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Moose
2008-Apr-28, 09:53 PM
I don't think it is that decisive in calculations at that level, but I will defend your right to say so.

Speaking of which, that was a simple scenario. In the real world, you get to approximate. Taking the integral to get an exact volume would be handy if you were looking at planning out a maintenance budget over decades.

And I'm talking something like this (http://www.nubricks.com/archives/392/worlds-largest-swimming-pool/). Not some two-bit backyard pool you can guestimate for a few extra bucks a month.

Bogie
2008-Apr-28, 09:54 PM
*shrug* It's meaningless, Bogie, 1 = 0.999~. You've been shown why. I'm not interested in getting into an ATM discussion with you.I wish you were. There are things that are considered ATM that that will prove to be reality.

Bogie
2008-Apr-28, 09:58 PM
Speaking of which, that was a simple scenario. In the real world, you get to approximate. Taking the integral to get an exact volume would be handy if you were looking at planning out a maintenance budget over decades.

And I'm talking something like this (http://www.nubricks.com/archives/392/worlds-largest-swimming-pool/). Not some two-bit backyard pool you can guestimate for a few extra bucks a month.Wow. I am picturing myself in one of the ocean front rooms above the pool, posting on BAUT :D. Ain't life grand!

Nowhere Man
2008-Apr-28, 10:03 PM
What do you get if you subtract 0.99999999... from 1? Show your work.

1 - 0.99999999... = ?

Fred

geonuc
2008-Apr-28, 10:07 PM
What do you get if you subtract 0.99999999... from 1? Show your work.

1 - 0.99999999... = ?

Fred
Zero.

1 = 0.99999~ (see any of the above proofs)
1 - 1 = 0

Tango Cat
2008-Apr-28, 10:08 PM
You didn't cover an infinite distance.

Precisely, that is the point. I covered 0.9 of the distance, 0.99 of the distance, 0.999 of the distance, 0.9999 of the distance, and so on, as much as you like. A million 9s, a billion 9s, a googolplex 9s, I covered them all. I can travel all of these distances, without traveling infinitely far, and I can do it in the real world. So you say that 0.999~ and 1 are different because it is impossible to attain infinity, and yet when I give an example in which I travel 0.9 of the way, 0.99 of the way, 0.999 of the way, 0.9999 of the way, and so on forever, you say it is not infinity.

Where we part company is at the concept that something can be infinitely small.

I should be glad to hear what you feel the difference between 0.999~ and 1 is, keeping in mind that nothing can be infinitely small.

We are in agreement. :)

I am glad :)

Bogie
2008-Apr-28, 10:12 PM
What do you get if you subtract 0.99999999... from 1? Show your work.

1 - 0.99999999... = ?

Fred

Zero.

1 = 0.99999~ (see any of the above proofs)
1 - 1 = 0Yes, in mathematical terms. But in reality you cannot produce anything that is equal to .999~ because infinity can not be attained in the real world. Lucky that we have minds that can grasp math so we can talk about the difference between math and reality.

Kaptain K
2008-Apr-28, 10:19 PM
This has stopped being a "Q&A" thread and become a "game" for Bogie to play until it is locked or somebody blows his cool and gets banned! I'm not gonna play anymore!

Bogie
2008-Apr-28, 10:20 PM
Precisely, that is the point. I covered 0.9 of the distance, 0.99 of the distance, 0.999 of the distance, 0.9999 of the distance, and so on, as much as you like. A million 9s, a billion 9s, a googolplex 9s, I covered them all. I can travel all of these distances, without traveling infinitely far, and I can do it in the real world. So you say that 0.999~ and 1 are different because it is impossible to attain infinity, and yet when I give an example in which I travel 0.9 of the way, 0.99 of the way, 0.999 of the way, 0.9999 of the way, and so on forever, you say it is not infinity.

I should be glad to hear what you feel the difference between 0.999~ and 1 is, keeping in mind that nothing can be infinitely small.So how much distance did you cover, 20 feet. And to you that is an infinite distance? Are you making this discussion about how much you object to my position instead of acknowledging that my position is that you can't attain infinity in the real world? But I agree you can use mathematical concepts to equate your 20 foot trip to an infinite number of smaller and smaller increments. Two different things I think.

Bogie
2008-Apr-28, 10:24 PM
This has stopped being a "Q&A" thread and become a "game" for Bogie to play until it is locked or somebody blows his cool and gets banned! I'm not gonna play anymore!You have me wrong.

Kaptain K
2008-Apr-28, 10:36 PM
You have me wrong.

**! You have been shown, by many people in almost as many ways, that you are wrong. Either admit it or admit that you're playing a game!

Tango Cat
2008-Apr-28, 10:52 PM
So how much distance did you cover, 20 feet. And to you that is an infinite distance?

No, to me that is a finite distance - you are the one claiming that a journey of 0.9 of the way, 0.99 of the way, 0.999 of the way, etc. can't terminate at one on the grounds that "you can't attain infinity in the real world." But to keep it simple, suppose I traveled 1 foot. On the way, would you say I made it as far as:

0.9 ft?
0.99 ft?
0.999 ft?
0.9999 ft?
0.99999 ft?
0.999999 ft?
0.9999999 ft?

and so on? Do let us know. How many 9s can I add on, before I can not have traveled that far, on the grounds that "you can't attain infinity in the real world."

Are you making this discussion about how much you object to my position instead of acknowledging that my position is that you can't attain infinity in the real world?

What difference does my motive make? If you know what you're talking about, you should have no problem defending your position, regardless of my motive.

What is your motive?

But I agree you can use mathematical concepts to equate your 20 foot trip to an infinite number of smaller and smaller increments. Two different things I think.

Keep it simple by making a one foot trip instead of 20. How many 9s do I need to add to the distance traveled before I cease to have traveled that far in the "reality" you like to discuss?

In my mathematics, the mathematical problem and the physical problem coincide perfectly. It is in your alternative mathematical system that there is a deviation. If you adopt the math the rest of the world uses, you won't have that problem.

Bogie
2008-Apr-28, 10:55 PM
I think I have over stayed my welcome.

worzel
2008-Apr-28, 11:01 PM
Here is the post I thought you missed.
http://www.bautforum.com/off-topic-babbling/14593-do-you-think-0-9999999-1-infinite-9s-57.html#post1229174

Like I said, I didn't miss it. To say that there is a choice between two different answers is nonsensical. Either both answers are the same, in which case you agree that 0.999~ does equal 1; or they are not, in which case you believe that arithmetic is inconsistent and therefore believe that x=y whatever they are (including 0.999~ and 1).

But that's beside the point, which you missed with your retreat into questioning step (7) just as you did when you originally questioned step (8). The proof only requires that you accept that step (7) equals 0.999~. And that result is a necessary consequence of steps (2) and (6).

So rather than hand waving away step (7) with the illogical objection that it has two different answers, you must either state which of steps (2) and (6) is incorrect or point to the logical mistake in inferring step (7) from steps (2) and (6).

Tango Cat
2008-Apr-28, 11:17 PM
Like I said, I didn't miss it. To say that there is a choice between two different answers is nonsensical. Either both answers are the same, in which case you agree that 0.999~ does equal 1; or they are not, in which case you believe that arithmetic is inconsistent and therefore believe that x=y whatever they are (including 0.999~ and 1).

But that's beside the point, which you missed with your retreat into questioning step (7) just as you did when you originally questioned step (8). The proof only requires that you accept that step (7) equals 0.999~. And that result is a necessary consequence of steps (2) and (6).

So rather than hand waving away step (7) with the illogical objection that it has two different answers, you must either state which of steps (2) and (6) is incorrect or point to the logical mistake in inferring step (7) from steps (2) and (6).

Next thing you know, we'll have some mathematical sophist trying to tell us that 3/6 is equal to 1/2 :)

Is this sort of thing common here?

Kaptain K
2008-Apr-28, 11:19 PM
Next thing you know, we'll have some mathematical sophist trying to tell us that 3/6 is equal to 1/2 :)

Is this sort of thing common here?

Praise Deity, no!

Bogie
2008-Apr-28, 11:32 PM
Next thing you know, we'll have some mathematical sophist trying to tell us that 3/6 is equal to 1/2 :)

Is this sort of thing common here?

Praise Deity, no!

Okie dokie, you had me gone and then you had to show off.

What could be easier to understand than math and reality are two different things.

Infinity in reality is unattainable. In math it is definitional.

Silly arguments comparing my point to the mathematical infinites based on an infinite number of smaller steps to cover a foot in distance seems the kind of argument to be expected from someone who was ignoring my point and insisting that their point was somehow enlightening.

And to rejoyce in getting rid of someone who went to lengths to explain something that they consider meaningful is uncalled for.

hhEb09'1
2008-Apr-28, 11:45 PM
Okie dokie, you had me gone and then you had to show off.Don't leave.

Bogie, what's 0.444... divided by 2?I appreciate the efforts to help me grow my understanding and I'm sure that my answer of .222... will lead to another step or two that would convince someone of normal competence that 9/9 equals .999~. But I will just have to be deemed wrong on this and add it to the long list of misconceptions about reality that I must be harboring.Your intuition is correct! :)

The next question is: "What is 0.999... divided by 3?"

Tango Cat
2008-Apr-28, 11:46 PM
What could be easier to understand than math and reality are two different things.

Infinity in reality is unattainable. In math it is definitional.

I would have thought that it would be extremely easy to understand that if you insist of defining your system of maths so that it deviates from reality, then it will deviate, and if you define your system of maths so that it conforms to reality, then it will conform. But this idea appears to be controversial to some.

Silly arguments comparing my point to the mathematical infinites based on an infinite number of smaller steps to cover a foot in distance seems the kind of argument to be expected from someone who was ignoring my point and insisting that their point was somehow enlightening.

Claiming that you know about reality and then refusing to answer even the simplest questions about it, dismissing a series of questions as a "silly argument" is regrettably the kind of tactic I'm beginning to expect from you.

You claim you know about reality. Would like us to believe you? If so, maybe you will consider answering the questions I asked?

And to rejoyce in getting rid of someone who went to lengths to explain something that they consider meaningful is uncalled for.

I would consider it meaningful if a person who claims to know how things are in reality would answer a few simple questions about that reality. It seems you feel differently.

Why don't you just cool off, and answer the questions I asked? Are we not supposed to ask you questions about your claims? Are we supposed to just accept them blindly?

Frog march
2008-Apr-28, 11:51 PM
I don't know why you keep on going on about "reality", bogie.

is maths not part of reality?

Bogie
2008-Apr-29, 12:10 AM
This response is to the last several posts dissing me about leaving:

Don't leave.Your intuition is correct! :)

The next question is: "What is 0.999... divided by 3?"Sorry, the gang has spoken. Did you all miss that I acknowledged that in a mathematical context 1 and .999~ are equal, at least five times?

Here are several places where, if you took the time to read them you would see that:

http://www.bautforum.com/off-topic-babbling/14593-do-you-think-0-9999999-1-infinite-9s-58.html#post1229443
http://www.bautforum.com/off-topic-babbling/14593-do-you-think-0-9999999-1-infinite-9s-59.html#post1229478
http://www.bautforum.com/off-topic-babbling/14593-do-you-think-0-9999999-1-infinite-9s-59.html#post1229505
http://www.bautforum.com/off-topic-babbling/14593-do-you-think-0-9999999-1-infinite-9s-59.html#post1229539
http://www.bautforum.com/off-topic-babbling/14593-do-you-think-0-9999999-1-infinite-9s-59.html#post1229544

But who among you has acknowledged my point?

Let it go.

To keep portraying me as one who doesn't get the point when I repeated acknowledged it is an unworthy tactic in itself.

I feel no obligation to those who can not see that I agreed with the mathematically point, and who insist on some kind of capitulation beyond that.

I don't expect many of you to acknowledge my point because I have low expectations, but read back.

Kaptain K
2008-Apr-29, 12:16 AM
Still playing games!

Bogie
2008-Apr-29, 12:20 AM
Still playing games!Argumentative.

Kaptain K
2008-Apr-29, 12:22 AM
Statement of fact!

Neverfly
2008-Apr-29, 12:29 AM
I'm not seeing any games being played. And this is a topic that seems to have been spinning people around to 68 pages and just under 1800 posts in this thread.

Remember the zero multiplied by infinity thread?

Moose
2008-Apr-29, 12:42 AM
Is this sort of thing common here?

There's a reason this thread has 40-odd pages.

This sort of thing is common enough inside the ATM forum, although it tends to be buried under layers and layers of hand-waving and obfuscation, and you'll see similar logical mistakes used in CT as well. Of course, the rules discourage that sort of thing.

It's sort of refreshing to see it laid bare to this extent, though. It's hard to find cover when you're talking about elementary school arithmetic.

Bogie
2008-Apr-29, 12:50 AM
There's a reason this thread has 40-odd pages.

This sort of thing is common enough inside the ATM forum, although it tends to be buried under layers and layers of hand-waving and obfuscation, and you'll see similar logical mistakes used in CT as well. Of course, the rules discourage that sort of thing.

It's sort of refreshing to see it laid bare to this extent, though. It's hard to find cover when you're talking about elementary school arithmetic.Talk about hand waving. You seem to be reveling in the misconception that I don't get the point. I posted these examples:

http://www.bautforum.com/off-topic-babbling/14593-do-you-think-0-9999999-1-infinite-9s-58.html#post1229443
http://www.bautforum.com/off-topic-babbling/14593-do-you-think-0-9999999-1-infinite-9s-59.html#post1229478
http://www.bautforum.com/off-topic-babbling/14593-do-you-think-0-9999999-1-infinite-9s-59.html#post1229505
http://www.bautforum.com/off-topic-babbling/14593-do-you-think-0-9999999-1-infinite-9s-59.html#post1229539
http://www.bautforum.com/off-topic-babbling/14593-do-you-think-0-9999999-1-infinite-9s-59.html#post1229544

You might want to acknowledge that I have the point instead of making a false characterization for the benefit of having a fresh example of a rare event.

Moose
2008-Apr-29, 01:06 AM
You seem to be reveling in the misconception that I wasn't paying attention, Bogie. I'm quite aware of each time you've said that we're right but that you've explicitly chosen to not agree for reasons that have shifted from post to post.

My favorite was the one where you stopped just a hair's-breadth from the inescapable conclusion that 1/9 + 8/9 somehow did not equal 9/9.

Bogie
2008-Apr-29, 01:12 AM
You seem to be reveling in the misconception that I wasn't paying attention, Bogie. I'm quite aware of each time you've said that we're right but that you've explicitly chosen to not agree for reasons that have shifted from post to post.

My favorite was the one where you stopped just a hair's-breadth from the inescapable conclusion that 1/9 + 8/9 somehow did not equal 9/9.You are fantasizing now. I have agreed. I have posted links to my post doing so.

If you are paying attention then you have an alternative agenda because even a low attention span should permit you to derive from my posts that I have acknowledged that in the mathematical context that 1 and .999~ are equal.

Say you see it unless you have another agenda, and if so, just state what it is so we can all understand your reticence to do so.

Moose
2008-Apr-29, 01:29 AM
To recap: a five minute summary of Moose's fantasies:

The choice that you present, 1= 9/9 = .999~ is two possible outcomes to choose from. To me 9/9 =1, not .999~. Thanks for letting me be wrong on this.

9/9 does not equal .999~ it equals 1 if you choose the simplest of the two possibilities.

8/9 plus 1/9 equals 9/9 or .999~ and I choose 9/9 which equals 1.

What if I die before I agree. Problem solved, right :p

I noticed that Bogie has already cast his vote in the poll. I guess his mind is made up.

And not only that but I'm willing to be wrong on this.

No. I'm really sure that 9/9 = 1 and not .9999~.

I disagree then. 8/9 + 1/9 can be 9/9 so there is the choice as to whether it is .999~ as your statement says or if it is equal to 9/9.

here probably is a sub-conscious reason that I refuse to say the 9/9 equals both .999~ and 1. I could worry about it or I could chose to have a mental block. I chose the mental block .

Yes we can. But when we get to 8/9 + 1/9, and say it =.999~, my feeble mind says 8/9 + 1/9, simple fractions, equals 9/9 which reduces to 1, not .999~. Sorry but that is my problem.

OK. Now can you let me live with the problem that 3 time 0.333~ = 0.999~ is not as simple as reducing 9/9 to 1. I'm choosing the simplest of two methods of resolving the equations which is reducing 9/9 to 1.

Wait, wait, wait. You believe that 1 and .999... are two different numbers but 9/9 is equal to both of them?

SeanF, why is it so important that you convince me.

So I agree that under the rules of math with the proof that 1 and .999~ are equal, but I don't agree that in the real world that is true.

Do you believe [irrational numbers] exist in the "real world?"

Nick, do you really care what I think about them or are you interesting in classifying me in regard to my mathematical acumen?

And the subject shifted from there to irrelevant hand-waving about the nature of infinity and whether or not Bogie is playing games.

As for my agenda, in this thread, I'm an avowed part of the pro-arithmetic lobby.

[Edit: And on that note, I'm off to dream about the \$620 tax refund I appear to have coming.]

Bogie
2008-Apr-29, 01:37 AM
To recap: a five minute summary of Moose's fantasies:
And the subject shifted from there to irrelevant hand-waving about the nature of infinity and whether or not Bogie is playing games.

As for my agenda, in this thread, I'm an avowed part of the pro-arithmetic lobby.

[Edit: And on that note, I'm off to dream about the \$620 tax refund I appear to have coming.]I was merely developing my case that math and reality are not the same thing. You failed to acknowledge that I agreed many times that mathematically 1 and .999~ are equal, and failed to acknowledge that I pointed out that in reality, infinity is not attainable because an infinite progression never ends.

Hard Knocks
2008-Apr-29, 01:47 AM
:eek:

run away! dont look back!

Bogie
2008-Apr-29, 01:56 AM
:eek:

run away! dont look back!Good advice. I think I'll take it. Thank you.

Neverfly
2008-Apr-29, 02:21 AM
I voted no too.

I'm certainly not going to sit here and argue about it.

A math wizard might explain to me exactly how I am wrong- but other than that- one is an infinity (Undefined) and one is a number.

Grizzle
2008-Apr-29, 02:44 AM
A math wizard might explain to me exactly how I am wrong- but other than that- one is an infinity (Undefined) and one is a number.

A small army of people have already explained exactly how this supposedly undefined quantity is defined. Will one more really make a difference?

Nick Theodorakis
2008-Apr-29, 02:48 AM
FWIW, my ten-year-old son understands the proof(s) just fine and has no trouble with the concept that 0.999~ = 1.

Nick

hhEb09'1
2008-Apr-29, 02:53 AM
This response is to the last several posts dissing me about leaving:I myself did not diss you. I just offered another question (besides asking you not to leave). I don't mind if you don't answer it, it makes my argument look stronger.

Sorry, the gang has spoken.There is no gang. And if there were, I wouldn't be a part of it. :)
Did you all miss that I acknowledged that in a mathematical context 1 and .999~ are equal, at least five times?What other context is there? 0.999... is clearly a mathematical construction.

But who among you has acknowledged my point?I'm not certain of your point. I'll go back to those five posts and look them over (unfortunately, I just tried, and the permalink bug seems to have bitten me. This'll take longer than I thought. I'll do it tomorrow.)

Grizzle
2008-Apr-29, 02:55 AM
FWIW, my ten-year-old son understands the proof(s) just fine and has no trouble with the concept that 0.999~ = 1.

Nick

Good for him. Tell him some grouchy old guy on the web thinks he's smart.

Neverfly
2008-Apr-29, 03:03 AM
Good for him. Tell him some grouchy old guy on the web thinks he's smart.

And another Grumpy old man on the net thinks you've got understanding issues of your own...:

A small army of people have already explained exactly how this supposedly undefined quantity is defined. Will one more really make a difference?
Nope. One more won't.

Essentially, it's a justification for imagination. .9999>infinity is just that- an imaginary construct. It doesn't equal anything.

Nick Theodorakis
2008-Apr-29, 03:18 AM
Good for him. Tell him some grouchy old guy on the web thinks he's smart.

Not to worry; he already knows he's smart. ;)

Last year, at the "math bowl" (math contest in Indiana for middle schoolers) his team was one of the few to get a certain problem right because he happened to know that 1 is not a prime number (oh dear, do we need another poll?).

Nick

Neverfly
2008-Apr-29, 03:19 AM
You know
This thread and the
Zero Multiplied by Infinity thread

Both had these new posters up and register and start posting in that (or this) thread blustering about how they think everyone else is an idiot and sharing their Almighty Wisdom with us scabs.

That endearing quality, although charming, has no real effect except for continuing a quarrel over values that do not exist.

Grizzle
2008-Apr-29, 03:25 AM
Not to worry; he already knows he's smart. ;)

Lot's of people know they're smart. Some of them actually are.

Last year, at the "math bowl" (math contest in Indiana for middle schoolers) his team was one of the few to get a certain problem right because he happened to know that 1 is not a prime number (oh dear, do we need another poll?).

Think that one would go better than this one? :)

Neverfly
2008-Apr-29, 03:46 AM
Why do these strange Math threads always turn into heated peeing for IQ contests?

Paul Leeks
2008-Apr-29, 04:05 AM
they are not equal!

is 1.9999... equal to 2.0?

No!

Musashi
2008-Apr-29, 05:48 AM
Actually, yes.

Neverfly
2008-Apr-29, 05:58 AM
I'm not going to go through 1800 posts so... THis may have already been asked.

But:

In order to compare (is this equal to this) two numbers, don't they both need to be defined?

This is a bit of a pointless debate- in my mind.

If you take .9999>Stretching off to infinity- and you consider that because it stretches into infinity- there is always another decimal place- it must be "equal" to one with a 'proof.'

But that same argument can work the other way too, that it's less than 1 to the nth degree.

The difference being that .9999> Stretching into infinity is a construct mathematically but not necessarily a reality.

If I have one apple in front of me - I have ONE apple (No semantics please about how many atoms are in the apple or how many seeds etc).
If I take a knife and cut it into fractions- it does not matter how I measure them. Each fraction will be a Finite Amount.

If I cut .33333>stretching into infinity from an apple (1/3) I still have a finite amount of apple.
The .33333>stretching into infinity is a construct of the mathematics and not a TRUE value- it only "exists" because of the construct of the number system.
Say I do the math using Anteran (Alien) numeric system and math- that infinity will disappear. It only exists within the construct- not in reality.

.99999> is no more "equal" to one than omega, delta or orange. It is a construct. We only consider it as equal because of the construct.

If I put .333333333333333333333333333333333333333333333333 into my calculator and don't specify infinity- and multiply it by 3, I will not get 1 as an answer. Because in dividing 1 by 3 I get the construct- I would need to multiply 3 by an infinite number to get back to 1.

The answer is that .9999>infinity is NOT "equal" to 1. We only are constrained to consider it such because of the nature of the construct.

Neverfly
2008-Apr-29, 06:23 AM
If I put one mirror in front of me and one mirror behind me and take a look, I will see an infinite number of mirrors and..."me's" (God Forbid...):neutral:

What I am seeing is a construct of the perception.
There is not an infinite amount.

With our system, we can divide any number by any other number. Except zero.

Ok well... It ain't a perfect system.

Occasionally you will get these artifacts like dividing by zero or multiplying zero by infinity or dividing one by three.

ETA:

Limit theory says that as your series advances to an infinite number of terms, the difference between the result of your progression and the value it's approaching (the limit) also becomes infinitely small. This property lets you treat the (infinite) series as perfectly equivalent and utterly consistent with the limit itself.

In other words, it lets us fudge the number so we can get on with our day.
Like a Service patch to a computer program.

agingjb
2008-Apr-29, 07:58 AM
Limits, as usually defined, don't use infinities (or infinitesimals). Typically the definition will say something of the form "for any δ there is an ε such the difference between the value of an expression containing that ε and the proposed limit is less than δ" (and, please, some mathematician can refine this).

But this sort of construction is what rigorous analysis is based on (including much of the mathematics used in astronomy). Naturally, anyone is free to reject it. I'd regard such a rejection as ATM; others may disagree.

Neverfly
2008-Apr-29, 08:21 AM
But this sort of construction is what rigorous analysis is based on (including much of the mathematics used in astronomy). Naturally, anyone is free to reject it. I'd regard such a rejection as ATM; others may disagree.

It is neither ATM nor a rejection.

It is not a basis. Nor is it fundamental- It is a Quirk of The Construct- it is recognition of the Quirk in the Construct- Nothing More.

And these massive heated debates over these simplistic details seems nothing more to me than some mathematicians defending their "precious..." because God-forbid it could be flawed....:rolleyes:

This is probably wrong of me but I'm wrong a lot anyway soo....: Click the who voted button and look at WHO voted that it is not equal. That list includes many mathematically studied people, mainstreamers and folks who are well rehearsed in BBT, GR, SR etc. Stupendous man is one I recall seeing off memory.

Then- you are going to see posters jumping in with a superiority complex all of the sudden, making snide remarks about how even a child can understand such and such (Yeah Right! A kid will nod his head and go along with it- doesn't mean they truly understand it) and that it speaks badly or reflects badly on the forum considering how the votes went and what's wrong you you people that you don't know math... and basically acting like anyone who doesn't agree with them is intellectually challenged or is uneducated. Happens everytime in these particular threads over something so absurd.
So frankly, I've been watching the debate with Bogie- and the outlandish assertions being made. It's just plain silly.

Start a thread about how to divide anything by zero- Pop some popcorn and sit back and watch the brawl. It would be entertaining if it wasn't so sad.

agingjb
2008-Apr-29, 09:25 AM
I see. Is it the general view on this site that mathematics has no "mainstream"?

Moose
2008-Apr-29, 09:39 AM
Essentially, it's a justification for imagination. .9999>infinity is just that- an imaginary construct. It doesn't equal anything.

No, that's not true at all. 1 = 0.999~ is exactly as real, and for exactly the same reasons, as the number 2.

But here's the thing, Nev, you said just now you didn't want to read this thread to understand the proofs, but just a few days ago, you made the argument in About BAUT (and to me via PM) that you agree with Mugs' positions on newbies doing their research first. That's not consistent. You can't want to require something of newbies you aren't willing to do yourself.

Bogie, if you have something to say to me, say it in public where the mods can see it. I've already asked you to knock it off with the PMs.

Moose
2008-Apr-29, 09:57 AM
I see. Is it the general view on this site that mathematics has no "mainstream"?

I don't think that's true. I think the overwhelming consensus would be that there is a math mainstream. Where you'd get argument, apparently, is whether limit theory is part of that mainstream or not. I think you'd get a great deal of correlation between the 1 = 0.9999~ question and the question: "Does math have to be perfectly internally consistent in all respects in order to have validity?"

Thing is, like the Monty Hall problem, the concepts involved are subtle, non-intuitive, and apparently unsatisfying, even after they have been demonstrated with layperson maths. I think there's an not-quite-spoken temptation to try and dismiss limit theory as "just a theory" in the creationist sense of the term, something that really can't stick because of how internally consistent math has to be in order to function at all.

I guess what I don't understand is the personal stake people invest in the rejection of limit theory.

Neverfly
2008-Apr-29, 09:57 AM
No, that's not true at all. 1 = 0.999~ is exactly as real, and for exactly the same reasons, as the number 2.

I made about three posts. Please read- each and every word of them- then get back to me.

I'm not being rude.

I have a reason for asking you to do that.

But here's the thing, Nev, you said just now you didn't want to read this thread to understand the proofs, but just a few days ago, you made the argument in About BAUT (and to me via PM) that you agree with Mugs' positions on newbies doing their research first. That's not consistent. You can't want to require something of newbies you aren't willing to do yourself.

This entire statement is flawed. Here is why:
You are comparing apples to oranges. Yes, folks should do their research.
Reading back I have no idea how many through 1800 posts is a waste of my time. WHY?
Because I would see theorems and proofs that I have seen before- because I know what those theorems and proofs actually are.
Patches.:neutral:

.9999> is not EQUAL to one. Define "Equal."

We TREAT it as if it was equal- because we need to to patch up the quirk in the System.

I find all the arguments against this- and rather crude insults like calling it ATM to state what I have stated as only mundane protectiveness of the system.

It does not mean that math breaks down.
It does not mean the system is fundamentally flawed.
These minor quirks have no true effect on precise calculations even.

The argument is absolutely Trivial.

If you draw a square.
Mark each side as 1 unit.
Draw a diagonal line across the square.
Calculate the length of that line.
Your answer will be a number that stretches into infinity:neutral:
Yet the line is finite. Theoretically- it is infinite- always stretching for the other side of the square- but never quite reaching it.

Well, let's say it was one inch. Change inches to something else- and that infinity- disappears.:doh:
Quirk of the System. It's inevitable.
Stop being overly protective about such a trivial little detail.

1800 posts and over 60 pages is ridiculous!

Neverfly
2008-Apr-29, 10:01 AM
I see. Is it the general view on this site that mathematics has no "mainstream"?

You've been a member here since 2005.

You tell me.

Meantime- did you not read my post about "it is NOT rejection"?

Sheesh...

It's about recognizing the quirks in the system.

I really do not see what is so hard about this.
You know what infinite pi is?
7/3rds.

Sheesh.

hhEb09'1
2008-Apr-29, 10:02 AM
I really do not see what is so hard about this.It's not hard.

You know what infinite pi is?
7/3rds.
That one, you're going to have to explain. I don't geddit :)

Moose
2008-Apr-29, 10:07 AM
It's not hard.That one, you're going to have to explain. I don't geddit :)

Neither do I. :confused:

Neverfly
2008-Apr-29, 10:20 AM
LOL OOPS!!

22/7ths.

It's an example- it follows the same characteristics of predictability. Irrational.

Moose
2008-Apr-29, 10:26 AM
I made about three posts. Please read- each and every word of them- then get back to me.

I'm out of time this morning. I won't have time to fine-comb it until much later today, possibly only this evening.

hhEb09'1
2008-Apr-29, 10:32 AM
LOL OOPS!!
22/7ths.After I posted, that occurred to me as a possibility.

Have you brought the popcorn to the new math thread (http://www.bautforum.com/universe-today-story-comments/73419-why-cant-we-launch-trash-into-space.html#post1229792) I'm participating in? :)

It's an example- it follows the same characteristics of predictability. Irrational.22/7 is rational, of course. Do you mean, pi is irrational?

Or, are you talking about our discussion here? :)

Neverfly
2008-Apr-29, 10:38 AM
After I posted, that occurred to me as a possibility.

Have you brought the popcorn to the new math thread (http://www.bautforum.com/universe-today-story-comments/73419-why-cant-we-launch-trash-into-space.html#post1229792) I'm participating in? :)22/7 is rational, of course. Do you mean, pi is irrational?

Or, are you talking about our discussion here? :)

Pi but the discussion works too:p

I haven't slept tonight (obviously) so I'm a little irrational to o and my posts are getting fast and vague.

Disinfo Agent
2008-Apr-29, 10:42 AM
I would like to offer another argument in support of the equality between 1 and 0.999~, this time of a practical nature. I hope the doubters will find it more persuasive than the mathematical proofs, since one of the recurring objections to this result is that "0.999~ exists in math, but not in the real world".

Does thinking about infinity make your head hurt? Then don't. You can always interpret statements like the following:

1/3 = 0.333...
2/3 = 0.666...
or
1/7 = 0.142857142857...

simply as shorthand ways to give approximations. So, for example, "1/7 = 0.142857142857..." means that 0, 0.1, 0.14, 0.142 and so on are approximations to 1/7, each with one more correct digit than the previous one. Indeed, the only way we have in practice to speak of numbers with infinite expansions is through approximations, right?

Similarly, you can think of "0.999... = 1" as a shorthand way of indicating that the number 1 may be progressively approximated by the following list of values: 0, 0.9, 0.99, 0.999, 0.9999, and so on.

What about 0.999..., with infinite nines? Many of the people who doubt this is equal to 1 claim it is some other number, close to 1, but different. Let's assume, purely for the sake of the argument, that it is true that 0.999... is some number x<1.

Then, (0, 0.9, 0.99, 0.999, 0.9999, ...) is clearly a list of approximations to x as well as to 1. How would you distinguish x from 1 in practice? Any list of convergent approximations to x will also be a list of convergent approximations to 1. Or, in other words: any measurement of 0.999... you make, no matter how precise, might as well be a measurement of 1. They are indistinguishable!

Rather than the equality 0.999... = 1 being a fancy theoretical convention, it's actually the inequality 0.999... ≠ 1 which would be unrealistic. There are not only sound theoretical reasons, but also good practical reasons, to accept that 1 and 0.999... are the same number.

Neverfly
2008-Apr-29, 10:46 AM
Disinfo agent- what you just said didn't make any sense at all! :neutral:

Here: Let me try!

Folks... if I say a car is the same as a horse, sure that may seem silly. But if you think about it this way: A car does not equal a horse- well that just would be unrealistic!

So saying a car is the same as a horse makes complete sense!

Disinfo Agent
2008-Apr-29, 10:51 AM
Disinfo agent- what you just said didn't make any sense at all! :neutral:Good morning. Please try reading it again. :)

hhEb09'1
2008-Apr-29, 10:53 AM
Pi but Then, what is the "characteristics of predictabiliy" connection to 22/7?

But if you think about it this way: A car does not equal a horse- well that just would be unrealistic!Why would you say that?

Neverfly
2008-Apr-29, 10:57 AM
Then, what is the "characteristics of predictabiliy" connection to 22/7?
Tough to explain and I can't find a good website on it... I'll deal with this question later.
http://en.wikipedia.org/wiki/Proof_that_22/7_exceeds_%CF%80
Either way- it was a sidepoint anyway....

Why would you say that?

Rather than the equality 0.999... = 1 being a fancy theoretical convention, it's actually the inequality 0.999... ≠ 1 which would be unrealistic.

Ivan Viehoff
2008-Apr-29, 11:31 AM
In order to compare (is this equal to this) two numbers, don't they both need to be defined?

....

The answer is that .9999>infinity is NOT "equal" to 1. We only are constrained to consider it such because of the nature of the construct.
Hi there N.
Your first comment is spot on. I've just flicked through this topic, and there is a clear lack of definition here.

Now quite clearly 1.0000... is a different object from the object 0.9999.... in the same way that two fifty cent coins are a different object from a dollar bill, or two other fifty cent coins. So to that extent you are correct. It is only when we apply meaning, and treat them as real numbers that they become equal to each other. Two fifty cent coins have the same value as a dollar bill, or two other fifty cent coins, and that is the sense in which 1 is the same as 0.9999.... So we need some definitions.

In this case, the "same value as" comes from the ordering of the real numbers. For example, in axiomatic set theory, the real numbers are generally constructed as Dedekind cuts. A Dedekind cut is a subset of the rational numbers which contains everything less than a limit. Consider having a stick marked with all the rational numbers (the closer you magnify the more you can see). Then cut the stick at a location. Plainly we can cut the stick at an irrational location as well as a rational location. This is the Dedekind cut - all of the rational numbers to the left of the cut point. The Dedekind cuts are unique. But when we represent those Dedekind cuts as decimal expansions, then there becomes non-uniqueness of representation for the terminating decimals. Now 0.999... is a proper decimal represention of the Dedekind cut whose value is 1. In fact, it is a common technique in considering the set theory of the real numbers to deal with the non-uniqueness of decimal representations by banning all decimal representations that terminate, or equivalently end 0000..., by only including the non-terminating ones that end 9999...., So that way every everything increases to its limit. In that manner of doing things, 0.9999.... is 1, because 1.000... is banned.

So, there is a sense in which they are different, and, a very important and practical sense in which they are equal, ie, as real numbers they have the same value.

Ivan

geonuc
2008-Apr-29, 11:34 AM
Wow, this thread is a mess. But I've read most of it. :neutral:

Lot's of things to disagree with - most have been covered by others. But, Moose, you posted this earlier today:

But here's the thing, Nev, you said just now you didn't want to read this thread to understand the proofs, but just a few days ago, you made the argument in About BAUT (and to me via PM) that you agree with Mugs' positions on newbies doing their research first. That's not consistent. You can't want to require something of newbies you aren't willing to do yourself.

I don't think that's a fair point. This thread is a mess and is very repetitive and full of not very interesting spitting contests. Reading through this is not similar to what some (Neverfly and Mugs, in this case) ask of newbies before coming to BAUT with questions.

Neverfly
2008-Apr-29, 11:37 AM
Ivan Viehoff, from what you just said, It basically boils down to: We all agree- we are just squabbling like children over how we agree by definition.
Eh?

Disinfo Agent
2008-Apr-29, 11:40 AM
.9999> is not EQUAL to one. Define "Equal."Two numerical expressions are equal when they represent the same number. The number represented by "1" is different from the number represented by "0.9999", but equal to the number represented by "0.9999..."

You must not forget that, at the end of the day, "1", "2", "4", "1/2", "2/4", "0.999...", or "IV" are just symbols standing in for numbers. They are not the numbers themselves; numbers are abstract notions.

Neverfly
2008-Apr-29, 11:41 AM
You must not forget that, at the end of the day, "1", "2", "4", "1/2", "2/4", "0.999...", or "IV" are just symbols standing in for numbers. They are not the numbers themselves; numbers are abstract notions.

This is exactly the same point I have been making too.:neutral:

Disinfo Agent
2008-Apr-29, 11:43 AM
THis is exactly the same point I have been making too.:neutral:Then what's the problem with accepting that "1" and "0.999..." may represent the same number?

geonuc
2008-Apr-29, 11:45 AM
This is exactly the same point I have been making too.:neutral:
I thought you voted 'no'?

I voted yes: the symbols in question (1, 0.9999~) are different, but they represent the same mathematical quantity.

Neverfly
2008-Apr-29, 11:47 AM
Then what's the problem with accepting that "1" and "0.999..." may represent the same number?

I have no problem with that. And I have said so.
We must use them to represent the same number.
As Ivan just commented- its the quirk of the system.

But the Defining of it with the question "is .9999> equal to 1?"- is why I'm making a point to say "No"

Because Purists want to maintain that they are equal at all points- and they seem to forget that the numbers, the mathematics - are constructs of our imagination used to represent reality. This is why I used the analogies that I used- such as two mirrors facing eachother.

We need to remember the reality.

Why do I make the point?
When we forget that reality- and we are deep in our calculations- not remembering the point can lead you astray.

Disinfo Agent
2008-Apr-29, 11:57 AM
But the Defining of is .9999> equal to 1- is why I'm making a point to say "No"I beg your pardon? :confused:

Bogie
2008-Apr-29, 12:00 PM
No, that's not true at all. 1 = 0.999~ is exactly as real, and for exactly the same reasons, as the number 2.

But here's the thing, Nev, you said just now you didn't want to read this thread to understand the proofs, but just a few days ago, you made the argument in About BAUT (and to me via PM) that you agree with Mugs' positions on newbies doing their research first. That's not consistent. You can't want to require something of newbies you aren't willing to do yourself.

Bogie, if you have something to say to me, say it in public where the mods can see it. I've already asked you to knock it off with the PMs.You are implying that I said something in PMs that the mods would object to. You have my permission to reveal all of our PMs to the mods.

I do have some comments about the thread and you that I won't have any problem saying in public. You did a hatchet job on me by going back and snipping quotes from the four different conversations that I had going on about proofs. That post took things out of sequence and out of context, and you did it after I had responded several times to you in a courteous fashion. I presented my position and your comments are worth discussing in the context of a subtle attack that was unprovoked.

Neverfly
2008-Apr-29, 12:01 PM
I beg your pardon? :confused:

Edited above post and Added quotation marks;)

Moose
2008-Apr-29, 12:43 PM
That post took things out of sequence and out of context,

No. The sequence of the quoting is complete and demonstrably and absolutely chronological, and covered nearly all of your posts before the subject got changed into these irrelevancies.

In any case, if you want to be right about this, all you have to do is do one thing. This challenge has been brought up many times in this thread, and it'll be brought up once more:

All you have to do to overthrow 1 = 0.9999~ is find or describe a number such that 0.9999~ > x > 1. If you can tell us what X is, you win.

In any case, I've already spent far too much time on this thread.

hhEb09'1
2008-Apr-29, 12:48 PM
Tough to explain and I can't find a good website on it... I'll deal with this question later.Fair enough

Bogie, what's 0.444... divided by 2?

I appreciate the efforts to help me grow my understanding and I'm sure that my answer of .222... will lead to another step or two that would convince someone of normal competence that 9/9 equals .999~. But I will just have to be deemed wrong on this and add it to the long list of misconceptions about reality that I must be harboring :).

The next question is: "What is 0.999... divided by 3?"

I'm still interested in Bogie's answer to this question.

HenrikOlsen
2008-Apr-29, 01:06 PM
Ok, lets try to separate numbers from their representation, since it seems like a lot of this confusion is caused by mistaking the two.

We have on one hand the abstract concept of the number one, easily recognized by it's many diverse and interesting properties (multiply it with itself and the result is itself, and similar properties).

Then we have various ways to write that number 1, 1.000~, 9/9, 0.999~, -e^(i*Pi) and many others which are all ways to represent that number, and most of the arguments in this thread has been about showing that they are.

We say two representations are equal if they are representations of the same number.

Bogie
2008-Apr-29, 01:17 PM
No. The sequence of the quoting is complete and demonstrably and absolutely chronological, and covered nearly all of your posts before the subject got changed into these irrelevancies.

In any case, if you want to be right about this, all you have to do is do one thing. This challenge has been brought up many times in this thread, and it'll be brought up once more:

All you have to do to overthrow 1 = 0.9999~ is find or describe a number such that 0.9999~ > x > 1. If you can tell us what X is, you win.

In any case, I've already spent far too much time on this thread.I thought this post would satisfy those who seemed to think I didn't agree:

http://www.bautforum.com/off-topic-babbling/14593-do-you-think-0-9999999-1-infinite-9s-58.html#post1229341

However I brought up an issue about math and reality being two different things and that set you off.

You dismissed my comments as irrelevant and failed to acknowledge that I had agreed to the equality. I was wondering why the conversation got antagonistic and figured you were the antagonizer. I felt that accusations of trolling constituted an antagonism that was uncalled for.

I'm still interested in Bogie's answer to this question.I’m not sure that you realize that I agreed.
http://www.bautforum.com/off-topic-babbling/14593-do-you-think-0-9999999-1-infinite-9s-58.html#post1229341

Is that satisfactory?

And why is the topic of a difference between math and reality I mentioned several times so controversial? Am I wrong about that?

And my several statements about being OK about being wrong on the issue meant that if the thread participants thought it was irrelevant that there was a difference between math and reality, then I was OK with being wrong on that side issue in the eyes of the participants who rejected that position.

Disinfo Agent
2008-Apr-29, 01:21 PM
I would agree that Bogie accepted the equality. However, like many people who started out as doubters, I think he still has the impression that there's something more to the issue than just mathematics.

And why is the topic of a difference between math and reality I mentioned several times so controversial? Am I wrong about that?It's not controversial, we just disagree that that difference makes any difference.

worzel
2008-Apr-29, 01:22 PM
Bogie, until now you haven't actually acknowledged that you were wrong. You have instead tried to hand wave away your shift in position (from the answer to the question posed by this thread being "no" to "yes) as a development of your argument.

What reality are you talking about? Where can we go and measure these curious 0.999~ creatures so that we can compare them to the 1s (and where are they?).

Bogie
2008-Apr-29, 01:41 PM
Bogie, you haven't actually acknowledged that you were wrong. You have instead tried to hand wave away your shift in position (from the answer to the question posed by this thread being "no" to "yes) as a development of your argument.

What reality are you talking about? Where can we go and measure these curious 0.999~ creatures so that we can compare them to the 1s (and where are they?).Right from the start my position was that in reality infinity cannot be achieved. Now there was one or two who said that was wrong and presented arguments that approaching an object constituted an infinite number of smaller and smaller increments but I reject that argument because it address a different infinity. I gave an example that if space was infinite, you could never reach the end of it. That was the type of infinity that I equated to approaching a limit but never being able to attain it like for example .999~ in math is defined as equal to one, but in reality you can't achieve the limit by approaching it in an infinite progression.

As for a shift in position I was swarmed by five people with their own version of how to prove the equality. I wanted to develop my position that there was a difference between the mathematical definition of .999~ and my view of it outside of math and in real situations, reality. I had to get to the final argument that all five wanted to present to show the proof of the equality before I could say "Yes, but that is math not reality".

There was an argument that eventually the increments get infinitely small and I reject that anything can be infinitely small; in comparison to what. I mentioned the infinitely dense zero volume point start to the universe and that I viewed it as an impossibility. Granted there are those who see it as possible but I am not one.

So why are you so bent on me saying I was wrong? I just explained that I took exception on the basis of math vs. reality when it comes to an infinite progression. I am not wrong about that and I didn't get to the point where I could explain why I was saying that .999~ and 1 are different until the five or one of the five had finished their version of the proof.

I'm not as dumb as you give me credit for though I admit that I sometimes pretend to be :).

hhEb09'1
2008-Apr-29, 01:46 PM
I'm still interested in Bogie's answer to this question.
I’m not sure that you realize that I agreed.
http://www.bautforum.com/off-topic-babbling/14593-do-you-think-0-9999999-1-infinite-9s-58.html#post1229341

Is that satisfactory?I've looked at the links. They don't answer the question that I'm interested in hearing your response to: "What is 0.999... divided by 3?"

But after reviewing the links, I'd expand the question to asking whether there were two different answers, depending upon context--and what the answer was in each context, if so.

Bogie
2008-Apr-29, 01:52 PM
I would agree that Bogie accepted the equality. However, like many people who started out as doubters, I think he still has the impression that there's something more to the issue than just mathematics.

It's not controversial, we just disagree that that difference makes any difference.You have been a pretty good arbitrator on this thread. I understand that you disagree with me about whether the difference makes any difference but how is it helpful to suggest that there is more to the issue with me than has been aired?

I have addressed all of the issues put to me I think. No one has acknowledged that my position is relevant. I'm not sure if the unaired issue is mine or yours.

Frog march
2008-Apr-29, 01:52 PM
Neverfly,

if 1 and 0.999~ are different, then there must be a number between them, in fact there would have to be an infinite number of numbers between them, can you give one of those number between them?

I suppose that you would just say that 0.999~ is just a construct. If so does it not represent any number?

if 0.999~ does represent a number/value then even if you have to give an answer to my first question as a construct, it should be possible.

Bogie
2008-Apr-29, 02:01 PM
I've looked at the links. They don't answer the question that I'm interested in hearing your response to: "What is 0.999... divided by 3?"

But after reviewing the links, I'd expand the question to asking whether there were two different answers, depending upon context--and what the answer was in each context, if so.That is a resolved issue. I have agreed with the equality and raised and issue that has itself not been responded to.

I don't feel obligated to answer your questions since I have no problem with the equality and I have not seen any indication that anyone considers my issue relevant. I'll just be unsubscribing now.

Disinfo Agent
2008-Apr-29, 02:09 PM
[...] how is it helpful to suggest that there is more to the issue with me than has been aired?I'm a bit confused. Are you referring to what I wrote in my previous post?

If so, I only meant that you believe that while 0.999~ may equal 1 in mathematics, the two may differ in some other more physical context. Am I mistaken?

worzel
2008-Apr-29, 02:18 PM
Right from the start my position was that in reality infinity cannot be achieved.
Right from the start you said that 1.000~ and 0.999~ are not the same. Right here (http://www.bautforum.com/off-topic-babbling/14593-do-you-think-0-9999999-1-infinite-9s-55.html#post1228651).

As for a shift in position I was swarmed by five people with their own version of how to prove the equality. I wanted to develop my position that there was a difference between the mathematical definition of .999~ and my view of it outside of math and in real situations, reality. I had to get to the final argument that all five wanted to present to show the proof of the equality before I could say "Yes, but that is math not reality".
As you seemed to be admitting that you were initially wrong I moved on to ask you where in reality these 0.999~s are. Where are they in reality?

So why are you so bent on me saying I was wrong?
Why are you so bent on not saying you were wrong while claiming that you meant something other than 1.000~ <> 0.999~ when that was exactly what you initially said?

The context of this thread is maths. Your initial post was a mathematical statement. You even stated which step (step 7) in one mathematical proof offered you found at fault (and have yet to answer my follow up question on that). So how could you have not been talking about maths?

2008-Apr-29, 02:32 PM
I would agree that 1 is not equal to .999~ but by the definition of what they are.
.999~ is not an absolute, it is a process. You cannot resolve .999~
I thought a good example is pi. Pi is not an absolute, it is also a process - unless they find the end of course :) - but as a number it is unworkable. So we modify pi by rounding up or down at some decimal place and make it an absolute.
The distance between .999~ and 1 isn't really the issue, as the reality of accuracy in the real word doesn't care. It does fine without it. As circles go to show!

hhEb09'1
2008-Apr-29, 02:34 PM
I have agreed with the equality and raised and issue that has itself not been responded to.I think my question is relevant to your issue.

hhEb09'1
2008-Apr-29, 02:36 PM
I would agree that 1 is not equal to .999~ but by the definition of what they are.
.999~ is not an absolute, it is a process. You cannot resolve .999~
I thought a good example is pi. Pi is not an absolute, it is also a process - Is 0.333... an absolute, or a process, or both?

2008-Apr-29, 02:42 PM
Is 0.333... an absolute, or a process, or both?

If it is .3333~ then yes it is a process. You cannot resolve it. At some point you must modify it by rounding down to make the number workable, thus making it an absolute.

SeanF
2008-Apr-29, 02:43 PM
Hey, Bogie, I've been trying to think of something that could show the issue within "reality," and this is the best I could come up with:

Let's say you have an apple. You take away 9/10 of the apple, leaving 1/10. Then you take away 9/10 of the remaining part. Then you take away 9/10 of that remaining part. etc., etc., etc.

Now, it is true that it would take an infinite number if iterations to remove all the apple, and that "can't be done." But do you see that as long as there is any apple remaining, you have not done the infinite iterations? If there is any bit of that apple remaining, no matter how small, then you've done only a finite number of nines, not infinite.

In other words, the only way you can take away .999~ of the apple is to take away the whole apple. Therefore, the only meaning that .999~ can possibly have - in reality - is the same meaning as 1.

hhEb09'1
2008-Apr-29, 02:54 PM
If it is .3333~ then yes it is a process. You cannot resolve it. At some point you must modify it by rounding down to make the number workable, thus making it an absolute.So, in your opinion, would you say that 1/3 does not equal 0.333~ ?

Chuck
2008-Apr-29, 03:02 PM
.999~ is not a process, it's a decimal point followed by an infinity of nines. All of the nines are already there. We don't have to add nines to the string forever because we're already done. The difference between .999~ and 1 is zero so they represent the same number.

It's the same for .333~ and 1/3. The infinity of threes is already there and need not be rounded off.

2008-Apr-29, 03:04 PM
No! It doesn't. 1/3 is an absolute if you use it as a fraction. 3/3 = 1.
.333~ is a process. You cannot resolve it. So you modify it.

The point I'm making is that in the real world it doesn't matter. Trying to give someone .9999~ of an apple isn't going to happen. So you give them the whole apple. You modified the amount you would give them, but it is such an absurdly small amount it doesn't make a difference to them or you.

I can write 1/3 on paper and have my absolute. I can't do that with .3333~. I have to use a symbol because the number is unworkable. It is purely in the definition of what they are that they differ enough to be worth commenting on.

Frog march
2008-Apr-29, 03:13 PM
what makes you say it is a process?

it is just a way of writing that the infinite number of 9s is already there.

worzel
2008-Apr-29, 03:15 PM

If the symbol ~ means "an infinity of" the I can write out infinite 3s just by writing 0.333~

Is it true that 2n=n+n? Is that not an infinite number of arithmetical statements?

2008-Apr-29, 03:27 PM
what makes you say it is a process?

it is just a way of writing that the infinite number of 9s is already there.

True. Should have said a symbol representing infinity.

I say it is a process because it never ends. You cannot physically write the number down. You have to use the symbol infinity to represent the process.

If I do 10/3 in long division it will never end. I cannot give you an exact answer. But I can say it is 3.3333~ and get away with it, because the difference in distance between the number and the process is irrelevant to reality.
.9999~ just happens to stand out because the process leads to 1.

hhEb09'1
2008-Apr-29, 03:31 PM
No! It doesn't. was that an answer to my question "would you say that 1/3 does not equal 0.333~ ?"

A post intervened, maybe. :)

Chuck
2008-Apr-29, 03:38 PM
If I wanted to write down 1/3 in decimal form digit by digit then I'd never be done, but there's no need for me to write them. I can just say "A decimal point followed by an infinity of threes." Now I'm done.

Disinfo Agent
2008-Apr-29, 03:38 PM
I say it is a process because it never ends. You cannot physically write the number down.You cannot write down the symbol for the number in full, agreed. But, as we've just been poiting out, a number is not the same as the symbols for it. And there are other ways to represent zero-point-nine-repeating we can use, such as 0.(9). This one we can easily write down.

So, not being able to write down a symbol in full is not as serious as it may seem at first glance. As long as we are perfectly clear on what we mean by the symbol, we can abbreviate it for convenience.

It's not the string of nines that matters the most to us, but the number it represents.

2008-Apr-29, 03:39 PM
was that an answer to my question "would you say that 1/3 does not equal 0.333~ ?"

A post intervened, maybe. :)

By definition no. By laying one on the other it doesn't matter. The difference is not worth anything.

01101001
2008-Apr-29, 03:44 PM
I say it is a process because it never ends.

It's not a process. It's a name. It's just a string of characters that designate a value.

But, since you say 0.333~ is a process, then may I ask you what value is designated by another string of characters, the "process" 1 divided by 3 (i.e. 1/3)? When that long-division process ends, what is the result?

hhEb09'1
2008-Apr-29, 03:53 PM
was that an answer to my question "would you say that 1/3 does not equal 0.333~ ?"

A post intervened, maybe. :)By definition no. By laying one on the other it doesn't matter. The difference is not worth anything.I should know better than asking a question that includes a question within it! BAUT doesn't support recursion, as is well known, so I accept your answer. :)

Now, if Bogie would only answer the question "What is 0.999... divided by 3?"

Neverfly
2008-Apr-29, 08:20 PM
Neverfly,

if 1 and 0.999~ are different, then there must be a number between them, in fact there would have to be an infinite number of numbers between them, can you give one of those number between them?

I suppose that you would just say that 0.999~ is just a construct. If so does it not represent any number?

if 0.999~ does represent a number/value then even if you have to give an answer to my first question as a construct, it should be possible.

Yes, there is a number between them. That's the problem- that number is undefined.

I've made four? I think? posts on it at this point. It's a quirk of the system.
The same with the other examples some of you are shoving up Bogie's nose.

In order to facilitate- We assign the regular representation- therefor 1 is equal to0.9999>Infinity.
It isn't equal- but close enough and the number between is undefinable.

To answer the last question- No. It is not possible. It would be nice if the system was perfect making it possible- it is not.

And That- my friend- is the very basic grounds for why this argument continues.

It isn't about the numbers. It's about those who want to believe the system is perfect- and those who don't.
For anyone calling one person Right or another person Wrong- therein lies your problem.

Bogie is not wrong.Neither is Moose. that's part of the quirk.

I was enjoying tolling this thread, not arguing in it. So aside from providing GrapesofhHeb with his request, I'm going to return to doing that now. Any other questions folks might ask are probably already addressed somewhere in this and previous posts.

Kaptain K
2008-Apr-29, 08:27 PM
It's about those who want to believe the system is perfect- and those who don't.
No, it is about those who understand arithmetic and those who don't!

Neverfly
2008-Apr-29, 09:40 PM
No, it is about those who understand arithmetic and those who don't!

It's hardly in the realm of arithmetic, Kaptain K.

Limit theory and several other theorems discussed are not covered in arithmetic.

Now you are just being absolutely silly and making outrageous claims hoping it will magically give weight to your arguments. It won't.
You are baiting- nothing more.

Disinfo Agent
2008-Apr-29, 09:46 PM
Well, it's a little above basic arithmetic, but you don't really need limits to do it either. This shows up when we teach decimal expansions for rational and irrational numbers for the first time, which where I come from is between the 5th and the 7th years of schooling. (I know that this does not cover all proofs, but it covers the simplest, which seem to be the most persuasive for many people anyway.)

Tobin Dax
2008-Apr-29, 11:17 PM
It's hardly in the realm of arithmetic, Kaptain K.

Limit theory and several other theorems discussed are not covered in arithmetic.

Now you are just being absolutely silly and making outrageous claims hoping it will magically give weight to your arguments. It won't.
You are baiting- nothing more.

The proof can be shown by arithmetic, as posters have done with Bogie for the last few pages. And unlike tricks showing that 1+1=3, there are no trick steps here. The only possible exception to only using arithmetic is saying that 1/9=0.11111~ (or 1/3=0.3333~). Simple long division will show that, though. So Kaptain K is right: It's about people who don't understand arithmetic. These values are the same answer because you can get both answers the same way.

Neverfly
2008-Apr-29, 11:28 PM
The proof can be shown by arithmetic, as posters have done with Bogie for the last few pages. And unlike tricks showing that 1+1=3, there are no trick steps here. The only possible exception to only using arithmetic is saying that 1/9=0.11111~ (or 1/3=0.3333~). Simple long division will show that, though. So Kaptain K is right: It's about people who don't understand arithmetic. These values are the same answer because you can get both answers the same way.

This isn't quite correct.
You can't.

If you take a calculator and pop in 0.33333333333333333 and multiply it by 3, you are not going to get 1 as an answer. You are going to get 0.99999999999999.
If you punch in 1 and hit enter and then 3 and hit divide- you will get0.333333>infinity as the answer.

Now it depends a bit on the calculator and what algorithms it uses- but generally this is what will happen.

Kaptain K
2008-Apr-29, 11:39 PM
So, now we're down to it doesn't work on my calculator.Therefore, it must be wrong!

Sheeeeeesh. :wall:

Neverfly
2008-Apr-29, 11:42 PM
So, now we're down to it doesn't work on my calculator.Therefore, it must be wrong!

Sheeeeeesh.

No, Kaptain K.

It doesn't work on most any calculator that way.

Ditch the calculator and do it on paper. You will get the same result.

The point is something that you seem to be missing- which I have covered in several posts at this point.

Let me be frank: You are being Stubborn.

Kaptain K
2008-Apr-29, 11:47 PM
Let me be frank: You are being Stubborn.
Let me be frank: You are Wrong and Stubborn!

Bearded One
2008-Apr-29, 11:53 PM
This isn't quite correct.
You can't.

If you take a calculator and pop in 0.33333333333333333 and multiply it by 3, you are not going to get 1 as an answer. You are going to get 0.99999999999999.With a good calculator if you divide 1 by 3 and then multiply the result by 3 you get 1 again. Doesn't work on the simple calculators though. The problem is the decimal is infnite, that's what the ~ or ... means. You simply can't enter that on a calculator.

Moose
2008-Apr-29, 11:54 PM
If you take a calculator and pop in 0.33333333333333333 and multiply it by 3, you are not going to get 1 as an answer. You are going to get 0.99999999999999.

Okay, Neverfly? You are firmly in my professional territory now, and I know I'm not the only professional programmer or engineer here. What you said is literally accurate, but an absolute strawman as far as this discussion is concerned. Don't equate consumer electronics with math. They're not the same thing.

You cannot approximate 1/3 into a floating point variable and expect to inverse it. That variable type will chop off the end because that's precisely what it was designed to do with irrationals. Approximate.

This is an identical argument that would claim that '1000 + 1000' cannot exist because it doesn't fit in a small int (1 byte) variable which holds numbers only as large as 256.

If an approximation isn't good enough (it usually is), programmers know how (at least I've done it from time to time) to not lose any precision at all over some non-abusive number of divisions. It's expensive, though, so you do it only when you need that precision.

0.333~, however, is not an approximation. It's a very exact value, because every single 3 in the expansion is already in place. That's what the tilde means. Every 3 is already where it needs to be. It is absolutely not a process unless you are explicitly dealing with an approximation of the infinite expansion. If you want to deal with the real thing, however, 0.333~ = 1/3.

Nev, start here (http://en.wikipedia.org/wiki/Floating_point#Representable_numbers.2C_conversion _and_rounding) and work your way down. Especially pay attention to sections 8 and 9.

Moose
2008-Apr-30, 12:00 AM
With a good calculator if you divide 1 by 3 and then multiply the result by 3 you get 1 again. Doesn't work on the simple calculators though. The problem is the decimal is infnite, that's what the ~ or ... means. You simply can't enter that on a calculator.

Yup. This is because a relatively new innovation (it predates my degree) is to have your floating point variable contain a few more digits than you'll display. It means your divisions will be more accurate than your rounding point. In an electronic device that uses this method, 1 / 3 * 3 will result in 1. The cake is a lie, though, because it's still storing 999999999|999 x 10^-12. (The pipe char represents the rounding point, and 12:9 are common float sizes in non-dollar-store consumer gadgets.)

Consumer electronics don't do math. They do approximations of math.

Neverfly
2008-Apr-30, 12:19 AM
Nev, start here (http://en.wikipedia.org/wiki/Floating_point#Representable_numbers.2C_conversion _and_rounding) and work your way down. Especially pay attention to sections 8 and 9.

Sections 8 (Accuracy problems) and 9 (Minimizing Accuracy Problems) Cover just what I said Moose.:neutral:

Next, the calculator is an example- in which I also suggested algorithms and pen and paper.

Th OP is a fundamental problem. Not a true question.

The debate is not about Who is Right and Who is Wrong- But about How one views the quirks in the system.

No one is right or wrong on this one.

We are discussing a Construct- that does not always represent things perfectly.

Cut an apple into thirds- and mathematically you end up with infinities.
But the apple remains finite.
The construct of the math results in infinities- not the cutting of the apple.

ETA: Bearded one- I'm using a Hewlett Packard 48G. I can specify infinities and reach the result you described- or I can use a basic algorithm and reach an irrational answer. So it doesn't matter as much whether you are using a "good" calculator or a cheap pocket calc.
ETA²: Bearded one your following post just agreed with my ETA:p except you're talking TI and I'm talking HP.

Bearded One
2008-Apr-30, 12:21 AM
Yup. This is because a relatively new innovation (it predates my degree) is to have your floating point variable contain a few more digits than you'll display.The give away on most is that the final 1 contains a decimal point: 1. That tells you here are hidden digits. Now my TI-89 can work in an "exact" mode. In that mode if you enter 1/3 it tells you the answer is one third (1/3). In extended calculations it carries it forth symbolically, maintaining accuracy. You can force it to display a decimal approximation if needed.

Moose
2008-Apr-30, 12:24 AM
Sections 8 (Accuracy problems) and 9 (Minimizing Accuracy Problems) Cover just what I said Moose.:neutral:

Then you should be well aware that floating point arithmetic is an approximation of math, and not math. Sections 8 and 9 are very firm on that point. It's simply not relevant to this discussion.

Neverfly
2008-Apr-30, 12:27 AM
Then you should be well aware that floating point arithmetic is an approximation of math, and not math. Sections 8 and 9 are very firm on that point. It's simply not relevant to this discussion.
I typed up a disagreement...

...then realized you're right.

Moose
2008-Apr-30, 12:29 AM
The give away on most is that the final 1 contains a decimal point: 1. That tells you here are hidden digits. Now my TI-89 can work in an "exact" mode. In that mode if you enter 1/3 it tells you the answer is one third (1/3). In extended calculations it carries it forth symbolically, maintaining accuracy. You can force it to display a decimal approximation if needed.

That's neat. I don't do much precision work with calculators. Just computers, occasionally, so if I care about telling the difference, I actually do have to take a closer look at the variable to know if my result is going to be rounded or not.

Moose
2008-Apr-30, 12:41 AM
...then realized you're right.

Thanks for saying. Lots of people would have stuck to their guns to save face way past the point that was possible.

Here's the thing Nev: the big mistake Bogie and later you made in this thread is the instinctive (and understandable) desire to treat iterative approximations of math as if they were the same thing as math.

Engineers, programmers, physicists are perfectly content to use approximations in their arithmetic to save time and effort (and programmers really don't have a choice in the matter.)

Approximations are 'good enough' for everything but formal proofs or deriving new math. It stops being math, though, and becomes approximations of math just as soon as you accept a loss of precision.

Formal math requires absolute precision at all times. Pi cannot be written out as anything other than Pi or some equivalent algebraic fractional: c/d. Not and remain math. 3.14 is an approximation of Pi. Not Pi. In formal math, and you've seen proof after proof of this, 1 = 0.999~ because the tilde in 0.999~ says explicitly that we have not accepted a loss of precision.

The moment you let yourself think of some finite approximation of 0.999~, however, (your "process" idea), it stops being math and becomes an approximation of math. When you're using approximations, then 0.999...9 < x < 0, and x has an infinite number of possible solutions.

Is this a bit clearer now?

blueshift
2008-Apr-30, 12:55 AM
I can prove it in four steps.

1.) .999 = 1 - .001 Now subtract one from each side and you get:
2.) .999 - 1 = -.001 Next multiply each side by 1000:
3.) 999 - 1000 = -1 Finally divide each side by a minus one resulting in:
4.) 1 = 1

Voila!

2008-Apr-30, 12:59 AM
Moose

The moment you let yourself think of some finite approximation of 0.999~, however, (your "process" idea), it stops being math and becomes an approximation of math. When you're using approximations, then 0.999...9 < x < 0, and x has an infinite number of possible solutions.

What I was saying was that .999~ is a process because to resolve the number you must work wiith every digit, which you can't do if it is to infinity.
By rounding it up or down at some point we modify the process of resolving it to make a workable number.

However, I also understand where you are saying that .999~ can be the complete number with all the 9s accounted for by the tilde.

Neverfly
2008-Apr-30, 01:06 AM
Is this a bit clearer now?

It was clear all along....:doh:

What was unclear- to me and maybe others- because I failed to see and clarify it- Was the difference between the two.

Yes, You're absolutely right. And 0.99999> = 1. I was talking about the quirks in the system that initially create the discrepancy- treating it like math.

When you said that pi is not 3.14159 - it is π - and THAT is how it is used in math - I literally slapped my forehead ---
Duh Ross! That's true!
We are on the same page now. Your statement agrees with mine.
0.99999999> is not 1
3.14159 is not pi.
Those are results of our approximations and they are not math.

Click!* I can't believe I didn't realize my error in recognition earlier!

Moose
2008-Apr-30, 01:12 AM
Right.

.9999....9, the finite approximation, is not equal to 1.

.9999~, the infinite progression where no loss of precision is accepted, is equal to 1.

Neverfly
2008-Apr-30, 01:14 AM
Right.

.9999....9, the finite approximation, is not equal to 1.

.9999~, the infinite progression where no loss of precision is accepted, is equal to 1.

Exactly!
Definitely on the same page now;)

Nowhere Man
2008-Apr-30, 01:18 AM
I think everyone here could use a heapin' helping of Eli Maor's To Infinity and Beyond. (http://www.amazon.com/Infinity-Beyond-Eli-Maor/dp/0691025118/ref=sr_1_4?ie=UTF8&s=books&qid=1209518122&sr=1-4) (And yes, it came out long before Toy Story.) Some for the information, others for the story.

Fred

afterburner
2008-Apr-30, 03:20 AM
So then we all agree that a 9 is actually an 8.9999...
and an 8 is a 7.9999...
and a 7 is a 6.9999...
etc.

I wonder what would happen if we substitute all of the numbers in our 0.999...with the substitutes above, and all those substitutes with similar substitutes to infinity...:think:

Neverfly
2008-Apr-30, 03:33 AM
So then we all agree that a 9 is actually an 8.9999...
and an 8 is a 7.9999...
and a 7 is a 6.9999...
etc.

I wonder what would happen if we substitute all of the numbers in our 0.999...with the substitutes above, and all those substitutes with similar substitutes to infinity...:think:

Nothing would change.

The agreement is based on the understanding of the principles.

Simply put- I did not retract my claims.
I just realized that my claims were a Literal Interpretation of the OP which is actually irrelevant.
The resultant .99999>infinity - as I previously said - is a product of the construct (The arithmetic rounding things off). In that regard- you could say I was correct- and the reason I was pushing the case was because I was seeing a Group mentality- kind of a minor 'gang up' against folks that were pointing out was I was pointing out.

Although I realized that I was referring to an illusion, what I failed to account for was that the illusion was a construct within a construct.

What clarified that for me was when Moose mentioned that pi was not the approximation we use, it was pi.
Click* I thought about how when I write out my own equations or formulas, I don't put 3.14159 in- I put in the symbol for pi- I use pi- not the approximation. The approximation does not show up until I start calculating--- the construct.
The difference explains why .99999> infinity is a product of that and it isn't being used to represent 1. It IS 1. Mathematically, it IS 1.

Jerry
2008-Apr-30, 03:41 AM
So then we all agree that a 9 is actually an 8.9999...
and an 8 is a 7.9999...
and a 7 is a 6.9999...
etc.

I wonder what would happen if we substitute all of the numbers in our 0.999...with the substitutes above, and all those substitutes with similar substitutes to infinity...:think:

Remember, all math is an abstraction, and defininig limits as being equal to finite numbers is also an abstraction - one that is easy to prove using the rules of elementry calculus. We could define 0.9999... as something else, just as the US congress once tried to define Pi as a rational number.

Neverfly
2008-Apr-30, 03:49 AM
just as the US congress once tried to define Pi as a rational number.

Bob Dole spoke out against that citing that it was "apples to Oranges."

Nowhere Man
2008-Apr-30, 03:58 AM
just as the US congress once tried to define Pi as a rational number.

Are you thinking of the Indiana state legislature? (http://en.wikipedia.org/wiki/Indiana_Pi_Bill) Or the bogus story about Alabama? (http://www.snopes.com/religion/pi.asp)

Fred

Frog march
2008-Apr-30, 05:31 AM
I think I said this years ago, on this thread, but if you use base 2 then the number 0.11111111111~ =1 too.

and if you use Hexadecimal then 0.FFFFFF~=1

just for interest.

kucharek
2008-Apr-30, 10:38 AM
OMG - what have I done! Haven't been here after my initial (in a relative way) post and now found this thread again grown by half a dozen pages... Bad dragon.

Moose
2008-Apr-30, 11:07 AM
OMG - what have I done! Haven't been here after my initial (in a relative way) post and now found this thread again grown by half a dozen pages... Bad dragon.

We're working on a petition that call for you to stand in front of a pie (Pi) throwing firing squad, that is, 3 adults and a toddler (.14 of an adult), each with 0.999~ pies of the kind of your choice.

Would you like to sign our petition and support a 'worthy' cause? ;)

SeanF
2008-Apr-30, 01:50 PM
Psst, Moose...

You cannot approximate 1/3 into a floating point variable and expect to inverse it. That variable type will chop off the end because that's precisely what it was designed to do with irrationals. Approximate.
1/3 isn't an irrational. :)

Moose
2008-Apr-30, 02:27 PM
1/3 isn't an irrational. :)

True, my bad. Still, to computers, 1/3 may as well be. The same precision issue applies.

Disinfo Agent
2008-Apr-30, 02:50 PM
Yup. This is because a relatively new innovation (it predates my degree) is to have your floating point variable contain a few more digits than you'll display. It means your divisions will be more accurate than your rounding point. In an electronic device that uses this method, 1 / 3 * 3 will result in 1. The cake is a lie, though, because it's still storing 999999999|999 x 10^-12. (The pipe char represents the rounding point, and 12:9 are common float sizes in non-dollar-store consumer gadgets.)

Consumer electronics don't do math. They do approximations of math.I asked for (0.999 999 999 9)-1 in my calculator a while ago, when I was fooling around with continued fractions, and got 1 as the result.

I can prove it in four steps.

1.) .999 = 1 - .001 Now subtract one from each side and you get:
2.) .999 - 1 = -.001 Next multiply each side by 1000:
3.) 999 - 1000 = -1 Finally divide each side by a minus one resulting in:
4.) 1 = 1

Voila!And how would you extend that to .999~?

So then we all agree that a 9 is actually an 8.9999...
and an 8 is a 7.9999...
and a 7 is a 6.9999...
etc.

I wonder what would happen if we substitute all of the numbers in our 0.999...with the substitutes above, and all those substitutes with similar substitutes to infinity...:think:Basically, you've fallen into that trap of confusing symbol with number everyone has been talking about.

9 = 8.9999...
8 = 7.9999...
7 = 6.9999...
etc.They represent the same number, yes.

I wonder what would happen if we substitute all of the numbers in our 0.999...with the substitutes above, and all those substitutes with similar substitutes to infinity...:think:Not in this case, because that would mean replacing the first 9 with an infinity of digits, the second nine with another infinity of digits, and so on. The decimal notation does not allow you to do this. It accepts at most one denumerable infinity of digits, but no more. "0.89...89...89...~" does not exist as a decimal expansion.

tdvance
2008-Apr-30, 05:44 PM
What it really comes down to is the definition of the decimal system for specifying numbers.

First of all, 1 is not the symbol "1", it is just "there" and the symbol 1 represents it. Similarly, 1.1 represents a number that happens to be the same number represented by 2 - 9/10, for example.

The convention accepted by mathematicians (such as myself) around the world is:

a decimal number has a finite number of digits to the left of the decimal point, and either a finite or infinite number of digits to the right.

I think there's no disagreement here about the finite cases, so I'll skip that.

For the case of an infinite number of digits to the right of the decimal point (let's make the left--integer--part zero for simplicity, same principle holds in general though)--

suppose the number looks like:

0.a1a2a3a4a5... (where the ... is shorthand for the unwriteable truth--keep adding digits and never stop).

This is defined to be the number obtained by computing

sum from i=1 to infinity of ai(1/10)i

Now, a sum to infinity is defined as a limit of the partial sums, explicitly

The limit, as n goes to infinity, of the value of:

sum from i=1 to n of ai(1/10)i

Weierstrass gave what is the (now, universally accepted among mathematicians) definition of a limit. It's pretty technical, but it means something close to this in this case:

if f(n) is some function on the integers, the limit of f(n) as n goes to infinity is that number x (if it exists) such that, for any small but positive e you choose, we can always find an n so that |f(n) - x| < e (vertical bars mean "absolute value") and that this holds for all larger n too. It can be proven, if there is a limit, is must be unique (there cannot be two or more limits to any convergent sequence).

In the case of 0.9999999....

the sum above translates to the limit of f(n), as n goes to infinity, where
f(n) is 0.9999...9, with exactly n 9's.

This is a convergent sequence--it turns out there is a unique real number, and it happens to be 1, such that for any small positive e you choose, there is an n with |f(n)-1| < e, and for all larger n too.

in fact, if e is 0.000001, then choose n to be 7 or greater and it works. Any positive e has an n that works. The definition requires e to be positive, so don't say "e=0"--that's not the definition.

No other number has this property.

Thus, by definition, not by the proofs you see in algebra books based on arithmetic methods taught in grade school but never actually proven, but by the definition of an infinite decimal,

0.9999... is equal to 1.

The arguments about a "last 9" are spurious--there is no last 9. That's what "unending" means.

And you can't just "do it again" to get another representation of 1--the decimal representations of 1 are 1 and 0.9999..., not counting the cases where you do 1.0, 1.00, or 01 or 001, etc. There are no others. You can't do 0.999 forever and end in 8, because forever means "no end". If you stick an 8 in the sequence of 9s, it has to be a finite number of digits away.

worzel
2008-Apr-30, 11:27 PM
Yes, it really comes down to definition. But then in a sense so does every theorem of every formal system.

But mathematicians choose their definitions to make numbers behave the way they think they should. And people doubting the equality have their own ideas about how numbers should behave. The utility of the arithmetic proofs offered is that they prove the equality by using only arithmetical identities that presumably the doubters do accept.

The arithmetical proofs also demonstrate that if the equality didn't hold then arithmetic would be inconsistent. So claiming it doesn't hold is claiming that arithmetic is inconsistent, which is a much grander claim than I think most doubters realize they are making.

Neverfly
2008-Apr-30, 11:34 PM
Yes, it really comes down to definition. But then in a sense so does every theorem of every formal system.

But mathematicians choose their definitions to make numbers behave the way they think they should. And people doubting the equality have their own ideas about how numbers should behave. The utility of the arithmetic proofs offered is that they prove the equality by using only arithmetical identities that presumably the doubters do accept.

The arithmetical proofs also demonstrate that if the equality didn't hold then arithmetic would be inconsistent. So claiming it doesn't hold is claiming that arithmetic is inconsistent, which is a much grander claim than I think most doubters realize they are making.

This may apply to some "doubters" Worzel, but not all.

TheHalcyonYear
2008-May-01, 03:11 AM
2004?? This thread has been going for 4 years and people still doubt this fundamental limit theorem :rolleyes:

Kaptain K
2008-May-01, 03:28 AM
2004?? This thread has been going for 4 years and people still doubt this fundamental limit theorem :rolleyes:

No! It was dead until resurrected recently! :eek:

Moose
2008-May-01, 09:35 AM
The thread's been raised at least six times. It's a bit like a movie zombie, turns up when you least expected, needs braaaaaaiiinnnnsss, and no matter how fast you run away, you can never quite seem to get away from it.

Bogie
2008-May-01, 02:17 PM
tdvance made what looks like a pretty authoritative post that said Thus, by definition, not by the proofs you see in algebra books based on arithmetic methods taught in grade school but never actually proven, but by the definition of an infinite decimal,

0.9999... is equal to 1. (http://www.bautforum.com/off-topic-babbling/14593-do-you-think-0-9999999-1-infinite-9s-64.html#post1230848) Does everyone agree with that statement? If so we can put this thread to rest :).

Disinfo Agent
2008-May-01, 02:21 PM
Tdvance was being a tad hyperbolic, IMHO. All those justifications for the equality are valid, for the reasons worzel explained above, not just the one tdvance wrote.

Bogie
2008-May-01, 06:14 PM
Yes, it really comes down to definition. But then in a sense so does every theorem of every formal system.

But mathematicians choose their definitions to make numbers behave the way they think they should. And people doubting the equality have their own ideas about how numbers should behave. The utility of the arithmetic proofs offered is that they prove the equality by using only arithmetical identities that presumably the doubters do accept.

The arithmetical proofs also demonstrate that if the equality didn't hold then arithmetic would be inconsistent. So claiming it doesn't hold is claiming that arithmetic is inconsistent, which is a much grander claim than I think most doubters realize they are making.Are we making a distinction between definitions to make numbers behave, and "rules".

Bogie
2008-May-01, 11:51 PM
Are we making a distinction between definitions to make numbers behave, and "rules".OK, as stated I agree that if you invoke the rule or definition then the equality is proven because I accept the necessity of such definitions and rules to make math work without contradiction. By definition .999…~ = 1.

Now the arithmetic Proof:

You ask your student, Do you agree the 3/9 = .333…~

If they say “Yes” then you ask,

Do you agree that 6/9 = .666…~

If they say “Yes” then you ask,

Do you agree that 3/9 + 6/9 = 9/9

If they say “Yes” then you ask,

Do you agree that .333…~ + .666…~ = .999…~

If they say “Yes” then you ask,

Do you agree that 9/9 = 1

Then you point out that they must agree that .999…~ = 1 based on their own admissions.

But you have not proven that .999…~ = 1, your subject has simply agreed to accept infinite decimal extensions along the way without proof.

Now the next step is to prove that an infinite decimal extension of .999…~ equals one.

Let’s start with .9 and add .09, you get .99

Then add .009 and you get .999,
Then add .0009 and you get .9999,
Then add .00009 and you get .99999,

And you can keep going forever and never get a total of 1.

So the proof in reality is not possible and requires a rule.

Where am I wrong?

Moose
2008-May-02, 12:03 AM
Where am I wrong?

The second half of your post in its entirety is not correct. You should consult my last few posts to Neverfly for the explanation why. You're still equating approximate math with formal math. They're not equivalent.

But you have not proven that .999…~ = 1, your subject has simply agreed to accept infinite decimal extensions along the way without proof.

Now the next step is to prove that an infinite decimal extension of .999…~ equals one.

Let’s start with .9 and add .09, you get .99

Then add .009 and you get .999,
Then add .0009 and you get .9999,
Then add .00009 and you get .99999,

And you can keep going forever and never get a total of 1.

So the proof in reality is not possible and requires a rule.

Bogie
2008-May-02, 12:17 AM
The second half of your post in its entirety is not correct. You should consult my last few posts to Neverfly for the explanation why. You're still equating approximate math with formal math. They're not equivalent.Which part of the following is approximate math? What formal math are you using in place of the approximate math?
Originally Posted by Bogie
But you have not proven that .999…~ = 1, your subject has simply agreed to accept infinite decimal extensions along the way without proof.

Now the next step is to prove that an infinite decimal extension of .999…~ equals one.

Let’s start with .9 and add .09, you get .99

Then add .009 and you get .999,
Then add .0009 and you get .9999,
Then add .00009 and you get .99999,

And you can keep going forever and never get a total of 1.

So the proof in reality is not possible and requires a rule.

Moose
2008-May-02, 12:49 AM
Approximation math:

Let’s start with .9 and add .09, you get .99

Then add .009 and you get .999,
Then add .0009 and you get .9999,
Then add .00009 and you get .99999,

And you can keep going forever and never get a total of 1.

This is approximate math because what you're really describing is a finite series of approximations. You say "goes on forever", but you're still fixated on the idea that such a series must ultimately end. It doesn't. And that absolutely means you're never going to physically write it out. Which is the whole point of limit theory.

Limit theory deals with what happens when you've got an honest to goodness infinite series on your hands.

Formal math:

Now the arithmetic Proof:

You ask your student, Do you agree the 3/9 = .333…~

If they say “Yes” then you ask,

Do you agree that 6/9 = .666…~

If they say “Yes” then you ask,

Do you agree that 3/9 + 6/9 = 9/9

If they say “Yes” then you ask,

Do you agree that .333…~ + .666…~ = .999…~

If they say “Yes” then you ask,

Do you agree that 9/9 = 1

This remains within the realm of formal math, even if it's casually stated. It's still correct and complete, though, because you have not relinquished any precision at all.

Ultimately, your argument doesn't hold together because you're trying to express 0.999~ as a finite progression of increasingly better approximations. 0.999~ is an infinite progression. It's not an approximation. 0.999~ = 1 because every single one of the infinite number of 9s are explicitly specified by the tilde. That's what the tilde means. And only when you retain the absolute precision that math requires does 0.999~ = 1 hold true.

[Edited for clarity. Also, I'm low on time this evening. I have to get a few things done before I turn in, so I'm unlikely to answer until tomorrow sometime.]

Bogie
2008-May-02, 12:57 AM
Approximation math:

This is approximate math because what you're really describing is a finite series of approximations. You say "goes on forever", but you're still fixated on the idea that such a series must ultimately end. It doesn't. And that absolutely means you're never going to physically write it out. Which is the whole point of limit theory.

Limit theory deals with what happens when you've got an honest to goodness infinite series on your hands. Limit theory is definitional; uses rules to replace the impossible so that the math works.

Formal math:

This remains within the realm of formal math, even if it's casually stated. It's still correct and complete, though, because you have not relinquished any precision at all.

Ultimately, your argument doesn't hold together because you're trying to express 0.999~ as a finite progression of increasingly better approximations. 0.999~ is an infinite progression. It's not an approximation. 0.999~ = 1 because every single one of the infinite number of 9s are explicitly specified by the tilde. That's what the tilde means. And only when you retain the absolute precision that math requires does 0.999~ = 1 hold true.That is true by definition. It's a rule. Can't be proven.

Moose
2008-May-02, 01:07 AM
Limit theory is definitional; uses rules to replace the impossible so that the math works.That is true by definition. It's a rule. Can't be proven.

1+1 = 2 is definitional. It's just as much a provable rule as anything else in math. And the proof is in its absolute consistency throughout the entire set that is formal math. Just like limit theory.

Within the rules of formal math (and only within the rules of formal math), 0.999~ = 1 because 0.111~ = 1/9 (which you've asserted but not approximated). And because 0.333~ = 1/3 (which you've also asserted but not approximated). Each one is a limit, and in fact, they work absolutely identically to one another.

0.999~ = 1 is ultimately an inescapable consequence of defining 1+1 to be equal to 2. Math is a symbolic numbering system. Nothing more. It's never pretended to be anything else.

Bogie
2008-May-02, 01:12 AM
1+1 = 2 is definitional. It's just as much a provable rule as anything else in math. And the proof is in its absolute consistency throughout the entire set that is formal math. Just like limit theory.

Within the rules of formal math (and only within the rules of formal math), 0.999~ = 1 because 0.111~ = 1/9 (which you've asserted but not approximated). And because 0.333~ = 1/3 (which you've also asserted but not approximated). Each one is a limit, and in fact, they work absolutely identically to one another.

0.999~ = 1 is ultimately an inescapable consequence of defining 1+1 to be equal to 2. Math is a symbolic numbering system. Nothing more. It's never pretended to be anything else.You are saying math is a symbolic numbering system, nothing more? And the symbols are used to represent numbers. When you use a symbol to represent something it is definitional. You mentioned definitions and rules several times. It is a rule, a definition that enables the equality in the OP.

Neverfly
2008-May-02, 01:55 AM
You are saying math is a symbolic numbering system, nothing more? And the symbols are used to represent numbers. When you use a symbol to represent something it is definitional. You mentioned definitions and rules several times. It is a rule, a definition that enables the equality in the OP.

Essentially yes.

But it isn't the definition that enables the "equality."

It is that the result that makes .9999~ appear to be unequal is from the approximate math.

ETA To clarify: .9999~ does not exist. It's an illusion created by the construct (approximate math).

It cannot be and is not equal with 1- but that doesn't matter.
The reason it does not matter is because it's only a result of approximate math- it is actually 1 that got distorted in the process of using the approximate math (Construct).

So Bogie- you are correct that .999~ - all by itself- is not equal to 1. But that becomes irrelevant when you recognize that it's a product of distortion.

A photograph of an orange- depicts an orange. If you say it cannot be an orange because the photograph is flat and an orange is round, people will keep repeating that it is still an orange in the picture. It's just that it got 'distorted' in representation.

Bogie
2008-May-02, 02:16 AM
Neverfly, in the statement .999…~, the .999 are numbers, the … represents numbers, and the tilde represents an infinite decimal extension.

It is as simple as this. You have to use notations and definitions to express the infinity, and only the infinity makes the equality under the rules of limit theory.

Early in the series of posts that I participated in, I said if it is a rule, then I agree with the equality.

It is a rule if you allow that definitions, notations, representations are rules.

Neverfly
2008-May-02, 02:19 AM
Neverfly, in the statement .999…~, the .999 are numbers, the … represents numbers, and the tilde represents an infinite decimal extension.

It is as simple as this. You have to use notations and definitions to express the infinity, and only the infinity makes the equality under the rules of limit theory.

Early in the series of posts that I participated in, I said if it is a rule, then I agree with the equality.

It is a rule if you allow that definitions, notations, representations are rules.

See, that's just it. It isn't a rule or definitions that make the equality.

It is the reality.

The approximate math- the Construct- presents the illusion of inequality- the Limit theory is to account for that illusion.

Kaptain K
2008-May-02, 02:39 AM
To paraphrase Emerson, Lake and Palmer:

Welcome back my friends to the thread that never ends...

Neverfly
2008-May-02, 02:46 AM
<Kicks Kaptain K outta the thread>

Bogie
2008-May-02, 03:17 AM
Does everyone agree:
if we start with .9 and add .09, you get .99
Then add .009 and you get .999,
Then add .0009 and you get .9999,
And you can keep going forever and never get a total of 1.
Does everyone agree that you cannot prove the equality this way?

OK, let’s just throw out the possibility of proving the equality that way.

So let’s use the arithmetic proof and get the student by their own admission to accept the equality without having to prove an infinite decimal extension. Essentially we are tricking them into accepting the equality without having to address the proof of the infinite decimal extension as follows:

You ask your student, Do you agree the 3/9 = .333…~
If they say “Yes” then you ask,
Do you agree that 6/9 = .666…~
If they say “Yes” then you ask,
Do you agree that 3/9 + 6/9 = 9/9
If they say “Yes” then you ask,
Do you agree that .333…~ + .666…~ = .999…~
If they say “Yes” then you ask,
Do you agree that 9/9 = 1

Then you point out that they must agree that .999…~ = 1 based on their own admissions.

But you have not proven that .999…~ = 1, your subject has simply agreed to accept infinite decimal extensions along the way without proof.

So what are we going to do?

Let’s define in limit theory that the limit “1” can be reached using the infinite decimal extension. And let’s use the notation .999…~ to represent that the limit 1 is equal to the notation .999…~.

Have we proved it or have we defined it to be true?

TheHalcyonYear
2008-May-02, 04:01 AM
Are people really unable to understand that this simple equality is true? Do the astronomers and scientists here at least realize that this is true by nature of limit theory??

Neverfly
2008-May-02, 04:16 AM
TheHalcyoneyear, This thread has generated over 1900 posts now.

It really is a more Subtle strangeness and a bit more complex than understanding something like a Rock is Hard.

Snide remarks are more inaccurate than anything else...

TheHalcyonYear
2008-May-02, 05:53 AM
TheHalcyoneyear, This thread has generated over 1900 posts now.

It really is a more Subtle strangeness and a bit more complex than understanding something like a Rock is Hard.

Snide remarks are more inaccurate than anything else...
It may be a but subtle, but it is fundamental to the understanding of calculus. One cannot much hard science without a understanding of the calculus and it surprises me that so many here don't seem to understand something so basic.

My remarks are not meant to be snide, they express my surprise at seeing such a fundamental and well understood proposition being questioned in a supposedly scientific forum.

worzel
2008-May-02, 05:54 AM
So let’s use the arithmetic proof and get the student by their own admission to accept the equality without having to prove an infinite decimal extension. Essentially we are tricking them into accepting the equality without having to address the proof of the infinite decimal extension as follows:

You ask your student, Do you agree the 3/9 = .333…~
If they say “Yes” then you ask,
Do you agree that 6/9 = .666…~
If they say “Yes” then you ask,
Do you agree that 3/9 + 6/9 = 9/9
If they say “Yes” then you ask,
Do you agree that .333…~ + .666…~ = .999…~
If they say “Yes” then you ask,
Do you agree that 9/9 = 1

Then you point out that they must agree that .999…~ = 1 based on their own admissions.

But you have not proven that .999…~ = 1, your subject has simply agreed to accept infinite decimal extensions along the way without proof.
The proof works because the student already tacitly accepts that infinite decimals can express rational numbers.

So what are we going to do?

Let’s define in limit theory that the limit “1” can be reached using the infinite decimal extension. And let’s use the notation .999…~ to represent that the limit 1 is equal to the notation .999…~.

Have we proved it or have we defined it to be true?
Virtually every proof you'll ever see does not start from the axioms of the formal system and progress from there using only the primitive logical rules of inference. They typically rely on previous well known theorems and derived rules of inference. That's all that's going on with using 1/3=0.333... No proof can be said to have been checked until a few hard core mathematicians have made sure that the proof can be broken down to the basics, but virtually no one reading the proof would ever do that.

It's no more a trick than using a proof that ¬A is false to prove A without including a proof that (¬(¬A))=>A using only the axioms

(p => (q => p))
((p => (q => r)) => ((p => q) => (p => r)))
(( ¬p => ¬q) => (q => p))

and the inference rule

p,(p=>q) |= p

Using your logic you could reject any proof by contradiction as being "merely by definition" if it didn't include this proof (or similar) and boldly state that "in reality [ whatever that's supposed to mean ] it still isn't true."

The real problem here is that people like you just don't get that maths is just a formal system. Everything that is provable is "true" by definition. Whether it is true in reality or not has absolutely no meaning unless you first state what you think the formal system is modelling: maths does not model numbers; numbers are the formal system.

To say "oh yeah well that's just true by definition" as if that some how diminishes the truth of the statement is to completely and utterly miss the whole point.

It reminds me this:

Facts are meaningless. You could use facts to prove anything that's even remotely true! - Homer Simpson

Neverfly
2008-May-02, 06:12 AM
It may be a but subtle, but it is fundamental to the understanding of calculus. One cannot much hard science without a understanding of the calculus and it surprises me that so many here don't seem to understand something so basic.

My remarks are not meant to be snide, they express my surprise at seeing such a fundamental and well understood proposition being questioned in a supposedly scientific forum.

I disagree- I managed to successfully get through Calculus with an A (many years ago mind you).
It's an understanding that can by tricky due to the illusionary effects.

I mean c'mon- I was arguing the point earlier- even though I even knew better.

It wasn't until Moose reminded me of HOW I work with equations that It all came back to the front of my brain.

Mainly because a lot of folks don't USE the higher maths on a daily basis- and participate on the forum- it isn't strange to think that old habits will die hard.

worzel
2008-May-02, 06:14 AM
ETA To clarify: .9999~ does not exist. It's an illusion created by the construct (approximate math).
No it's not. It's an infinity of 9s after a decimal point. The set of positive numbers is infinite; is saying "the positive numbers" likewise an illusion?

It cannot be and is not equal with 1- but that doesn't matter.
The reason it does not matter is because it's only a result of approximate math- it is actually 1 that got distorted in the process of using the approximate math (Construct).
It does matter, it's not approximate, and it does equal 1. But hey, at least 0 out of 3 is consistent :)

So Bogie- you are correct that .999~ - all by itself- is not equal to 1.
No he's not. I thought you came round to accepting the equality.

But that becomes irrelevant when you recognize that it's a product of distortion.
It's no more a distortion than 1.000~ or 4/4 is a distortion of 1.

A photograph of an orange- depicts an orange. If you say it cannot be an orange because the photograph is flat and an orange is round, people will keep repeating that it is still an orange in the picture. It's just that it got 'distorted' in representation.
That's really just saying that syntactically they're different. That doesn't change the fact that mathematically they are equal. Your logic here could apply to any equation where the right hand side weren't character for character a copy of the left hand side.

I.e. you could use the same logic on 2+2=4, but would you claim that 2+2 is a distortion of 4 and that's why it doesn't really matter that 2+2 doesn't really equal 4?

Neverfly
2008-May-02, 06:17 AM
worzel ,
I understand exactly what you are saying- But I was talking to Bogie and describing it progressively.

Now, I'm not going to argue the semantics with you- when I understand the other side of how it's being viewed- and you, clearly, do not understand.

worzel
2008-May-02, 06:28 AM
neverfly, how is it helpful to say that 0.999~ does not equal 1? It is not just semantics.

I certainly don't understand how you can say that they are not really equal after agreeing that they are.

Neverfly
2008-May-02, 06:31 AM

No it's not. It's an infinity of 9s after a decimal point. The set of positive numbers is infinite; is saying "the positive numbers" likewise an illusion?
It doesn't exist because it is a 1.

It does matter, it's not approximate, and it does equal 1. But hey, at least 0 out of 3 is consistent :)
You didn't even understand what I said, but whipped out your hammer to -what? Show off your smarts? You failed because you swung the hammer without understanding what was said.

What was said was that the .999~ is 1- however, using the same Approximate Math that Moose talking about earlier is how it resulted.

Moose also linked to a Wiki article- I suggest you read it too like I did.

No he's not. I thought you came round to accepting the equality.
Yes, he is. He is right
He is also wrong.
Again, you failed to understand exactly what I was saying (You are capable of that mistake as much as I am you know) and reead it in your eyes- and not Bogies- the reason I specifically Worded it the WAY that I did, is to put it in Bogie's terms. I'm not talking to you worzel- and Bogie is the one asking so I'm using it in the terms Progressively- to get him from point A to point B.

You don't have to agree with How I'm doing it- but you can refrain from swinging your hammer until you have understood EXACTLY what it is that I am saying.

It's no more a distortion than 1.000~ or 4/4 is a distortion of 1.

That's really just saying that syntactically they're different. That doesn't change the fact that mathematically they are equal.
Gee.... NO! Really?! How odd- I thought I said that too:doh:

Your logic here could apply to any equation where the right hand side weren't character for character a copy of the left hand side.

I.e. you could use the same logic on 2+2=4, but would you claim that 2+2 is a distortion of 4 and that's why it doesn't really matter that 2+2 doesn't really equal 4?
What you just said makes no sense and has absolutely nothing to do with what I said.
Don't even TRY telling me you understood me now- that last line totally blows that idea outta the water. You clearly misunderstood.

worzel
2008-May-02, 06:32 AM
I understand the other side of how it's being viewed- and you, clearly, do not understand.
Other than that they are obviously syntactically different, what other side are you talking about that you but not I understand?

Neverfly
2008-May-02, 06:33 AM
(snip)

I certainly don't understand (snip)

I agree. So let me do it my way.

worzel
2008-May-02, 06:39 AM

It doesn't exist because it is a 1.
Whaaaat?

So then 2/2 doesn't exist? Explain why not rather than giving me your rather ridiculous commentary.

And you still haven't explained why "the positive numbers" is only approximate.

Yes, he is. He is right
He is also wrong.
This is maths, not zen buddhism

You don't have to agree with How I'm doing it- but you can refrain from swinging your hammer until you have understood EXACTLY what it is that I am saying.
Look, if you keep saying that 0.999~ does not equal 1 in anything other than a purely syntactic sense then I am going to disagree with you.

worzel
2008-May-02, 06:41 AM
(snip)

I certainly don't understand (snip)
I agree. So let me do it my way.
If you're gonna start quoting me like that you might as well pm yourself.

Neverfly
2008-May-02, 06:48 AM
Whaaaat?

So then 2/2 doesn't exist? Explain why not rather than giving me your rather ridiculous commentary.

And you still haven't explained why "the positive numbers" is only approximate.

This is maths, not zen buddhism

Look, if you keep saying that 0.999~ does not equal 1 in anything other than a purely syntactic sense then I am going to disagree with you.

Knock yourself out. All you basically are doing is arguing with how you don't like my wording. Frankly I don't care if you disagree or not. All I have been saying to you is to back up off my back a bit and let me explain it to Bogie in the terms he and I are using.

If you're gonna start quoting me like that you might as well pm yourself.

You have been distorting my posts and applying meaning to them that I did not say.

So rock on.

Frog march
2008-May-02, 07:21 AM
0.9999~ is a bit like a car that in pieces in boxes and stored in a shed.
all the pieces are there to make the car, not a piece is missing.

Frog march
2008-May-02, 07:25 AM
further to that analogy

to say that 0.999~ is not equal to 1 is to say that at infinity there is an '8', but even if you could get to infinity, by definition, all the numbers are '9's.

that is, all the boxes are full, not one is missing a part.

Ling
2008-May-02, 07:36 AM
This is maths, not zen buddhism

Maybe we need a poll option that says they are both equal and not equal :)

Neverfly
2008-May-02, 07:39 AM
Maybe we need a poll option that says they are both equal and not equal :)

LOL
That might spoil all the fun.

worzel
2008-May-02, 08:32 AM
Knock yourself out. All you basically are doing is arguing with how you don't like my wording.
If that is the case then presumably you mean that you are simply pointing out that they are syntactically different. If that is so then you can hardly say that I have failed to understand "the other side".

Frankly I don't care if you disagree or not. All I have been saying to you is to back up off my back a bit and let me explain it to Bogie in the terms he and I are using.
Well you've been posting that they are not really equal despite really being equal for a while now in an attempt to help Bogie, with little success it seems. But I'll respect your wish and back off if you really are just saying that they are syntactically different. But my parting shot is that I think that Bogie already knows that and it is irrelevant to his problem with the equality.

You have been distorting my posts and applying meaning to them that I did not say.
I apologize if that is the case. That wasn't my intent and I didn't think that I had. Just to be clear, could you confirm categorically that you are merely making the point that they are syntactically different.

Neverfly
2008-May-02, 08:34 AM
I apologize if that is the case. That wasn't my intent and I didn't think that I had. Just to be clear, could you confirm categorically that you are merely making the point that they are syntactically different.

Acepted and thanks for not letting my attitude bait you.

Yes, I can confirm that. That is why I used the analogy of the orange in a photograph.

Moose
2008-May-02, 09:35 AM
I apologize if that is the case. That wasn't my intent and I didn't think that I had. Just to be clear, could you confirm categorically that you are merely making the point that they are syntactically different.

Nev, 0.999~ is no more illusory than the number 1 itself. Think about it (and this applies to Bogie as well). If I tell you '1', well, what is that? '1'? '1' what? '1' does not apply to the real world. It's a symbolic construct, a definition if you will, and the very cornerstone of our numbering system. If anything can be said to be a rule, this is it. Ultimately, it's math.

Bogie, no, I don't agree that 0.999~ = 1 is a rule, at least not in the same way you seem to mean it. It's a logical consequence of the real number set, a natural property of 1, and a critically important discovery that (ultimately) let us put people on the moon.

'1' has no literal basis in the natural world, but as a symbolic construct, it has vital real world applications including 'Dude, hold my beer.' Infinity (a symbolic construct), and limit theory (a consequence of that construct) are exactly the same in that respect.

Ultimately, it doesn't matter how you parse the words, 0.999~ = 1 because it's a direct consequence of the self-consistency of the real number set.

Moose
2008-May-02, 09:44 AM
Yes, I can confirm that.

Okay, so you're saying that 0.999~ = 1, but are being represented differently? I really don't see why that's important to you. Math isn't useful if we don't work with equalities that have different representations. 2+2 = 4 is useful. 4 = 4 is self-evident. (Both are illusory.)

Neverfly
2008-May-02, 09:49 AM
I'd like to know this myself, Nev, because you appear to have completely fallen off what I'd thought we'd settled yesterday (or was it the day before?)

Worzel is correct. TheHalcyonYear is correct. And I would like you to go back and try to understand what they're saying.

No I didn't.

Let me try to be very clear here...

You are telling folks that something that looks different than something else is the same thing.

And you expect this isn't going to cause confusion?

Worzel was NOT correct, TheHalcyonYear was not- and now you are not too.

You are not incorrect in your argument- you are incorrect in your stubborness to accept that your argument is causing (partly) some of the confusion.

That is what I am trying to address.

I have been pretty clear- and it's more than frustrating to have several people jump on my back when they misconstrue what I am saying because they are afraid that it's backsliding- It is not backsliding. I am breaking the argument down into chunks- because it is the very appearance of it that is causing the confusion- I should Know- because it threw ME for a loop for a while and once I got where I went wrong- I was able to look backwards at what caused me to fall off the wagon in the first place.

Moose, everything you just said in that post, ultimately causes confusion too.

It surprised you when I suddenly clicked- because what got through wasn't what you Expected to get through- and I am trying to use that to my advantage now in my posts.

Does that make any sense?

You guys have the mentality now- after having argued the point- to react sharply if someone says something you don't like out of the fear that it will cause others to think they are unequal.

I assure you that I understand why they are equal and what it is that's causing the confusion- So I'm trying a different path to cover it.
As long as the other person clicks into place- they will then be able to understand it and the semantics used to demonstrate will no longer be relevant.

Neverfly
2008-May-02, 09:52 AM
Okay, so you're saying that 0.999~ = 1, but are being represented differently? I really don't see why that's important to you. Math isn't useful if we don't work with equalities that have different representations. 2+2 = 4 is useful. 4 = 4 is self-evident. (Both are illusory.)

Exactly- and that is part of what's causing the confusion.
We are accustomed to using normal numbers.

Only folks that have proceeded to higher math get into these concepts.
I know Bogie has proceeded into higher math too.

So what's going to clarify this one is (Strangely) a step backwards- to cover the known- then steps forward to cover how math works and what causes our habitual acclimation to numbers to jump up and confuse us occasionally.

I have no idea at this point if I'm wording this post as I intend for it to read...:doh:

Moose
2008-May-02, 09:57 AM
Tell me you're not suggesting a full deconstruction of math down to its first principles and reinvent the wheel by proving the entire works just to establish why 0.999~ = 1, are you? I can think of a few things I'd rather be doing with my time. Dental work, for example.

Frog march
2008-May-02, 10:10 AM
Neverfly,

if 1 and 0.999~ are different, then there must be a number between them, in fact there would have to be an infinite number of numbers between them, can you give one of those number between them?

I suppose that you would just say that 0.999~ is just a construct. If so does it not represent any number?

if 0.999~ does represent a number/value then even if you have to give an answer to my first question as a construct, it should be possible.
Yes, there is a number between them. That's the problem- that number is undefined.

do you still agree with this, previous, reply?

Neverfly
2008-May-02, 10:11 AM
Tell me you're not suggesting a full deconstruction of math down to its first principles and reinvent the wheel by proving the entire works just to establish why 0.999~ = 1, are you? I can think of a few things I'd rather be doing with my time. Dental work, for example.

Oh brother...
No.
And it wouldn't matter if it did would it?
Since those posts wouldn't be directed at you- they would be directed at those who disagree that .9999~ =1.

Or you know what..
Nevermind.

I'm outta this thread- since you want to give me a hard time about how I'm relaying myself considering that all of you have utterly failed repeatedly and your method isn't working but you are going to stand by your guns stubbornly and claim it's your way or the highway- Have Fun.

Moose
2008-May-02, 10:21 AM
Nev, I think a lot of the confusion is because your grammar hasn't been very clear in this thread. This isn't a spelling/grammar flame, but it's been very hard to figure out what it is you're trying to say. It's why we keep asking you to confirm what we think might be your underlying point.

Neverfly
2008-May-02, 10:27 AM
Nev, I think a lot of the confusion is because your grammar hasn't been very clear in this thread. This isn't a spelling/grammar flame, but it's been very hard to figure out what it is you're trying to say. It's why we keep asking you to confirm what we think might be your underlying point.
No, the grammar has been fine.

I've watched my words distorted into totally different meanings and it had nothing to do with grammar, Moose.

It has to do with some of you thinking, in your own minds, that I am suddenly promoting that .9999~ does not equal 1.
That is why.

The evidence- suggests that those explaining why .9999~ = 1 must not have been very clear either.

The specific posts I have been making recently are addressed to those that look at the two numbers and say they must not be equal. It is not addressed to those that see they are equal.

You seem to be having as hard a time understanding that as some folks may seem to be having a hard time seeing they are equal.

worzel
2008-May-02, 11:05 AM
Worzel was NOT correct
Exactly where was I not correct? The only possibility I can think of is in understanding what you're going on about, and that's down to you.

You've recently accepted that you were merely talking about 0.999~ and 1 being different syntactically (as I suspected you were all along). Quite frankly that is obvious, uninteresting and as shallow to discuss as 2+2 not being syntactically the same as 4.

If that's all you've been saying all along then you've been wasting everyone's time. If you have anything other than that to justify why you're still trying to tell Bogie that 0.999... does not equal 1 then you need to be a clearer and less hostile in stating it

It has to do with some of you thinking, in your own minds, that I am suddenly promoting that .9999~ does not equal 1.
That is why.
That might have something to do with the fact that you keep saying it after agreeing that they are equal :rolleyes:

hhEb09'1
2008-May-02, 11:08 AM
But you have not proven that .999…~ = 1, your subject has simply agreed to accept infinite decimal extensions along the way without proof.

So what are we going to do? Ask them what .999...~ divided by 3 is? :)

wb, btw

Neverfly
2008-May-02, 11:24 AM
If that's all you've been saying all along then you've been wasting everyone's time.
Whatever- and this isn't hostile?
It is NOT directed at You was it? It was directed at those saying the two are unequal.

So you don't have to read it- and you can just get off my back.
Frankly, by you jumping into the middle of what I was saying and clearly misunderstanding it - You were the one wasting my time. I've spent God only knows how many posts having to argue with you- when we agree that .999~ = 1, only because you are stubbornly refusing to admit that you misunderstood that ---
I tried a Different Direction than your failed direction in explaining it.

Frankly Worzel, most of your posts in this thread recently have come across as a bit hostile. You have little room to talk about Hostile. I'm just not a faker- I call it like it is.

If you have anything other than that to justify why you're still trying to tell Bogie that 0.999... does not equal 1 then you need to be a clearer and less hostile in stating it
Several of you could be much less hostile in stating that they are equal too.
I, for one, got tired of seeing Pompous posts from people in this thread rolling their eyes about how those [Insert whatever word you use to politely call someone an idiot here]s don't understand- and how this reflects badly on the board if so many [whatever word you chose here again]s don't get it.

I have also been clear- but as in psychology- you only notice the things in my posts that catch your eye- and you ignore the rest of what I said.
I'm willing to bet that when you were typing up your "rebuttal" to my post, you felt a bit confused about what side of the fence I was on.
This led you to default that I must be on the "unequal" side, so you responded as such even when your response made no sense at all and you ended up distorting what I said.

That might have something to do with the fact that you keep saying it after agreeing that they are equal :rolleyes:
Which should suggest to you that you are probably misunderstanding it considering that I know they are equal.

If some of you can be so quick to accept that others are missing something - then you should be just as willing to accept you may be missing something here too- Your responses are just as confusing half the time.
The other half the time, your posts are stuck on stubborn.

In the recent posts, Only Moose- has been patient in explaining and been polite. TheHalcyonYear, Kaptain K and worzel have been aggressive, sarcastic at times, hostile and it presents the appearance of being demeaning whether you intended to or not.

Your explanations don't seem to be working. I tried a Different approach- and you jumped on my back. I'd appreciate it if you got OFF my back considering that your sharp, unwarranted and unnecessary reaction has served only to spark an argument and totally distract from the topic.

If you're seeing hostility in my posts, I'm reacting to a build up of hostility that I have seen going on for a while in this thread.

And if you don't like my attitude- TOUGH!
I don't like your attitude either!

Neverfly
2008-May-02, 11:27 AM
Ask a teacher if the same method works for every student.

geonuc
2008-May-02, 11:31 AM
I have to agree with Worzel on one point: if any of you that voted 'no' did so because of symbology (i.e., you would have also voted no with respect to 2+2=4), then you've wasted our time.

To have any substance, the question clearly must be read to ask if the values represented by the two terms are equal.

This isn't directed at Neverfly, many people have been arguing in this thread. Although I'd point out to Nev - the poll shows you voted no. Would you change your vote if you could?

Neverfly
2008-May-02, 11:33 AM
This isn't directed at Neverfly, many people have been arguing in this thread. Although I'd point out to Nev - the poll shows you voted no. Would you change your vote if you could?

Yes. I'm sure a great many of other folks would too.

The point is that I ALSO have seen where exactly I got off the path. Because of that- My recent posts have been directed at recognizing Where exactly that happened- So That I can Address that point specifically to others that must have done the same. I know that Bogie for example is in the exact same position I was. That is why he was implicated by name. Or singled out I should say.

Except that I'm getting pounced in the process and cannot effectively do that at this point.

Frog march
2008-May-02, 11:46 AM
I thought bogie's position was that in maths, it should do what it says on the tin(an ad slogan for some product in the UK), and that 0.999~ just didn't do that...

Bogie
2008-May-02, 12:04 PM
The proof works because the student already tacitly accepts that infinite decimals can express rational numbers.A student accepting tacitly that infinite decimals can express rational numbers in no way changes that there are definitions and rules that must be used and followed to establish that.

Virtually every proof you'll ever see does not start from the axioms of the formal system and progress from there using only the primitive logical rules of inference. They typically rely on previous well known theorems and derived rules of inference. That's all that's going on with using 1/3=0.333... No proof can be said to have been checked until a few hard core mathematicians have made sure that the proof can be broken down to the basics, but virtually no one reading the proof would ever do that.This is not a revelation. Give me credit for having progressed through some schooling though it formally ended when I got my finance degree in '67, yikes. They still taught these things back then you know.

It's no more a trick than using a proof that ¬A is false to prove A without including a proof that (¬(¬A))=>A using only the axioms

(p => (q => p))
((p => (q => r)) => ((p => q) => (p => r)))
(( ¬p => ¬q) => (q => p))

and the inference rule

p,(p=>q) |= p

Using your logic you could reject any proof by contradiction as being "merely by definition" if it didn't include this proof (or similar) and boldly state that "in reality [ whatever that's supposed to mean ] it still isn't true."I am guilty of generalizing a little bit but early on I said that you can't prove an infinity without a rule or definition. You may recall the Moose replied, "It's not a rule, it just is. And suggested that even a grade schooler knew more than I do. Suggested I might be trolling by not admitting that .999...~ = 1 did not require a definition or rule.

The real problem here is that people like you just don't get that maths is just a formal system. You must be able to see that the real problem in a thread like this is people like you. You have some kind of a complex that gives you a feeling of superiority by suggesting that "people like me" are somehow deficient mentally. People like you don't actually follow what is being said but read in angles that will allow you to express your greater wisdom. We don't need any one instigating dispute just to try to impress people. So people like me see right through your thinly veiled conceit and elitism.
Everything[/i] that is provable is "true" by definition. Whether it is true in reality or not has absolutely no meaning unless you first state what you think the formal system is modelling: maths does not model numbers; numbers are the formal system.I knew you would come to your senses.

To say "oh yeah well that's just true by definition" as if that some how diminishes the truth of the statement is to completely and utterly miss the whole point.You are repeating yourself. I explained that it was true by rule, by definition and asked twice if it was based on a rule hoping someone would acknowledge. That question was ignored by two posters and answered by Moose as "No, it's not a rule". Can you see how the rest of the series of posts could develop from those circumstances? To belittle me for exchanges that took place is just conceit and failure to actually get a sense of the context.

It reminds me this:

Facts are meaningless. You could use facts to prove anything that's even remotely true! - Homer SimpsonYou are the one that misses the point.
"...when two opposite points of view are expressed with equal intensity, the truth does not necessarily lie exactly halfway between them. It is possible for one side to be simply wrong."

Richard Dawkins

If the opposite view points were the maths, definitions, rules, and paths to proofs and truth, there is probably a middle ground between us, but when the opposite view points are forum posting, understanding the context, and having some level of integrity you are simple wrong to go after me for the reasons that you proclaim. Not that there aren't reasons, but the ones you use are not valid.

Moose
2008-May-02, 12:15 PM
No, the grammar has been fine.

No, Nev, it really hasn't. You say I'm being stubborn and disagreeing with you, when the fact of the matter is: I can't agree or disagree with what you're saying because I'm having a great deal of trouble understanding what you're trying to convey. I can't make a judgment one way or another. When you're not running frustrated, your language is a lot clearer.

Let me give you just one example of what I mean:

And 0.99999> = 1

0.99999999> is not 1

This is pretty much why I had to reiterate what I thought you were saying in the very next post. Those looked, at the time, like direct contradictions caused by what I'd assumed was a typo, although there was enough context in the rest of the post to piece together an approximation of your meaning.

It took two full pages later (and several confused posts) for me to realize what you may have really been trying to say: that the representations take on different forms, but they remain mathematically equal. And even now, I'm not completely sure that's what you've been saying. But for now, until you clarify, that's all I can really go with to form a response:

I've been a part of this thread (and the other two) from the very first pages (and I can't tell you for certain I didn't jump the gun and vote 'no' before I came to understand the math, I sincerely don't remember), and I think there are two common misconceptions that have been leading to 'no' votes.

Common error #1, that an infinite progression must end at some point, which lead people to conclude that 0.999~ approximates 1, or is equal to 1 because of rounding the 'final' 9.

Common error #2, that an infinite progression is a partially-completed process and thus an approximation, rather than a fully functional, fully accurate, fully specified end result of that completed process, just like any other number in the rational set.

Geonuc brought up a third possibility that I hadn't really considered, but I never got the sense that too many people are making that one, even if the attempts to justify the inequality often drift onto those shoals.

It's a bit, I think, like getting oneself painted into a corner with the consequence that if 0.999~ <> 1, you also end up with the consequence that 1/9 + 8/9 <> 9/9. Neither one is an error of arithmetic (unless you somehow end up agreeing with the inequality beyond that point!) as much as they are simply the rock that punches a hole through the self-consistency keel.

Neverfly
2008-May-02, 12:23 PM
No, Nev, it really hasn't. You say I'm being stubborn and disagreeing with you, when the fact of the matter is: I can't agree or disagree with what you're saying because I'm having a great deal of trouble understanding what you're trying to convey. I can't make a judgment one way or another. When you're not running frustrated, your language is a lot clearer.

Let me give you just one example of what I mean:
That example has nothing to do with grammar.

It has to do with you thinking that I was saying a contradiction.

It took two full pages later (and several confused posts) for me to realize what you may have really been trying to say: that the representations take on different forms, but they remain mathematically equal. And even now, I'm not completely sure that's what you've been saying. But for now, until you clarify, that's all I can really go with to form a response:
How many times have I already clarified it? I've lost count.

Common error #2, that an infinite progression is a partially-completed process and thus an approximation, rather than a fully functional, fully accurate, fully specified end result of that completed process, just like any other number in the rational set.
Yes. That is the error I am trying to address now for others.

YOU who are on the equal side may have been confused by my post.
But they were not directed at YOU at this point.
I don't know how many times I must repeat that before it sinks in...
Those on the Unequal side will know what I'm talking about- at which point I can snag them and explain it. But having been in the shoes- I know somewhat what the view looks like from there.

Does THAT make sense?

I've explained THIS issue Four times now. If it still doesn't make sense than that fault is not mine- nor my grammar or anything else and I will just hold up a mirror.

In the meantime- I gotta get to work so won't be replying for a while.

Frog march
2008-May-02, 12:25 PM
if the software can remember that you voted, perhaps it should allow you to change your vote at a later stage.

Bogie
2008-May-02, 12:31 PM
Lol, I'm going to unsubscribe again.

One point being made is that it is lame for people like me to fall back on the definitions and rules which ignores the whole concept of math and proofs going back hundreds, even thousands of years. They suggest that such a person would question 2 + 2 = 4. Hmm, I don't think so.

The definition in limit theory is unique, i.e. the infinite series .999 is defined to equal 1 because the infinite series assumes that the limit is reached, i.e. it is reached by definition. I'll agree that 2 + 2 = 4 without calling up any rules or definitions.

Moose
2008-May-02, 12:38 PM
I am guilty of generalizing a little bit but early on I said that you can't prove an infinity without a rule or definition. You may recall the Moose replied, "It's not a rule, it just is. And suggested that even a grade schooler knew more than I do. Suggested I might be trolling by not admitting that .999...~ = 1 did not require a definition or rule.

Hold on. You're putting words and motivations in my mouth. Again.

It's not a rule. To call it a 'rule' implies arbitrariness, and that's a major oversimplification.

'1' is a definition of a value. '+' is a definition of an operator. The definition of a limit, that it's the consequence of an infinite process, is the rule, and not any particular application of that rule.

'1' is a definition. '+' is a definition. 1+1=2 is a consequence of those definitions. 0.999~ = 1 is also a consequence of those definitions, and limit theory, like any other theory, explains why this is true.

But like any theory, definition, axiom, or assumption in math, you have to come back and prove that they hold throughout the entire number space, that they work as advertised in all cases. That it's self-consistent.

In the case of 0.999~ = 1, you need no more than 6th grade arithmetic to successfully prove limit theory is self-consistent. It's completely above board where you can see it. It's why this proof tends to be far more compelling that some of the others.

It has nothing at all to do with "kids knowing more than you", something I do not, and have never believed. I've detailed in my immediately preceding post the mistake you're making (common error #2), and earlier, why your argument wrecks on basic fractions.

As for trolling, you seemed to be behaving like one, so I asked. (And it was a question.)

Now, as a last clarification as for what I mean (above) by arbitrariness being an oversimplification: consider the case of a new numbering system, where I reject commutativity and define 1+2 = 3 while 2+1 = 4. The numbering system fails (rather immediately) because self-consistency fails.

4 - 1 '=' 3
2+1 - 1 '=' 1 + 2
2+1 '=' 1 + 1 + 2
2+1 '=' 2 + 2
1 '=' 2 + 2 - 2
1 '=' 2, which clearly fails, because the hypothetical numbering system isn't self-consistent.

Now, that (commutativity and associativity) is 3rd grade arithmetic. The failure of this numbering system to function says nothing about my ability to do arithmetic, but everything about how well that numbering system holds.

Heck even the earliest numbering systems (positive, non-zero integers) were provably self-consistent as far as they went.

worzel
2008-May-02, 12:39 PM
Bogie, that quote of Dawkins couldn't be more apt. Never mind death and taxes, one thing you can be sure of is mathematical identities. If you understand that the equality is a necessary consequence of the mathematical system then that's all there is to the question.

Nev, whoever you choose to address, this is a public forum. If you make incorrect statements in an attempt to explain something to someone then you're gonna get called on it.

Bogie
2008-May-02, 12:44 PM
Hold on. You're putting words and motivations in my mouth. Again.

It's not a rule. To call it a 'rule' implies arbitrariness, and that's a major oversimplification.

'1' is a definition of a value. '+' is a definition of an operator. The definition of a limit, that it's the consequence of an infinite process, is the rule, and not any particular application of that rule.

'1' is a definition. '+' is a definition. 1+1=2 is a consequence of those definitions. 0.999~ = 1 is also a consequence of those definitions, and limit theory, like any other theory, explains why this is true.

But like any theory, definition, axiom, or assumption in math, you have to come back and prove that they hold throughout the entire number space, that they work as advertised in all cases. That it's self-consistent.

In the case of 0.999~ = 1, you need no more than 6th grade arithmetic to successfully prove limit theory is self-consistent. It's completely above board where you can see it. It's why this proof tends to be far more compelling that some of the others.

It has nothing at all to do with "kids knowing more than you", something I do not, and have never believed. I've detailed in my immediately preceding post the mistake you're making (common error #2), and earlier, why your argument wrecks on basic fractions.

As for trolling, you seemed to be behaving like one, so I asked. (And it was a question.)

Now, as a last clarification as for what I mean (above) by arbitrariness being an oversimplification: consider the case of a new numbering system, where I reject commutativity and define 1+2 = 3 while 2+1 = 4. The numbering system fails (rather immediately) because self-consistency fails.

4 - 1 '=' 3
2+1 - 1 '=' 1 + 2
2+1 '=' 1 + 1 + 2
2+1 '=' 2 + 2
1 '=' 2 + 2 - 2
1 '=' 2, which clearly fails, because the hypothetical numbering system isn't self-consistent.

Now, that (commutativity and associativity) is 3rd grade arithmetic. The failure of this numbering system to function says nothing about my ability to do arithmetic, but everything about how well that numbering system holds.

Heck even the earliest numbering systems (positive, non-zero integers) were provably self-consistent as far as they went.You seem to be agreeing with this post. (http://www.bautforum.com/off-topic-babbling/14593-do-you-think-0-9999999-1-infinite-9s-66.html#post1232357)

worzel
2008-May-02, 12:46 PM
I'll agree that 2 + 2 = 4 without calling up any rules or definitions.
Really? In some formal systems 2 + 2 = 22. You really need to consult the rules of the formal system to be able to figure the answer out. In fact, the question of what is 2 + 2 is utterly meaningless without specifying from which formal system the equation is constructed.

But of course, we all automatically assume that we're talking about maths when we see such constructs. That is why you can say 2 + 2 = 4 and not even realize that you're tacitly calling up the rules and definitions of maths (the very same ones which give rise to 0.999~ equaling 1).

hhEb09'1
2008-May-02, 12:48 PM
Bogie, what is .999...~ divided by 3, in your opinion?

Moose
2008-May-02, 12:57 PM
That example has nothing to do with grammar.

It has to do with you thinking that I was saying a contradiction.

It has everything to do with grammar. It's because your point was not expressed in a sufficiently clear and organized way. Like math, the tighter your grammar is, the clearer your intentions will be.

How many times have I already clarified it? I've lost count.

Step back a moment and think objectively about why you've had to do that. Why so many people appear to be unable to interpret your recent posts correctly. It really isn't some sort of conspiracy, Nev. I'd much rather speak directly to what you mean than waste a lot of time guessing.

Those on the Unequal side will know what I'm talking about- at which point I can snag them and explain it. But having been in the shoes- I know somewhat what the view looks like from there.

Does THAT make sense?

Yes, it does. But try to remember, so was I.

Bogie
2008-May-02, 01:02 PM
http://www.bautforum.com/off-topic-babbling/14593-do-you-think-0-9999999-1-infinite-9s-60.html#post1229650
That is still good advice Hard Knocks and I’m going to take it again.

#post1229650

Moose
2008-May-02, 01:14 PM
You seem to be agreeing with this post. (http://www.bautforum.com/off-topic-babbling/14593-do-you-think-0-9999999-1-infinite-9s-66.html#post1232357)

On the surface, yes (and we always have). Where we're disagreeing is on how significant self-consistency is.

I'll agree that 2 + 2 = 4 without calling up any rules or definitions.

2+2 = 10. Here's why:

Worzel's reply is right on the money, but even more importantly, you can't claim that 2 + 2 = 4 without defining the number space and the operator set. (Basically, what 2, 4, +, and = mean, then proving they're self-consistent with the rest of your numbering system.)

2 + 2 = 10, after all. In base-4.

The whole point of (formal) mathematics is to develop a numbering system such that the accepted standard notations, theorems and operators have all been proven and justified prior to your having used them.

2+2=4 and 0.999~ = 1 rest on the very same foundation: provable self-consistency within the set of 'real' numbers. They're equalities with exactly the same validity, the same precision, and the same rigor. You reject one (proven) equality, you necessarily and inescapably reject every equality.

Once these proofs are in place, then you can use the result, within reason, to approximate within the tolerances of the task you're trying to accomplish. 1+1 = 3 for sufficiently wide tolerances of 1 (or sufficiently wide tolerances of 3.)

Frog march
2008-May-02, 01:18 PM
http://www.bautforum.com/off-topic-babbling/14593-do-you-think-0-9999999-1-infinite-9s-60.html#post1229647
That is still good advice Hard Knocks and I’m going to take it again.

You can link directly to someone's post, by clicking on 'permalink' at the top right of someone's post, and then copy and paste the address from the URL bar.

hhEb09'1
2008-May-02, 02:13 PM
You can link directly to someone's post, by clicking on 'permalink' at the top right of someone's post, and then copy and paste the address from the URL bar.In my browser, that appears to be what he did, more or less :)

Frog march
2008-May-02, 02:33 PM
well, it just dumps me on page 60, and there was no one called Hard Knocks to be seen.

Moose
2008-May-02, 02:37 PM
The user list doesn't list anybody by that name. Bogie, were you responding to someone on another board?

JMV
2008-May-02, 03:05 PM
Have you guys changed the Number of Posts to Show Per Page setting in your User Control Panel to something other than Use Forum Default? I believe that messes up the permalink function.

This (http://www.bautforum.com/1229647-post1782.html) is the post he was referring to.

Moose
2008-May-02, 03:13 PM
Ah. There it is. I wonder how come I couldn't find Hard Knocks's profile through the alphabetical list or User Search facility?

Thanks JMV.

hhEb09'1
2008-May-02, 03:27 PM
Ah. There it is. I wonder how come I couldn't find Hard Knocks's profile through the alphabetical list or User Search facility?

Thanks JMV.I believe Fraser has limited that search to users with more than a certain number (five?) posts

PS: I could be wrong. I came up unsuccessful in finding any sort of documentation for that, just my (faulty) memory.

tdvance
2008-May-02, 05:21 PM

This has more relevance to the thread than appears on the surface--the permalink bug seems to conflate a "view" of data with the data itself--so if users change their view options, it fouls up the effect of clicking a permalink.

Likewise, many seem to conflate, say, computer representations of numbers with actual numbers (which, I'm sorry for sounding so mystical but I can't find a decent explanation better than this, are just "out there in the platonic ideal" somehow). The fact is, decimal representation is nothing more than a "view" of something that is just there. 1 and 0.999... are both representations of the same number--as I mentioned in a previous post, there is a precise definition of "decimal representation of a number", and from this definition, 0.999... and 1 have to represent the same number. IT's not as simple is "0.99... is defined to be 1, 1.99... is defined to be 2", i.e. an infinite sequence of definitions, but a single definition that provides for every case--finite decimals, infinite decimals.

You can't prove .9999... is 1 by computing .9, .99, .999--since mathematicians tend to expect finite-length proofs. But, we do an "end run" around that with Weierstrass's definition of the limit. We can thus work with infinite things using a finite number of symbols. The rules are funny--some things defy "common sense" based on experience with the finite, like for example, if you insist that 0 divided by 0 is a number (and yes, there's an infinity hidden in that), you have to give up the principle that a divided by b, times b, gets a back. Otherwise, it's easy to, for example, prove 1 = 2 that way. But, using the accepted definitions, my prior post gives a rather long explanation of why .999... has to be the same number as 1.

It's incorrect to argue that, because .9<1, .99<1, .999<1, and so on for all finite numbers of 9s, (that part is correct), that you can then conclude that .999... with an infinite number of 9s must also be less than 1. You don't jump from finite to infinity that way, but using Weierstrass's method.

(history--Newton and Leibnitz invented calculus independently, using largely intuition and manipulating infintessimals--quite correctly, that worried mathematicians because the infinitely large and the infinitely small were treated casually the way one would numbers like 1, 2, and 3, yet the methods somehow gave the right answers, leading mathematicians to think there was something deeper going on (within that mystical platonic ideal world I mentioned) that they didn't quite understand yet. It took Weierstrass to come up with a definition of the limit, and from that, precise definitions of the derivative, infinite summation, integrals, etc., thus finally putting calculus on a solid theoretical foundation as opposed to being a bunch of rules that just happened to correctly tell you where the planets were--it's because it works so well that Weierstrass's definitions are the accepted ones--they are not arbitrary as best as we can tell!).

01101001
2008-May-02, 05:50 PM
I believe Fraser has limited that search to users with more than a certain number (five?) posts.

Same recollection (modulo exact cutoff number).

More expedient is using Advanced Search on the member name, to obtain the article(s). Profile is Hard Knocks (http://www.bautforum.com/members/hard-knocks.html).

Edit: Fraser proposed post-count cutoff for member list in topic: Member List (http://www.bautforum.com/about-baut/69937-member-list.html).

kucharek
2008-May-02, 05:53 PM
We're working on a petition that call for you to stand in front of a pie (Pi) throwing firing squad, that is, 3 adults and a toddler (.14 of an adult), each with 0.999~ pies of the kind of your choice.

Would you like to sign our petition and support a 'worthy' cause? ;)

Naaaaa... I've attended bigger cake fights...
http://farm3.static.flickr.com/2178/2459925102_f2740d579a_m.jpg (http://www.flickr.com/photos/hk1963/2459925102/)
...and survived with only minor damages
http://farm4.static.flickr.com/3272/2459915364_ef81833d26_m.jpg (http://www.flickr.com/photos/hk1963/2459915364/)

I imagine this is 0.9999~=1 faction vs the 0.9999~!=1 faction:

:)

Moose
2008-May-02, 06:10 PM
I believe Fraser has limited that search to users with more than a certain number (five?) posts

Yeah, I seem to remember that discussion now. I hadn't realized he'd done it. Thanks.

hhEb09'1
2008-May-02, 06:56 PM
Same recollection (modulo exact cutoff number).

More expedient is using Advanced Search on the member name, to obtain the article(s). Profile is Hard Knocks (http://www.bautforum.com/members/hard-knocks.html).

Edit: Fraser proposed post-count cutoff for member list in topic: Member List (http://www.bautforum.com/about-baut/69937-member-list.html).Ah, thanks. Whew, senescence deferred

What was the OP again?

Donnie B.
2008-May-02, 07:51 PM
I imagine this is 0.9999~=1 faction vs the 0.9999~!=1 faction:

:)
Naaahh, no contest. The 'pro' side is armed with infinite pies (Pis?), while the 'con' side has just a smidgen fewer. ;)

It's incorrect to argue that, because .9<1, .99<1, .999<1, and so on for all finite numbers of 9s, (that part is correct), that you can then conclude that .999... with an infinite number of 9s must also be less than 1. You don't jump from finite to infinity that way...
This, I think, is the crux of the entire matter.

Red Dawn
2008-May-02, 08:29 PM
Naaahh, no contest. The 'pro' side is armed with infinite pies (Pis?), while the 'con' side has just a smidgen fewer. ;)

I wonder what the value of:

0.9+0.11-0.011+0.0011-0.00011+0.000011-0.0000011+...

is in con-land? Is it greater than one or less than one?

Bogie
2008-May-02, 09:48 PM
I wonder what the value of:

0.9+0.11-0.011+0.0011-0.00011+0.000011-0.0000011+...

is in con-land? Is it greater than one or less than one?When I unsubscribed my e-mail fell to near zero relative to the last few days.

So to continue my participation I wonder if it would be acceptable for me to lay claim to being the spokes person for the cons. In such a role I have only one request and that is that I am accepted as a member of the combined set of pros and cons without distinction and that efforts to get me to admit I am wrong or to convince me that I am wrong be consider in ill taste and shunned by the all pros and cons.

Let it be known and accepted that I am con on the basis that I accept the equality in the OP only if it is acknowledge that in limit theory there are rules that define the infinite set .999… as equal to one. I consider my position on that closed so that I can faithfully fulfill my con leadership role if that role is approved. I will serve if and for as long as there is at least one yes vote or a net positive yes vote should there be more that one vote. If there are zero votes I will serve at will.

Now quickly the issue will come up that I am the only con in an attempt to belittle my leadership role. It will be claimed that all others cons would change their vote in the poll from no to yes if they could.

That may be true but I prefer to believe that we are a minority group that deserves respect. Dissing us should be socially unacceptable and those guilty of minority slurs against us should be chastised by the entire group.

In reality I doubt that there are any cons left but me and if so none brave enough to acknowledge themselves and so my leadership position will likely go unchallenged for as long as it takes to remove a circle, one square at a time.

Is it acceptable to the pros and cons who are subscribed that I take on this role as I have described it?

If it is acceptable and there are no blackballs (any one with 20 or more posts in the thread can blackball this idea) my first duty will be to respond to Red Dawn.

Bogie
2008-May-02, 09:52 PM
If it is acceptable and there are no blackballs (any one with 20 or more posts in the thread can blackball this idea) my first duty will be to respond to Red Dawn.Since there are no objections yet, my first duty as Con Hero is to answer Red Dawn from the perspective of the Cons.

Let it be known that the cons see 0.9+0.11-0.011+0.0011-0.00011+0.000011-0.0000011+... as an algorithm for the slowest computer ever conceived.

Here is how it works:

Everything that enters into this champion slow computer is assigned a step in the 0.9+0.11-0.011+0.0011-0.00011+0.000011-0.0000011+... sequence. Every 0 and 1 is replaced by a step in the sequence, no step is used twice and the entire sequence up to that step becomes part of that 0 or 1 throughout the processing and in memory. Any new 1 processed by the computer is assigned the next positive step in the sequence and any new 0 processed is assigned the next negative step in the sequence. Very quickly the slowness of the processing will become legendary.

worzel
2008-May-02, 11:51 PM
Bogie, this is a public forum. We each speak for ourselves, not for each other. I'm sure there are lurking doubters who are following your conversation. You and they don't need anyone's approval to carry on, any more than I need yours to reply to you.

Let it be known and accepted that I am con on the basis that I accept the equality in the OP only if it is acknowledge that in limit theory there are rules that define the infinite set .999… as equal to one.
0.999... is not an infinite set, it is an infinite sequence that converges to 1. I.e. an infinite sum. It does so as a consequence of limit theory, not by definition except vacuously in that, in a sense, every theorem of a formal system is true by definition, as has been put to you many times now.

Also, the sequence must converge to 1 for infinite decimals to be mathematically consistent, not just because someone decided on a whim to do it that way, as has been put to you many times now.

I consider my position on that closed so that I can faithfully fulfill my con leadership role if that role is approved.
If you accept the equality on the basis of limit theory (which must hold for infinite decimals to be mathematically consistent) then please state on what basis you don't accept it. I just don't get what point you're trying to make if you agree that mathematically the equality holds. Do you think 0.9999~ has some platonic existence independent of the formal system from which it is constructed?

Also, could you just answer hhEb09'1's question, within whatever context you think the equality of the OP doesn't hold.

Bogie, what is .999...~ divided by 3, in your opinion?

I think you'll find his questions exceedingly illuminating if you'll humour him and just answer his questions as honestly and directly as you can.

Bogie
2008-May-03, 12:00 AM
I consider that a blackball. It was just a thought. Humor is often lost on the Internet.

worzel
2008-May-03, 12:15 AM
Get over yourself and make your point already. In what way do you disagree with the equality if you accept it mathematically?

Red Dawn
2008-May-03, 12:26 AM
Since there are no objections yet, my first duty as Con Hero is to answer Red Dawn from the perspective of the Cons.

You say answering me is your first duty, but then you didn't answer my question. Is it greater than one or less than one?

Let it be known that the cons see 0.9+0.11-0.011+0.0011-0.00011+0.000011-0.0000011+... as an algorithm for the slowest computer ever conceived.

It is not an algorithm, it is a mathematical expression. There exist algorithms to calculate it. Some algorithms come up with values closer and closer to the answer without ever reaching it; others terminate after a few steps.

hhEb09'1
2008-May-03, 12:35 AM
Also, could you just answer hhEb09'1's question, within whatever context you think the equality of the OP doesn't hold.He already answered (http://www.bautforum.com/off-topic-babbling/14593-do-you-think-0-9999999-1-infinite-9s-post1229233.html#post1229233) the question "What is 0.444... divided by 2?", and his answer was 0.222...

I know what my answer to "What is 0.999... divided by 3?" would be, but right now I am not sure what his would be.

TheHalcyonYear
2008-May-03, 02:28 AM
No I didn't.

Worzel was NOT correct, TheHalcyonYear was not- and now you are not too.

Actually you are right. Calculus is simply one big hoax that has been used to pull the wool over common folks for hundreds of years!!

Yup, you found us all out Neverfly. All the calculus books are in on it!! :)

Neverfly
2008-May-03, 02:34 AM
Actually you are right. Calculus is simply one big hoax that has been used to pull the wool over common folks for hundreds of years!!

Yup, you found us all out Neverfly. All the calculus books are in on it!! :)

This sarcastic post has absolutely nothing to do with what I actually said or what I was referring to.

Next time, actually read the entire post in question prior to putting your foot in your mouth...http://us.i1.yimg.com/us.yimg.com/i/mesg/emoticons7/23.gif

TheHalcyonYear
2008-May-03, 05:42 AM
This sarcastic post has absolutely nothing to do with what I actually said or what I was referring to.

Next time, actually read the entire post in question prior to putting your foot in your mouth...http://us.i1.yimg.com/us.yimg.com/i/mesg/emoticons7/23.gif
I never read your posts... waste of time... :rolleyes:

Neverfly
2008-May-03, 06:20 AM
I never read your posts... waste of time... :rolleyes:

Then do not reply to them.

You are fully aware by now, I am sure, that I'm not going to sit here and pander nicely to you.

Bogie
2008-May-03, 12:20 PM
You say answering me is your first duty, but then you didn't answer my question. Is it greater than one or less than one?Are those the only two choices? If there was a third choice that you don't know about that becomes available in Limit Theory by the way they define an infinite progression that approaches a limit as being equal to the limit would you be surprised?

Assuming you have read my posts you should know the answer is defined in Limit Theory and I shouldn't have to answer the same questions cloaked in various ways to get that message across. Who is really missing the point?

It is not an algorithm, it is a mathematical expression. There exist algorithms to calculate it. Some algorithms come up with values closer and closer to the answer without ever reaching it; others terminate after a few steps.This was humor; I feel it was obvious humor. As an iteration, each step switches between plus and minus. Parsing out each positive step to the computer 1's and each negative step to the 2's was the algorithm. It seemed like a perfect "Con" response. It was humor :).

I was blackballed by a 20 plus poster and so I am no longer the Con spokesperson. It would have been fun but then this is not about letting the former con leader have fun now is it. (Alert: humor coming)Its about the integrity of math in the minds of math Nazi's (humor :)). That's humor. Let it go ...~.

Disinfo Agent
2008-May-03, 01:04 PM
Is it greater than one or less than one?Are those the only two choices? If there was a third choice that you don't know about that becomes available in Limit Theory by the way they define an infinite progression that approaches a limit as being equal to the limit would you be surprised?The third and last possibility is that they are equal.

Bogie
2008-May-03, 01:13 PM
The third and last possibility is that they are equal.Yes sir. And if you want to say that there is only really one possibility I am OK with that :) if you are willing to acknowledge that it is always an equality by definition.

Disinfo Agent
2008-May-03, 01:19 PM
I don't entirely agree that it's by definition. That's not quite how we use that term in mathematics, IMHO. But if describing it as "by definition" makes it easier for you to accept the equality, I don't mind. The point is that in mathematics 0.999~ and 1 are exactly the same number.

Bogie
2008-May-03, 01:22 PM
I don't entirely agree that it's by definition. That's not quite how we use that term in mathematics, IMHO. But if describing it as "by definition" makes it easier for you to accept the equality, I don't mind. The point is that in mathematics 0.999~ and 1 are exactly the same number.(Humor of the Steve Martin type) You math people have a different meaning for everything :).

Frog march
2008-May-03, 01:28 PM
I think bogie means that if you define a car as being a carrot, then, by definition, it is a carrot, but otherwise they are completely different.

Disinfo Agent
2008-May-03, 01:29 PM
We do have slightly different, technical meanings for some common words. This happens in all the sciences; you could say it's an occupational hazard. However, I do not believe the expression "0.999~" is used at all outside mathematics, so in this particular case I would say we have precedence. We thought of it first. :)

hhEb09'1
2008-May-03, 02:54 PM
(Humor of the Steve Martin type) You math people have a different meaning for everything :).(Tim Allen) What is your meaning for 0.999... divided by 3? :)

Red Dawn
2008-May-03, 02:57 PM
Are those the only two choices? If there was a third choice that you don't know about that becomes available in Limit Theory by the way they define an infinite progression that approaches a limit as being equal to the limit would you be surprised?

Assuming you have read my posts you should know the answer is defined in Limit Theory and I shouldn't have to answer the same questions cloaked in various ways to get that message across. Who is really missing the point?

I know what I think. I was asking what you think. You said you would answer, but you didn't.

This was humor; I feel it was obvious humor. As an iteration, each step switches between plus and minus. Parsing out each positive step to the computer 1's and each negative step to the 2's was the algorithm. It seemed like a perfect "Con" response. It was humor :).

I was blackballed by a 20 plus poster and so I am no longer the Con spokesperson. It would have been fun but then this is not about letting the former con leader have fun now is it. (Alert: humor coming)Its about the integrity of math in the minds of math Nazi's (humor :)). That's humor. Let it go ...~.

I thought you'd probably dodge the question.

Bogie
2008-May-03, 05:28 PM
I know what I think. I was asking what you think. You said you would answer, but you didn't.

I thought you'd probably dodge the question.Are you saying that you thought I'd dodge the question and were surprised when I didn't. Or are you saying you couldn't find my answer in my posts?

Frog march
2008-May-03, 05:35 PM
there is a question that you are avoiding on this thread. Can't think what though.:whistle:

Bogie
2008-May-03, 05:37 PM
I think there is a question that you are avoiding on this thread. Can't think what though.:whistle:You mean the one about not knowing the difference between a carrot and a car?

Frog march
2008-May-03, 05:39 PM
did you think that was a question?