the first party, invite no-one.

To each of the following parties, invite only one guest; do this once for every guest in the hotel.

To the parties after those, invite just two guests, each time a different pair of guests. Include all possible pairs.

To the following parties, invite exactly three guests, each time a different trio of guests. Include all possible trios.

And so on, for every natural number of guests per party.

Next, start inviting an infinite number of guests for each party, but leave out a finite number of them. Do this for every possible combination of guests, and every natural number of guests left out per party.

The above parties are all countable. The ones in the next paragraph aren't.

Then, invite an infinite number of guests for each party, but leave out an infinite number of them, too. Do this for every possible (infinite) set of left out guests.

Finally, invite all guests to one last party, for good measure.

Now, suppose you wanted to use a different ballroom for each of those parties. That would make 2^aleph_0 ballrooms (http://mathworld.wolfram.com/PowerSet.html), which equals the continuum (http://mathworld.wolfram.com/Continuum.html). :)

[worzel]

The trouble is, fosley, you're making up your own maths. You would need to rigorously define your new concepts and then show that they are consistent. Even if your new system were consistent, it still wouldn't prove anything about the real number system. But I don't think it could be anyway. For instance, from your

<inf>-1/<inf>=<inf>
and
<inf>/<inf>=1

it follows that:


<inf> - 1 = <inf>-1/<inf> - <inf>/<inf>
= ( <inf>-1 - <inf> ) / <inf>
= -1 / <inf>
therefore
<inf> = 1 - 1/<inf>

Or in other words, infinity is infinitely close to 1, or 0.999... if you prefer.

[Fortis]

Since this is obviously not true, 0.9...9 cannot equal 1, although it can be indistinguishable.
I think that this is part of the problem. As you've written it, i.e. 0.9...9, it terminates, whereas 0.9... never terminates.

What do you think about the questions I asked, i.e. what number lies between 1 and 0.9...? :)

[jfribrg]

Since this is obviously not true, 0.9...9 cannot equal 1, although it can be indistinguishable.
I think that this is part of the problem. As you've written it, i.e. 0.9...9, it terminates, whereas 0.9... never terminates.

What do you think about the questions I asked, i.e. what number lies between 1 and 0.9...? :)

here we go again ](*,)

[Fortis]

Since this is obviously not true, 0.9...9 cannot equal 1, although it can be indistinguishable.
I think that this is part of the problem. As you've written it, i.e. 0.9...9, it terminates, whereas 0.9... never terminates.

What do you think about the questions I asked, i.e. what number lies between 1 and 0.9...? :)

here we go again ](*,)
Do you think that it may be more fruitful for each successive post to add an additional 9 to the sequence 0.9999... etc, and see if it reaches 1 before everyone is convinced by the other proofs that 1=0.9999...? ;)

[Musashi]

Sure.

I think 0.99999 = 1

[Inferno]

I can't believe this argument has been going for 27 pages.

Let's start a new page arguing whether 1 +1 = a window.

[Musashi]

No, I think it is important to determine if 0.999999 = 1

;)

[gzhpcu]

Being a converted "0.99999...=1" participant, I would just like to sum up with the infinite series:

for n=1 till n=infinity, the infinite series SUM[9/10^^n] = 0.9 + 0.09 + 0.009 .... converges to = 1

[Disinfo Agent]

So, I decided to use the word "absolute" to represent the opposite of zero, since that's pretty much what it means in non-mathematical contexts. That is:

[0=<nothing>
<abs>=<everything>
<inf>=<abstract set of numbers between 0 and <abs>>
Everything of what?

[fosley]

So, I decided to use the word "absolute" to represent the opposite of zero, since that's pretty much what it means in non-mathematical contexts. That is:

[0=<nothing>
<abs>=<everything>
<inf>=<abstract set of numbers between 0 and <abs>>
Everything of what?
<absolute>=The count of everything - every person, place, thing or idea - every point, every line, every curve, every number. Everything.

Because any lesser number can be trumped.

Now, absolute can also mean some other things. 1-1=0. In this case the answer is "absolute 0". Not "almost 0", but "exactly 0". So maybe there's a better word. It's just the word I chose.



Being a converted "0.99999...=1" participant, I would just like to sum up with the infinite series:

for n=1 till n=infinity, the infinite series SUM[9/10^^n] = 0.9 + 0.09 + 0.009 .... converges to = 1
Yes, but that begs the question. It's no different that saying "1/3=0.3... and 1/3 * 3 = 1 and 0.3... = 0.9... so 0.9...=1". The infinite series is damned close to accurate, so it works for anything we'll actually use. But to "prove" that 0.9...=1 using the infinite series, you must assume that the equation is perfect. Now, I'm also arguing that the equation cannot be perfect because 0.9... cannot be equal to 1.


<inf> - 1 = -1 / <inf>
therefore
<inf> = 1 - 1/<inf>
Yes, I see that my stated ideas of infinity could use some work. However, I also abandoned them for such reasons and went to the "absolute" idea.


What do you think about the questions I asked, i.e. what number lies between 1 and 0.9...?
There does not need to be anything between them. 0.9... can be right next to 1. It is therefore indistinguishable from 1 by the ideas of a continuum. However, as I stated, it cannot equal 1, because otherwise all numbers "equal" each other.


If do you it for infinity, the error is infinitly small (which is zero).
Circular logic, again. You are assuming that almost equal and equal are equivilent, then "proving" your point on the very assumption that is being challenged.

Here is another example of the difference:

Suppose that we remove everything from the universe except 1 atom. We know that we have 1 atom. We can go forever, and ever, and ever, and never find another atom. So, the universe is almost devoid of matter. It is actually 0.9... devoid (assuming an infinitely large universe, of course). But it is not 1.0 devoid of matter, because we have that 1 atom just sitting there all by itself. To claim that 0.9... = 1.0 is to claim that this atom no longer exists because it is the only one to exist. It simply doesn't work.

[A Thousand Pardons]

Circular logic, again. You are assuming that almost equal and equal are equivilent, then "proving" your point on the very assumption that is being challenged.



So, the universe is almost devoid of matter. It is actually 0.9... devoid (assuming an infinitely large universe, of course). But it is not 1.0 devoid of matter, because we have that 1 atom just sitting there all by itself. To claim that 0.9... = 1.0 is to claim that this atom no longer exists because it is the only one to exist. It simply doesn't work.
Isn't that also a circular argument? You're assuming that 0.9... devoid is not totally devoid.

[worzel]

<inf> - 1 = -1 / <inf>
therefore
<inf> = 1 - 1/<inf>
Yes, I see that my stated ideas of infinity could use some work. However, I also abandoned them for such reasons and went to the "absolute" idea.
You abandoned them because your ideas about infinity are inconsistent ?



What do you think about the questions I asked, i.e. what number lies between 1 and 0.9...?
There does not need to be anything between them. 0.9... can be right next to 1. It is therefore indistinguishable from 1 by the ideas of a continuum. However, as I stated, it cannot equal 1, because otherwise all numbers "equal" each other.
How are all numbers equal to each other if 0.99..=1? Do you mean if inf/inf=1? Then all numbers would be equal, but inf/inf is undefined in standard maths. So again, you're only only showing that your own ideas about infinity are inconsistent.



If do you it for infinity, the error is infinitly small (which is zero).
Circular logic, again. You are assuming that almost equal and equal are equivilent, then "proving" your point on the very assumption that is being challenged.
I wasn't trying to prove anything, I was just answering your question about what if you only knew arithmetic and your point that there must be an error when you stop - you seemed to think that because you can't do the expansion forever, that somehow forbids you from reasoning about doing it forever. I was just pointing out that if you do the expansion forever then the error is infinitely small, which is zero according to standard maths.

[mutineer]

What is the HIGHEST number that is LOWER than ONE ?
If it isn't 0.99999 ... recurring, what is it?

[Chuck]

There is no highest number lower than one. There are an infinite number of points between any two points. If a number is not one then there are an infinite number of points between that number and one.

[Lurker]

This is well known to be one. It's an infinite series that converges to one. Since the series is infinite it equals one.

[Dana_Mix]

The geometric series a+ar+ar^2...
converges if |r|<1 and equals a/(1-r)

It happens that for repeating digits, the answer is
the repented divided by a string of as many nines as the repented is long.
So .6161...=61/99, .612612=612/999, and of course .999... is 9/9, or 1.

[lti]

how is it that this is still being debated!? and why are the poll results so even! i would have thought that on a forum frequented by such people as u are it would be a very uneven poll result.

[mutineer]

how is it that this is still being debated!? and why are the poll results so even! i would have thought that on a forum frequented by such people as u are it would be a very uneven poll result.
I think it basically reflects the truism that a little learning is a dangerous thing. A truly simple mind and a truly intelligent mind both grasp that .9999... is unequal to 1. But it is possible to feed the simple mind with a series of propositions that skew if from using its common sense. Indoctination in scientology or marxism follows a similar course.

[Disinfo Agent]

So, I decided to use the word "absolute" to represent the opposite of zero, since that's pretty much what it means in non-mathematical contexts. That is:

[0=<nothing>
<abs>=<everything>
<inf>=<abstract set of numbers between 0 and <abs>>
Everything of what?
<absolute>=The count of everything - every person, place, thing or idea - every point, every line, every curve, every number. Everything.
I don't think there is such a count.
At least, Bertrand Russell showed in the early 20th century that the concept of a set that contains all sets -- the set whose cardinality would be the entity you're talking about, I expect -- leads to logical inconsistencies.


A truly simple mind and a truly intelligent mind both grasp that .9999... is unequal to 1. But it is possible to feed the simple mind with a series of propositions that skew if from using its common sense. Indoctination in scientology or marxism follows a similar course.
Don't you think you're being just a little bit arrogant?

After all, in this 28-page long thread, we've had several posters giving proofs that 0.999... =1, and we've even had a few explaining why that is reasonable.

Meanwhile, those who claim that 0.999... and 1 are two different numbers have been unable to produce a single proof of their claim.

Yet, we're the ones who are supposedly not "truly intelligent"? :roll:



how is it that this is still being debated!? and why are the poll results so even! i would have thought that on a forum frequented by such people as u are it would be a very uneven poll result.
I think it basically reflects the truism that a little learning is a dangerous thing.
That much is true. Clearly, some people went through high school without ever learning, or being taught, the true meaning of the decimal notation.

[Normandy6644]

how is it that this is still being debated!? and why are the poll results so even! i would have thought that on a forum frequented by such people as u are it would be a very uneven poll result.
I think it basically reflects the truism that a little learning is a dangerous thing. A truly simple mind and a truly intelligent mind both grasp that .9999... is unequal to 1. But it is possible to feed the simple mind with a series of propositions that skew if from using its common sense. Indoctination in scientology or marxism follows a similar course.

So political propaganda and mathematics are now the same thing? :roll:

[Gmann]

(with apologies to Sherri Lewis and Lambchop)

This is the thread that never ends,
it just goes on and on my friend.
Some people started posting it,
not knowing what it was.
Now they can't stop posting to it only just because,

This is the thread that never ends.... :D

(I'll be under the coffee table with my "guide to telling the difference between the devil's advocate, and the Beast) 8-[

[Glom]

Are we still on this? I thought we had settled it months ago. There was something like five proofs given to show mathematically 0.999r=1 and none to show they are not equal.

[JohnW]

Are we still on this? I thought we had settled it months ago. There was something like five proofs given to show mathematically 0.999r=1 and none to show they are not equal.
Exactly what I was thinking.

I know web-site polls are not very meaningful, but I was shocked that so many people on a science board didn't know this. Is mathematics education really that bad these days?

[Fram]

how is it that this is still being debated!? and why are the poll results so even! i would have thought that on a forum frequented by such people as u are it would be a very uneven poll result.
I think it basically reflects the truism that a little learning is a dangerous thing. A truly simple mind and a truly intelligent mind both grasp that .9999... is unequal to 1. But it is possible to feed the simple mind with a series of propositions that skew if from using its common sense. Indoctination in scientology or marxism follows a similar course.

Now that's a truly intelligent remark :roll:

[Moose]

how is it that this is still being debated!? and why are the poll results so even! i would have thought that on a forum frequented by such people as u are it would be a very uneven poll result.

I couldn't, for one, change my vote once I was satisfied by multiple proofs. I doubt I'm the only one. I suspect a new poll would look much different.

While my initial objection to 1 = 0.999... (rounding and/or carries from a programmer's perspective of "infinite", aka big but finite) was not satisfied by most of the proofs which had maths containing the same (IMO) issues.

I simply couldn't argue with this one:

1 = 1/3 + 1/3 + 1/3 = 0.333... + 0.333... + 0.333... = 0.999...

It's simple, true, and involves no possibility of rounding or carries to blur the issue.

[Moose]

Oh, and mutineer? :roll: Remember the Alamo, Dude.

[Lurker]

To mutineer and others that think that .999r is not equal to 1, could we see ANY sort of rigorously logical proof that these two quantities are NOT equal.

Oh and mutineer, when did the mathematics of series and limits become associated with scientology or marxism. :P


Edited to Add:

I do realize that mathematicshas has always been a bit fascist. Witness the fact that, as a chemistry professor once pointed out to me, 1+1 does not equal 3 no matter how large the values of 1 grow to be.

[Disinfo Agent]

I do realize that mathematicshas has always been a bit fascist. Witness the fact that, as a chemistry professor once pointed out to me, 1+1 does not equal 3 no matter how large the values of 1 grow to be.
Au contraire. Remember Orwell's Nineteen Eighty Four?
A fascist would tell you 1+1 is whatever he says it is. :)

[Lurker]

I do realize that mathematics has has always been a bit fascist. Witness the fact that, as a chemistry professor once pointed out to me, 1+1 does not equal 3 no matter how large the values of 1 grow to be.
Au contraire. Remember Orwell's Nineteen Eighty Four?
A fascist would tell you 1+1 is whatever he says it is. :)
I'll have you know I have made a career out of fluff posts!! I stand by every word of my statement!! [-(

So... if you line up an infinite number of Orwell's, then as they pass through the limit, what do they converge to?? :-k

:wink:

[Wally]

I can't believe this argument has been going for 27 pages.

Let's start a new page arguing whether 1 +1 = a window.

Nah. Let's not. It'd be a pain. 8)

[. . . runs and hides as tomatos get hurled in his general direction]

[Lurker]

Actually this whole argument comes as a real surprise to me. Waaaaaay, wayyyyyyy back before there were computers on desktops and Starbucks refused to toast your bagel, I was introduced to the fundamental theorem of calculus through a number of lectures on series and limits. One of the first "interesting" proofs we developed was the proof that .9999r was equal to 1.

So from my point of view this proof was the gateway to calculus and I have a hard time understanding how people can master the concept of the calculus without this sort of basic understanding of series and limits. I can still remember every time we wanted to integrate some area we would all laugh "we could chop it up in to a bunch of little strips, let the width of those strips become thinner and thinner, then as the width approaches 0 the area under the curve..."

I still have dreams about that class... but that's another story. 8-[

[worzel]

Actually this whole argument comes as a real surprise to me. Waaaaaay, wayyyyyyy back before there were computers on desktops and Starbucks refused to toast your bagel, I was introduced to the fundamental theorem of calculus through a number of lectures on series and limits. One of the first "interesting" proofs we developed was the proof that .9999r was equal to 1.
What I find curious about this whole debate is that people think that 0.99.. has some sort of external reality to be discovered or debated over. It seems to escape them that the real number system is perfectly well defined and within those definitions 0.99.. does equal 1. It is an understandable misunderstanding at first, but once you've seen one proof you can't believe it is not true without thinking that the real number system itself is inconsistent.

So if you're an inequality believer, you should be looking for that inconsistency within the real number system and not appealing to your own intuitions about how you think they should be defined.

[farmerjumperdon]

I for one do not think .999... has any external reality outside of this thread, or conversations like it.

Repetends are pretty much meaningless outside of thought experiments (what's the difference how many 9's there are once you get out 10 or 20 decimal places?).

I still contend that infinity exists only in theory. I mean it is a nice tool and all to help model and conduct experiments, but can it be proven that something physically touchable is infinite (something outside of mathematical theory and thought experiment)?

[Disinfo Agent]

I thought this was math we were discussing... My mistake, I guess.

[farmerjumperdon]

Suppose it could be. When I saw the title of the thread, I read it as simple arithmetic.

[Lurker]

I for one do not think .999... has any external reality outside of this thread, or conversations like it.

Repetends are pretty much meaningless outside of thought experiments (what's the difference how many 9's there are once you get out 10 or 20 decimal places?).

I still contend that infinity exists only in theory. I mean it is a nice tool and all to help model and conduct experiments, but can it be proven that something physically touchable is infinite (something outside of mathematical theory and thought experiment)?

Well you are welcome to that idea, but the Fundamental Theorem of Calculus is built on infinite series and limit theory. As an engineer and computer scientist, I have daily run-ins with both calculus, differential equations, and limit theory. These ideas are a part of my every day life as I use these mathematical tools to understand theories of computation, to deal with structural, to basic physics etc.

If you feel that they have no existance other than in theory that's up to you. I can't calculate the stresses in many structural elements without these tools. I cannot solve many basic physics problems with out these tools. They have a very real world existance to me.

[Grey]

Suppose it could be. When I saw the title of the thread, I read it as simple arithmetic.
Last I checked, arithmetic was math, too.


I still contend that infinity exists only in theory. I mean it is a nice tool and all to help model and conduct experiments, but can it be proven that something physically touchable is infinite (something outside of mathematical theory and thought experiment)?
Why would it make a difference if there's nothing physical that's infinite, since it is mathematical theory that we're discussing?

[farmerjumperdon]

Arithmetic only deals with positive numbers. No negatives and no variables. Add symbols and negative numbers and you have math.

As I said, the title of the thread was simply an arithmetic equation, and speaking strictly arithmetically, .999... does not equal 1. Expanding the scope of the discussion to the entire realm of mathematics changes things, and I should have recognized that.

Having said that and talking it through, whatever is used to convey repetend might be considered a symbol, taking the equation beyond the aritnmetical realm and, . . .

Never mind. :oops:

[A Thousand Pardons]

Arithmetic only deals with positive numbers. No negatives and no variables. Add symbols and negative numbers and you have math.
There are other versions of arithmetic. Your version is nonstandard, but still...


As I said, the title of the thread was simply an arithmetic equation, and speaking strictly arithmetically, .999... does not equal 1.

How does your version include "infinite 9s"?

Never mind. :oops:
LOL! OK!

Is that another convert? :)

[farmerjumperdon]

Yes, converted. Now about that Planet X thing.

BTW, my definition of arithmetic is from Websters, and is the only version given. To be exact . . . "the science of computing by positive, real numbers."

Mathematics on the other hand is . . . "the science dealing with quantities, forms, etc. and their relationships by the use of numbers and symbols."

[Lurker]

Yes, converted. Now about that Planet X thing.

HEY!! We got a forum for that discussion and this isn't it!!! [-(

:P

Sorry if the discussion gets a little heated, but it hits a raw nerve when someone talks like limit theory has no real meaning. Its just a character flaw... 8-[

[A Thousand Pardons]

BTW, my definition of arithmetic is from Websters, and is the only version given. To be exact . . . "the science of computing by positive, real numbers."

Weird. The online Merriam-Websters (http://www.m-w.com/cgi-bin/dictionary?book=Dictionary&va=arithmetic) says "nonnegative real numbers" so that version must have been before the invention of zero. :)

[fosley]

You abandoned them because your ideas about infinity are inconsistent ?
Umm. Yes. Although I think I've "fixed" that (by showing why it must be inconsistent--although I think I'm just going to quote someone else who said it before I got a chance to say it)


How are all numbers equal to each other if 0.99..=1? Do you mean if inf/inf=1? Then all numbers would be equal, but inf/inf is undefined in standard maths. So again, you're only only showing that your own ideas about infinity are inconsistent.
As I said before:
If A=B and B=C then A=C. I think we all know that one.

Now, if 1=0.9...9 and 0.9...9=0.9...98 then 1=0.9...98.

That is:
A=1
B=<the number right below 1>
C=<the number right below B>

So, if 1=<the number right below 1> and <the number right below 1>=<the number right below B> then 1=<the number right below B>.

Continue this pattern across the entire spectrum of numbers and every number suddenly equals one another. This is obviously false, so it seems to make sense that the original assertion (1=<the number right below 1>) is false.




<absolute>=The count of everything - every person, place, thing or idea - every point, every line, every curve, every number. Everything.
I don't think there is such a count.
At least, Bertrand Russell showed in the early 20th century that the concept of a set that contains all sets -- the set whose cardinality would be the entity you're talking about, I expect -- leads to logical inconsistencies.
I thought about that later, and I agree. The count of everything would include all the numbers in the count of everything, which doubles it's own size, which doubles the count, etc.

I guess we have to go with the idea of never-ending numbers. So, infinitesimal has to be 0.0<insert a never-ending amount of 0's here>1. Now, everyone keeps saying that you can't have such a number, which means that all you can do is have 0.0...0, which is obviously 0. But it is zero, completely zero, and nothing but zero. There is no point in creating a new word for zero that we call "infinitesimal". "Infinitesimal" should refer to the point just above zero, and if there is such a point, it has to be 0.0...01, however that works.

But if one insists that you can't do that because "never-ending" has no end, then what is right above zero?

Anyhow, the end result is this:
If there is no "lowest number >0" (so it rationally follows that there is no "highest number <1"), then 0.9...9 is obviously not 1, because there are numbers between them.
If there is a "lowest number >0" (and it follows that there is a "highest number <1), 0.9...9 is obviously not 1, because we just said it's less than 1 (unless the "highest number <1" is <0.9...9 in which case we have to figure out what it is, and whether 0.9...9 rounds to 1, or to the "highest number <1"--however, that whole scenario doesn't really make sense).


I simply couldn't argue with this one:

1 = 1/3 + 1/3 + 1/3 = 0.333... + 0.333... + 0.333... = 0.999...
I've said this before, but maybe it bears saying again:

You can divide 1 by 3 for ever and ever and ever without end, but you will always have a remainder that is equal to 1/3 of whatever place value you are on. Now, if you get to a number that is as small as you can possibly go, it doesn't make sense that you can have 1/3 of that number (since it's indivisible, by definition). But neither does it make sense that you can just throw away that extra 1/3 (if you can just throw away 1/3 of a number, at what point can you do so?). So I have to come to the conclusion that you cannot represent the number 1/3 using the decimal numbering system. It is obviously a real number, and you can easily represent it in ternary as 0.1, but you can't represent it using the decimal numbering system.


To mutineer and others that think that .999r is not equal to 1, could we see ANY sort of rigorously logical proof that these two quantities are NOT equal.

Meanwhile, those who claim that 0.999... and 1 are two different numbers have been unable to produce a single proof of their claim.


So, the universe is almost devoid of matter. It is actually 0.9... devoid (assuming an infinitely large universe, of course). But it is not 1.0 devoid of matter, because we have that 1 atom just sitting there all by itself. To claim that 0.9... = 1.0 is to claim that this atom no longer exists because it is the only one to exist. It simply doesn't work.
Isn't that also a circular argument? You're assuming that 0.9... devoid is not totally devoid.
Given:
The universe contains nothing, except 1 atom and space.
The atom remains a constant size, for whatever reason (maybe it's at absolute zero, since there's no heat in the universe--not that it matters, since this is hypothetical, but wouldn't that work?)
The universe is the size of 10 atoms.

The universe is 0.1 devoid of space and 0.9 devoid of matter, by volume.

Increase the size of the universe to 100 atoms: the universe is now 0.01 devoid of space and 0.99 devoid of matter.

Increase the size of the universe so that it's never-ending: the universe is now 0.0...01 devoid of space and 0.9...9 devoid of matter.

In order for the universe to be 0.0 devoid of space and 1.0 devoid of matter, there would have to be 0 atoms. However, there is 1 atom, so the universe simply is not devoid of matter. To say 0.9...9 devoid is equal to 1.0 devoid is to say that 1 atom is equal to 0 atoms (because a similar universe with 0 atoms would obviously be 1.0 devoid of matter), which is a self-evident untruth.

Nowhere did I rely on any assumptions, except the assumption that there is such a thing as "never-ending". Without that assumption, the argument that 0.9...9=1 is pointless, because the number doesn't exist. Furthermore, 0.<insert the max number of 9's here> is not 1, because it has to be less than 0.9...9 (because the non-existent fraction of it has been removed), which is already less than 1.

I suppose I also relied upon the assumption that the pattern [0.9, 0.99, 0.999, . . .]=0.9...9 after never-ending repetitions. However, I think that's pretty self-evident also.

[Lurker]

Congratulations!! After several hundred years of successful application of the calculus to all manner of problems you have just proven that the limit theory that it's built upon is incorrect. =D>


Or maybe we should rethink your proofs a bit... :-k


:wink:

[ToSeek]

Continue this pattern across the entire spectrum of numbers and every number suddenly equals one another. This is obviously false, so it seems to make sense that the original assertion (1=<the number right below 1>) is false.

That's not the original assertion, though. You're assuming that 0.999... is "the number right below 1". We're saying it's not. You can't use your assumption to prove your assumption.


If there is no "lowest number >0" (so it rationally follows that there is no "highest number <1"), then 0.9...9 is obviously not 1, because there are numbers between them.

So what numbers are between 0.999... and 1? I say there aren't any.

[Moose]

Now, if 1=0.9...9 and 0.9...9=0.9...98 then 1=0.9...98.

Sorry, but you're basing this on an incorrect assumption.

1 = 0.999.... (infinite progression), not 0.999...9. (finite progression.)

Infinity is infinite. You need to stop thinking about last digits. They simply don't apply. Infinity laughs in the face of your puny final digit.

The rest of your argument fails out of the gate, unfortunately, so there's no real point in my dissecting it.

[Edit to add an irresistable dorky wisecrack.]

[Donnie B.]

Infinity laughs in the face of your puny final digit.
Is it wrong to burst out in hysterical laughter when you're all alone and reading a video monitor?

:lol: =D>

[01101001]

You can divide 1 by 3 for ever and ever and ever without end, but you will always have a remainder that is equal to 1/3 of whatever place value you are on.
Question. You don't seem to have any problem knowing what "1/3" is. You're using it. It's that thing when added to itself twice equals 1, right?

Yet, 1/3 is a division problem. It is 1 divided by 3. Look at it: 1 divided by 3. 3 goes into 1.

"You can divide 1 by 3 for ever and ever without end", you say? Why, then, does "1/3" equal one-third of one, i.e. why is 1/3 + 1/3 + 1/3 = 1?

Why isn't the sum less than one, since it is the sum of three "forever divisions"?

[SciFi Chick]

Infinity laughs in the face of your puny final digit.
Is it wrong to burst out in hysterical laughter when you're all alone and reading a video monitor?

:lol: =D>

Absolutely not. People who do that are totally cool. :D

[Lurker]

Infinity laughs in the face of your puny final digit.
I'll sue!!! [-(

[Inferno]

Maybe this is one of those things where in certain circumstances 0.9999... does not equal one, while in others it does?

[SciFi Chick]

Maybe this is one of those things where in certain circumstances 0.9999... does not equal one, while in others it does?

Care to list a circumstance in which it's NOT equal to one?

[Lurker]

Maybe this is one of those things where in certain circumstances 0.9999... does not equal one, while in others it does?

Care to list a circumstance in which it's NOT equal to one?
Yeah... like when I take it to court!! [-(

And on the third full moon after the wolfbane blooms for us pagans... 8-[

:wink:

[SciFi Chick]

Maybe this is one of those things where in certain circumstances 0.9999... does not equal one, while in others it does?

Care to list a circumstance in which it's NOT equal to one?
Yeah... like when I take it to court!! [-(

And on the third full moon after the wolfbane blooms for us pagans... 8-[

:wink:

You've been doing Calculus too long, and I'm scared of what is going to happen to me this summer. :o

:wink:

[Lurker]

You've been doing Calculus too long, and I'm scared of what is going to happen to me this summer. :o

:wink:
You're doooooooommmmmmed!!! :o

Don't worry... just remember to make regular sacrifices to the limit gods and all will be fine..... 8-[

yup... I need a vacation... 8)

[lti]

note: we are not discussing whether 0.999999999 (with lots of 9s) is equal to 1. It isnt.

we are discussing whether 0.9 ... recurring is equal to 1. it is.

as we are unable to display the accepted notation for a recuring number with ASCII we have to rely on writing lots of 9s which may confuse people who counter that lots and lots of 9s is not 1

a real number is a number that can be written as a fraction with an integer as both a numerator and denominator. u all accept that 0.3 ... recurring can. 1/3

so why is three times that not 1/1?

i really didnt mean to say the same things which have already been said. I am just shocked at the level maths seems to have slid to world wide. especially amongst a group of (theoretically) pretty astute students of the sciences.

[Dana_Mix]

fosley, work out the average of .999... and 1


And your atom example doesn't fit here; there are a finite number of atoms. That's the same mistake you made with numbers.

[Fram]

Well, the first time you do 1/3, it's 0.333333.....3. The second time, it's again 0.333333.....3. But the third time, it's 0.333333.....4. Add them up, and you'll have 1. This way, 0.999999.....9 isn't equal to 1, division still works, and the world keeps on turning. Now all I need is a way to decide if 1/3 is on its first, second or third try :^o

[worzel]

How are all numbers equal to each other if 0.99..=1? Do you mean if inf/inf=1? Then all numbers would be equal, but inf/inf is undefined in standard maths. So again, you're only only showing that your own ideas about infinity are inconsistent.
As I said before:
If A=B and B=C then A=C. I think we all know that one.

Now, if 1=0.9...9 and 0.9...9=0.9...98 then 1=0.9...98.

That is:
A=1
B=<the number right below 1>
C=<the number right below B>

So, if 1=<the number right below 1> and <the number right below 1>=<the number right below B> then 1=<the number right below B>.
0.9...98 is not a decimal representation of a real number (if it were it would probably represent the real number 1 as well). There is no real number right below one (or any other real number). Between any two real numbers that aren't equal there are infinitely more real numbers. I think you should reflect on that, and the question put to you numerous times: what numbers lie between 0.99.. and 1?

[Disinfo Agent]

As I said before:
If A=B and B=C then A=C. I think we all know that one.

Now, if 1=0.9...9 and 0.9...9=0.9...98 then 1=0.9...98.
By definition, when we write 0.999..., we mean "zero, point, followed by a countable infinity of digits -- in other words, a sequence of digits -- , all of which are nines".

Now let me ask you: what do you mean by "0.9...98"?




<absolute>=The count of everything - every person, place, thing or idea - every point, every line, every curve, every number. Everything.
I don't think there is such a count.
At least, Bertrand Russell showed in the early 20th century that the concept of a set that contains all sets -- the set whose cardinality would be the entity you're talking about, I expect -- leads to logical inconsistencies.
I thought about that later, and I agree. The count of everything would include all the numbers in the count of everything, which doubles it's own size, which doubles the count, etc.
Did you ever hear about the Barber's Paradox (http://plus.maths.org/issue20/xfile/)? I think you'll like it. :)


Anyhow, the end result is this:
If there is no "lowest number >0" (so it rationally follows that there is no "highest number <1"), then 0.9...9 is obviously not 1, because there are numbers between them.
Unless 0.9...9 and 1 are the same number.

[Disinfo Agent]

I simply couldn't argue with this one:

1 = 1/3 + 1/3 + 1/3 = 0.333... + 0.333... + 0.333... = 0.999...
I've said this before, but maybe it bears saying again:

You can divide 1 by 3 for ever and ever and ever without end, but you will always have a remainder that is equal to 1/3 of whatever place value you are on.
Not if you carry out the division "for ever and ever and ever without end". :)

Here's how I divide 1 by 3:

first step:

1 | 3
___0

2nd. step:

1.0 | 3
__1_0.3

3rd. step:

1.00 | 3
__10_0.33
___1

4th. step:

1.000 | 3
__10__0.333
___10
____1

...

As I go through these successive steps, I get better and better approximations to the actual value of 1 divided by 3: 0, 0.3, 0.33, 0.333, etc. But if I stop at any one of the steps above, I end up only with an approximate value of 1 divided by 3, not the exact value.

If I want the exact value, I must divide a (countable) infinite number of times. Obviously, this is humanly impossible, but there are ways to get around it.

1 - I can accept (intuitively) that there's a pattern in the division that will always repeat, and, by allowing that pattern to extend to infinity, state that 1/3 = "zero, point, a countable infinity of threes".

2 - Or, if I'm very nitpicky, I can claim that only the steps I can actually carry out to their end are meaningful, and that in this case the algorithm can only produce approximate divisions, not the exact division. However, even if I decide to apply such strict logical standards, I can still reverse the less accurate argument given in (1), and accept the symbol "zero, point, a countable infinity of threes" as a suggestive notation for the limit of the sequence of approximations (0, 0.3, 0.33, 0.333, ...), which is 1/3. :P


Given:
The universe contains nothing, except 1 atom and space.
The atom remains a constant size, for whatever reason (maybe it's at absolute zero, since there's no heat in the universe--not that it matters, since this is hypothetical, but wouldn't that work?)
The universe is the size of 10 atoms.
Nitpick 1: assuming we're not talking about a 2-dimensional universe, you really should measure space by volume, too.
Nitpick 2: 'matter' is a bad choice of words. If such universe has only 1 atom, then presumably all its matter rests on that atom.

So, let's just talk about volume, and make this simpler. The universe has only one atom, with volume V, and the total volume of the universe is 10 times that of the atom.


The universe is 10% matter and 90% vacuum.

Increase the volume of the universe to 100 atoms: 1% matter and 99% vacuum.

Increase the volume of the universe so that it's never-ending: the universe is now 0.0...01 devoid of space and 0.9...9 devoid of matter.

In order for the universe to be 0.0 devoid of space and 1.0 devoid of matter, there would have to be 0 atoms.

However, there is 1 atom, so the universe simply is not devoid of matter. To say 0.9...9 devoid is equal to 1.0 devoid is to say that 1 atom is equal to 0 atoms (because a similar universe with 0 atoms would obviously be 1.0 devoid of matter), which is a self-evident untruth.

I kept the blue parts as you wrote them, because this is where we may diverge.

If I understand your scenario correctly, you're talking about a universe with infinite volume. In that case, the volume of its vacuum is infinity, which is 100% of infinity, and the relative volume of the atom is V/infinity=0, which we can regard as 0% of the volume of the universe.
I don't see any problem with this. Zero percent of a volume can be a nonzero volume, as long as the total volume is infinite. It does not mean that the universe is devoid of matter.

[SciFi Chick]

Several people keep commenting on the state of math in the world. I'd like to clear up a misconception. Math is doing just fine. Universities and high schools in the U.S. do not all require Calculus. I haven't taken Calculus yet, and you don't learn this before Calculus.

Now, if you want to all bemoan the fact that we aren't required to take Calculus, go right ahead, but just be clear on what is upsetting you.

I'd be interested in knowing how many people who originally voted incorrectly in this poll had Calculus. As I've mentioned, I haven't. :)

[worzel]

Several people keep commenting on the state of math in the world. I'd like to clear up a misconception. Math is doing just fine. Universities and high schools in the U.S. do not all require Calculus. I haven't taken Calculus yet, and you don't learn this before Calculus.

Now, if you want to all bemoan the fact that we aren't required to take Calculus, go right ahead, but just be clear on what is upsetting you.

I'd be interested in knowing how many people who originally voted incorrectly in this poll had Calculus. As I've mentioned, I haven't. :)
I think the problem is more that people don't know the difference between a mathematical proof and a vague argument based on one's own (mis)conceptions, not that poeple didn't already know this particular nugget.

[SciFi Chick]

Several people keep commenting on the state of math in the world. I'd like to clear up a misconception. Math is doing just fine. Universities and high schools in the U.S. do not all require Calculus. I haven't taken Calculus yet, and you don't learn this before Calculus.

Now, if you want to all bemoan the fact that we aren't required to take Calculus, go right ahead, but just be clear on what is upsetting you.

I'd be interested in knowing how many people who originally voted incorrectly in this poll had Calculus. As I've mentioned, I haven't. :)
I think the problem is more that people don't know the difference between a mathematical proof and a vague argument based on one's own (mis)conceptions, not that poeple didn't already know this particular nugget.

Either way - it's not indicative of the state of math. That was my point.

[Moose]

I'd be interested in knowing how many people who originally voted incorrectly in this poll had Calculus. As I've mentioned, I haven't. :)

No argument with anything you've said, I just wanted to mention that I have had calculus (12th grade and 1st year of university), and had been incorrect. This was mainly because I was looking at the issue through programmer-tinted glasses where infinity can only be roughly approximated, and rounding errors are a fact of life.

[SciFi Chick]

I'd be interested in knowing how many people who originally voted incorrectly in this poll had Calculus. As I've mentioned, I haven't. :)

No argument with anything you've said, I just wanted to mention that I have had calculus (12th grade and 1st year of university), and had been incorrect. This was mainly because I was looking at the issue through programmer-tinted glasses where infinity can only be roughly approximated, and rounding errors are a fact of life.

Ah... that makes sense. I did say I was curious after all. 8)

[worzel]

Several people keep commenting on the state of math in the world. I'd like to clear up a misconception. Math is doing just fine. Universities and high schools in the U.S. do not all require Calculus. I haven't taken Calculus yet, and you don't learn this before Calculus.

Now, if you want to all bemoan the fact that we aren't required to take Calculus, go right ahead, but just be clear on what is upsetting you.

I'd be interested in knowing how many people who originally voted incorrectly in this poll had Calculus. As I've mentioned, I haven't. :)
I think the problem is more that people don't know the difference between a mathematical proof and a vague argument based on one's own (mis)conceptions, not that poeple didn't already know this particular nugget.

Either way - it's not indicative of the state of math. That was my point.
I disagree. A good high school education in math should enable you to see the truth of a simple proof like the ones offered, and should enable you to see the illogic of arguing from intuition against a true statement from a formal system. If someone were to argue for using a different formal system, that would be one thing; but to not even realize that they are arguing against the formal system that half of their arguments rest upon is indicative of a poor understanding about what maths is, IMO.

[SciFi Chick]

Either way - it's not indicative of the state of math. That was my point.
I disagree. A good high school education in math should enable you to see the truth of a simple proof like the ones offered, and should enable you to see the illogic of arguing from intuition against a true statement from a formal system. If someone were to argue for using a different formal system, that would be one thing; but to not even realize that they are arguing against the formal system that half of their arguments rest upon is indicative of a poor understanding about what maths is, IMO.[/quote]

Just because a few people do not understand math does not mean education in math is suffering.

As has been pointed out several times, the poll is not indicative of people understanding the math proofs. Most people would change their vote now, after having seen the proofs.

[Disinfo Agent]

Several people keep commenting on the state of math in the world. I'd like to clear up a misconception. Math is doing just fine. Universities and high schools in the U.S. do not all require Calculus. I haven't taken Calculus yet, and you don't learn this before Calculus.
You should learn the gist of it when you learn about infinite decimals. That's arithmetic, not calculus.

[worzel]

Either way - it's not indicative of the state of math. That was my point.I disagree. A good high school education in math should enable you to see the truth of a simple proof like the ones offered, and should enable you to see the illogic of arguing from intuition against a true statement from a formal system. If someone were to argue for using a different formal system, that would be one thing; but to not even realize that they are arguing against the formal system that half of their arguments rest upon is indicative of a poor understanding about what maths is, IMO.Just because a few people do not understand math does not mean education in math is suffering.

As has been pointed out several times, the poll is not indicative of people understanding the math proofs. Most people would change their vote now, after having seen the proofs.
Fair point - if we're just talking about the pole result (which I suppose you were) then I agree with you. But there must have been a good 10% - 20% of the total number of voters trying to argue the inequality on this thread - and they're mainly quite articulate, whch I do find worrying.

[SciFi Chick]

Several people keep commenting on the state of math in the world. I'd like to clear up a misconception. Math is doing just fine. Universities and high schools in the U.S. do not all require Calculus. I haven't taken Calculus yet, and you don't learn this before Calculus.
You should learn the gist of it when you learn about infinite decimals. That's arithmetic, not calculus.

Arithmetic may be a foundation, but several calculus people have pointed out this the theorem they first learned in calculus. I hope you aren't expecting everyone to be able to recreate the work behind advanced math theorems just because they took arithmetic. Arithmetic merely allows one to be able to understand the proofs offered for the theorem.

In any case, here's (http://www.badastronomy.com/phpBB/viewtopic.php?t=20207) a chance for all Babbers to redeem themselves. 8)

[SciFi Chick]

Fair point - if we're just talking about the pole result (which I suppose you were) then I agree with you. But there must have been a good 10% - 20% of the total number of voters trying to argue the inequality on this thread - and they're mainly quite articulate, whch I do find worrying.

I wouldn't worry too much. People aren't good at every subject. I know engineers who cringe at the idea of financial math, when they shouldn't have any problem with it. I've met articulate, well-educated people who haven't heard of Milton's Paradise Lost. There are gaps in every person's knowledge, even the well-educated ones. Not everyone can be Maksutov. :wink:

[Disinfo Agent]

Arithmetic may be a foundation, but several calculus people have pointed out this the theorem they first learned in calculus. I hope you aren't expecting everyone to be able to recreate the work behind advanced math theorems just because they took arithmetic. Arithmetic merely allows one to be able to understand the proofs offered for the theorem.
Different people will end up being persuaded by different arguments, as this thread shows. However, by the time a high school student finishes the topic of the "rational numbers", (s)he should (IMO) have been told, and appropriately convinced of facts like "1 divided by 3 equals 0.333..., with infinitely many threes", and "0.999..., with infinitely many nines, equals 1". When (s)he later studies calculus, the teacher will be able to present more rigorous proofs of these statements, but the facts should be known to the student as soon as he starts manipulating infinite decimal expressions. Otherwise, he will be manipulating entities he doesn't really understand.

To be quite fair, these are fine details, and not knowing them will not hamper the student's progression in math. But he should know them, if only so that he won't later be tempted to imply that his teachers are "not truly intelligent". :wink:

[SciFi Chick]

You wouldn't happen to be a math teacher would you? :wink:

[Disinfo Agent]

Drat! You've uncovered my sinister secret agenda! :wink:

[fosley]

Before I begin: I am sometimes using the word "infinity" to refer to this concept of a never-ending number, even though that is not what it means. I am mainly doing this because it's a lot easier to just say infinity than try to describe this never-ending number, and it pretty much works the same way.


Nitpick 1: assuming we're not talking about a 2-dimensional universe, you really should measure space by volume, too.
Nitpick 2: 'matter' is a bad choice of words. If such universe has only 1 atom, then presumably all its matter rests on that atom.

So, let's just talk about volume, and make this simpler. The universe has only one atom, with volume V, and the total volume of the universe is 10 times that of the atom.


The universe is 10% matter and 90% vacuum.

Increase the volume of the universe to 100 atoms: 1% matter and 99% vacuum.

Increase the volume of the universe so that it's never-ending: the universe is now 0.0...01 devoid of space and 0.9...9 devoid of matter.

In order for the universe to be 0.0 devoid of space and 1.0 devoid of matter, there would have to be 0 atoms.

However, there is 1 atom, so the universe simply is not devoid of matter. To say 0.9...9 devoid is equal to 1.0 devoid is to say that 1 atom is equal to 0 atoms (because a similar universe with 0 atoms would obviously be 1.0 devoid of matter), which is a self-evident untruth.

I kept the blue parts as you wrote them, because this is where we may diverge.

If I understand your scenario correctly, you're talking about a universe with infinite volume. In that case, the volume of its vacuum is infinity, which is 100% of infinity, and the relative volume of the atom is V/infinity=0, which we can regard as 0% of the volume of the universe.
I don't see any problem with this. Zero percent of a volume can be a nonzero volume, as long as the total volume is infinite. It does not mean that the universe is devoid of matter.
First - "volume" restricts the process to 3D, where "size" is a bit more general (although "atom" sort of restricts it to 3D, logically). Either way, the statement works the same way. So whatever, just have to nitpick back.

Second - the only argument I can think of against using the word "matter" is more of an argument about using an "atom" as a unit. An atom is mostly empty space, so technically my numbers were incorrect. However, I'm pretty sure you got the assumption that the size of the atom was determined by the bounds of the atom, not the sum of the bounds of each of it's component parts.

Third - as already discussed, infinity is not a number. Infinity is an entire group of numbers, defined by being immeasureably large; on this note, infinitesimal works the same way, except it is immeasureably small. So, 1xSomeInfinity is an infinite number and 2xTheSameInfinity is an infinite number, but the former is 50% of the latter. They aren't the same number, even though they are both in "the infinite set of numbers". If the assumption is invalid for me, it's invalid for you, also.

So, the volume of the vacuum is infinity, which is 99.9...% of the infinite volume of the universe. Now, that puts the "does 99.9...%=100%?" into play, so--just to show SomeInfinity!=AnotherInfinity--let's do something a little different: let's say 10% of the volume of the universe is matter, and 90% is space. Now, the universe can go on forever, so there is infinite matter and infinite space, but there is still only 10% matter--not 100%.


I think you should reflect on that, and the question put to you numerous times: what numbers lie between 0.99.. and 1?
And I've answered this question already. It doesn't matter to my argument. If there is nothing between 0.9... and 1, then they can still be different by an infinitesimal amount. If there is anything between them, it just shows that 0.9... doesn't equal 1, because you can go higher than 0.9... without even getting to 1. So the answer to that question can only hurt you, even though it keeps getting asked as though that somehow shows me to be wrong.

Now, it has been stated by several people that you can't have a number that's right below 1, because there are an infinite number of numbers between any two numbers. On the other side, there's the logic that there must be a number right below 1, even if it can't be represented by our conventional numbering system. Again, one way invalidates the argument that 0.9...=1 and the other one doesn't invalidate anything.


Now let me ask you: what do you mean by "0.9...98
As I wrote just below that, I mean the number right below 0.9....


And your atom example doesn't fit here; there are a finite number of atoms. That's the same mistake you made with numbers.
There was always a finite portion of the number. In the case of 1 or 0.9..., the finite portion is 1, and the not-finite portion is infinitesimal. The difference is that I'm scaling the entire equation up by a "factor of infinity". The ratio of infinity:1 is the same as the ratio of 1:infinitesimal. The first part of each ratio is infinite times larger than the second part. And, to show the idea in simpler terms: the ratio 3:1 is the same as the ratio 1:1/3. Or, X/1:1 is the same as 1:1/X. And, it lends even more to the idea that you can't truly have a "smallest difference between two numbers", because you can still divide that atom by infinity. Then you scale up by a factor of infinity again, and divide again, etc.

My example works perfectly here.


By definition, when we write 0.999..., we mean "zero, point, followed by a countable infinity of digits -- in other words, a sequence of digits -- , all of which are nines".
"Countable" infinity is an oxymoron. An infinite number is any number so large that it cannot be counted. A never-ending infinite number is a really big uncountable number.


Did you ever hear about the Barber's Paradox? I think you'll like it.
I have heard that, but didn't remember it until I read the site. Now, I would like to note:

1. The paradox isn't really a mathematical paradox (like many other "mathematical" paradoxes). It can affect mathematics, but is really more of a philosophical paradox, because it includes concepts beyond the (relatively) small scope of mathematics. I don't know that anyone claimed otherwise, but I just thought I'd mention it. A set that includes all red bikes has gone beyond math--you can count the items in the set, but math has no idea what is meant by "red" or "bike".

2. These types of paradoxes are (usually) easily circumvented in real life. The site gave one example: make the barber a female. You could also say: he falls under the category of "men who don't shave themselves" until he shaves himself, at which point he never shaves himself again. Or (the most common solution), you could add the exception that he is allowed to shave himself.

(Neither of those points has anything to do with 1 or 0.9..., but I figure I can still respond.)

This is not an attempt to show you somehow "wrong" for something you never said. This is really just my abstract thoughts on something.



Anyhow, the end result is this:
If there is no "lowest number >0" (so it rationally follows that there is no "highest number <1"), then 0.9...9 is obviously not 1, because there are numbers between them.
Unless 0.9...9 and 1 are the same number.
The only way to do that is to say:
0.9...=1
and
<something else>=<the number before 1>

Now, it's my turn to ask: what comes right before 1, if it isn't 0.9... or something between 0.9... and 1? Although I have answered that below, and the answer is "some finite number", which destroys your argument.


1 - I can accept (intuitively) that there's a pattern in the division that will always repeat, and, by allowing that pattern to extend to infinity, state that 1/3 = "zero, point, a countable infinity of threes".

2 - Or, if I'm very nitpicky, I can claim that only the steps I can actually carry out to their end are meaningful, and that in this case the algorithm can only produce approximate divisions, not the exact division. However, even if I decide to apply such strict logical standards, I can still reverse the less accurate argument given in (1), and accept the symbol "zero, point, a countable infinity of threes" as a suggestive notation for the limit of the sequence of approximations (0, 0.3, 0.33, 0.333, ...), which is 1/3.
I accept (1). However, that pattern includes a remainder, so why would that remainder disappear unless the pattern changed? As for (2): if you define "the limit" for the sequence as being non-inclusive (that is, the number the sequence can almost--but not quite--reach), then I accept (2). If you define "the limit" as being inclusive, then I do not accept (2), for reasons stated.


1 = 0.999.... (infinite progression), not 0.999...9. (finite progression.)

Infinity is infinite. You need to stop thinking about last digits. They simply don't apply. Infinity laughs in the face of your puny final digit.
My final digit laughs right back in the face of your puny infinity.

You seem to have the idea that there are not two ends to the number. However, that is demonstrably false: you can (in a theory which your argument requires to be true) divide 1 an infinite number of times. That is, between 1 and 0, there are an infinite number of numbers. So, you start at 0, count by infinitesimals for the correct eternity (in this case a "never-ending" eternity), and end at 1. There are an infinite number of numbers in there, but there are two very distinct ends. So, 0.9... has an infinite number of 9's, but there are still two ends to the number; the logic carries to any infinity.

Unless you can't count by infinitesimals, since each infinitesimal adds exactly 0, which never gets you anywhere. So you can only count with finite numbers, which means there are a finite numer of numbers between 0 and 1. Because of this, never-ending decimal numbers can't really exist, so 0.9...=nothing, not 1.

Now, if 0.9... is less than 1, it doesn't have to be <the number just below 1>. There may still be an infinite number of numbers between 0.9... and 1 that can't be represented by our numbering system. But if it is right next to 1, it still doesn't equal 1, because if it did every number would equal every other number, as shown before.

[Archer17]

I quit reading your post about a million words into it .. how did you vote?

(For the record, I voted yes a while back)

[Stregone]

You are really thinking too hard dude.

[fosley]

I quit reading your post about a million words into it .. how did you vote?

(For the record, I voted yes a while back)
No. . . (I supposed that's different from "No...", which would be an "N" followed by infinite "o"'s. . . :D)



You are really thinking too hard dude.
Perhaps, but the search for the smallest number has gone on in my head since I was 4, and perhaps earlier. It's kept me awake countless nights (countless here=I lost track, not infinity :) ), and distracted me during countless days. I've tried using various logic to explain the idea, but the result is always the same - there is no smallest number, unless it's defined by physics (which requires that the universe is finite in size and there is a lower limit to how small you can go--the largest number would then be the number of smallest things you can fit in the entire universe, and the smallest number would be 1/the largest number--but even then you could start running the permutations of such objects, and other things of that nature, which pretty much means there is no smallest number). However, the search for the smallest number has resulted in several proofs (or "proofs"--however you feel like viewing it), such as what I'm showing here. I'm all about sharing, so I feel that all my sleepless nights should be useful somehow.

[fosley]

about a million words into it ..
So, just to have a fun nitpick: copying all the text from all my posts yields 27,796 bytes worth of ASCII text. Since each character is represented by 1 byte, that's a max of under 28,000 letters. Take away a lot of those for spaces and linefeeds (and, because linefeeds are actually a carriage-return/lineeed combination in Notepad, they each take 2 bytes instead of 1 byte - greedy linefeeds), and you get something like half the original number (could be wrong - I did a test of this once and I think that's what I came up with on average). So, if the average word is 7 letters long, there are somewhere between 2000 and 4000 words. This leads me to conclude that you probably didn't actually read a million words into anything I wrote.

:D :) 8) :-? :lol: :o :wink: 8-[ :D

Now, if I could just shell out some money and buy a copy of Microsoft Office or get un-lazy and find a program I wrote myself, I could just paste all the text in there and get an exact count. :roll:

[Stregone]

try this www.openoffice.org