PDA

View Full Version : Do you think 0.9999999~ =1 , that is infinite 9s.

Pages : 1 2 3 [4] 5 6 7 8 9 10 11

Lycus
2005-Mar-12, 07:08 AM
Now let me ask you: what do you mean by "0.9...98
As I wrote just below that, I mean the number right below 0.9....
Okay, then why wouldn't 0.9...985 be between those two numbers?

Archer17
2005-Mar-12, 07:11 AM

(For the record, I voted yes a while back)
No. . . (I supposed that's different from "No...", which would be an "N" followed by infinite "o"'s. . . :D)That's OK, I rounded it off. :wink:
So, just to have a fun nitpick: copying all the text from all my posts yields 27,796 bytes worth of ASCII text. Since each character is represented by 1 byte, that's a max of under 28,000 letters. Take away a lot of those for spaces and linefeeds (and, because linefeeds are actually a carriage-return/lineeed combination in Notepad, they each take 2 bytes instead of 1 byte - greedy linefeeds), and you get something like half the original number (could be wrong - I did a test of this once and I think that's what I came up with on average). So, if the average word is 7 letters long, there are somewhere between 2000 and 4000 words. This leads me to conclude that you probably didn't actually read a million words into anything I wrote. What I posted was sarcasm and was borderline (if not outright) rude and I apologize. I've been known to post some "epics" myself.

For the record, your post was 1225 words (I didn't count your "quotes"). I bookmarked WordPeddler (http://www.wordpeddler.com/wordcounter.htm) - it's a free "approximate" word-counter, just copy/paste your text into the box.

Grey
2005-Mar-12, 07:36 AM
Before I begin: I am sometimes using the word "infinity" to refer to this concept of a never-ending number, even though that is not what it means. I am mainly doing this because it's a lot easier to just say infinity than try to describe this never-ending number, and it pretty much works the same way.
I think that when you think of infinity, you're thinking that it's a number, and that's where your intuition is failing you.

Third - as already discussed, infinity is not a number. Infinity is an entire group of numbers, defined by being immeasureably large; on this note, infinitesimal works the same way, except it is immeasureably small. So, 1xSomeInfinity is an infinite number and 2xTheSameInfinity is an infinite number, but the former is 50% of the latter. They aren't the same number, even though they are both in "the infinite set of numbers". If the assumption is invalid for me, it's invalid for you, also.
This is wrong. You say it's not a number, but then you start trying to treat it like one, even saying "they aren't the same number". Infinity isn't a number at all. Mathematically, the set of even integers and the set of all integers are equal in size, even though that seems counterintuitive.

So, the volume of the vacuum is infinity, which is 99.9...% of the infinite volume of the universe. Now, that puts the "does 99.9...%=100%?" into play, so--just to show SomeInfinity!=AnotherInfinity--let's do something a little different: let's say 10% of the volume of the universe is matter, and 90% is space. Now, the universe can go on forever, so there is infinite matter and infinite space, but there is still only 10% matter--not 100%.
And although in any finite subset of such a universe, the volume taken up by matter is 9 times less than that not taken up by matter, in the infinite universe, these volumes have the same cardinality. Again, you're trying to treat infinity like a number that can be multiplied and divided, and it's not a number.

"Countable" infinity is an oxymoron. An infinite number is any number so large that it cannot be counted. A never-ending infinite number is a really big uncountable number.
It's clear that you aren't familiar with the terminology used in this area of mathematics. "Countable" refers to a set with the same cardinality as the integers.

1. The paradox isn't really a mathematical paradox (like many other "mathematical" paradoxes). It can affect mathematics, but is really more of a philosophical paradox, because it includes concepts beyond the (relatively) small scope of mathematics. I don't know that anyone claimed otherwise, but I just thought I'd mention it. A set that includes all red bikes has gone beyond math--you can count the items in the set, but math has no idea what is meant by "red" or "bike".
Not really. The features that define the objects in the set may be non-mathematical, but the behavior of sets doesn't really depend on those criteria. Set theory is a subfield of mathematics.

The only way to do that is to say:
0.9...=1
and
&lt;something else>=&lt;the number before 1>

Now, it's my turn to ask: what comes right before 1, if it isn't 0.9... or something between 0.9... and 1? Although I have answered that below, and the answer is "some finite number", which destroys your argument.
Can you demonstrate that there has to a be a finite number that comes right before 1, with no numbers between them?

You seem to have the idea that there are not two ends to the number.
There are not. If the number ended at any point, there would be a finite number of digits, which is not the case. You're doing it again, imagining that infinity is just a really big number, even though you've claimed that you believe it is not.

However, that is demonstrably false: you can (in a theory which your argument requires to be true) divide 1 an infinite number of times. That is, between 1 and 0, there are an infinite number of numbers. So, you start at 0, count by infinitesimals for the correct eternity (in this case a "never-ending" eternity), and end at 1. There are an infinite number of numbers in there, but there are two very distinct ends. So, 0.9... has an infinite number of 9's, but there are still two ends to the number; the logic carries to any infinity.

Unless you can't count by infinitesimals, since each infinitesimal adds exactly 0, which never gets you anywhere. So you can only count with finite numbers, which means there are a finite numer of numbers between 0 and 1. Because of this, never-ending decimal numbers can't really exist, so 0.9...=nothing, not 1.
You almost got it. You're correct that you can't count from 0 to 1 by infinitesimals, but then you assume that you must be able to count by finite numbers, implicitly assuming that you must be able to count from 0 to 1 by something. But that's not actually the case. The quantity of real numbers in any finite range is actually of greater cardinality than the integers. It's uncountably infinite, as opposed to countably infinite, so it's not correct to assume that it can be counted at all.

Disinfo Agent
2005-Mar-12, 02:47 PM
Second - the only argument I can think of against using the word "matter" is more of an argument about using an "atom" as a unit. An atom is mostly empty space, so technically my numbers were incorrect. However, I'm pretty sure you got the assumption that the size of the atom was determined by the bounds of the atom, not the sum of the bounds of each of it's component parts.
I’m not sure I understand the difference, actually, although I suspect it’s not relevant to the overall conversation.

Measuring the universe in “atoms”, or “spaces” occupied by atoms is a bad idea because the same number of atoms can occupy different volumes, depending on its mass density.

Third - as already discussed, infinity is not a number. Infinity is an entire group of numbers, defined by being immeasureably large; […]
Are you saying that there is more than one kind of infinity, or that infinity, for you, is a set rather than a number?

So, 1xSomeInfinity is an infinite number and 2xTheSameInfinity is an infinite number, but the former is 50% of the latter.
So far, so good...

They aren't the same number, even though they are both in "the infinite set of numbers".
No!
50% of Wallis’s infinity symbol from calculus is still Wallis’s infinity.
50% of Cantor’s aleph_0 is still aleph_0.
50% of the continuum is still the continuum.
In fact, I believe this is true for any infinity.

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

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

[…]

The only way to do that is to say:
0.9...=1
and
&lt;something else>=&lt;the number before 1>

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

You’re assuming that such a number exists. Who said it does?

By definition, when we write 0.999..., we mean "zero, point, followed by a countable infinity of digits -- in other words, a sequence of digits -- , all of which are nines"."Countable" infinity is an oxymoron.
Not in mathematics: ”countably infinite” (http://mathworld.wolfram.com/CountablyInfinite.html).

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

Neither of those points has anything to do with 1 or 0.9..., but I figure I can still respond.
The paradox is an example of what trouble one runs into when one tries to work with a ‘set of all sets’, or some such entity. Since you had previously tried to question the equality 0.9…=1 by appealing to this kind of absolute entities, I hoped you would see the connection.

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

2 - Or, if I'm very nitpicky, I can claim that only the steps I can actually carry out to their end are meaningful, and that in this case the algorithm can only produce approximate divisions, not the exact division. However, even if I decide to apply such strict logical standards, I can still reverse the less accurate argument given in (1), and accept the symbol "zero, point, a countable infinity of threes" as a suggestive notation for the limit of the sequence of approximations (0, 0.3, 0.33, 0.333, ...), which is 1/3.
I accept (1). However, that pattern includes a remainder, so why would that remainder disappear unless the pattern changed?
The remainder goes to zero. See this post (http://www.badastronomy.com/phpBB/viewtopic.php?p=432717#432717) of mine in a neighbouring thread.

As for (2): if you define "the limit" for the sequence as being non-inclusive (that is, the number the sequence can almost--but not quite--reach), then I accept (2). If you define "the limit" as being inclusive, then I do not accept (2), for reasons stated.
I’m not sure I understand that distinction between an “inclusive limit” and a “non-inclusive limit” that you’re trying to make. Do you think you could explain it a bit better?

Disinfo Agent
2005-Mar-12, 03:12 PM
Now, it has been stated by several people that you can't have a number that's right below 1, because there are an infinite number of numbers between any two numbers. On the other side, there's the logic that there must be a number right below 1, even if it can't be represented by our conventional numbering system.
What logic? Where did you get that idea from?

A Thousand Pardons
2005-Mar-12, 09:43 PM
Now, it has been stated by several people that you can't have a number that's right below 1, because there are an infinite number of numbers between any two numbers. On the other side, there's the logic that there must be a number right below 1, even if it can't be represented by our conventional numbering system.
What logic? Where did you get that idea from?
Of course then there's the logic that there is no number right below 1, too. :)

Chuck
2005-Mar-12, 11:34 PM
It's not logic that there must be a number right below 1, it's common sense. In everyday life you can line up a bunch of objects so close together that you can't get another between any two of them. Each object in the line except the last one has a next object. But all of the objects we regularly deal with have positive widths. A point on the number line doesn't so this common sense notion doesn't apply. No point can have a next point because if they're not in the same place then there's room between them for more. Since we don't deal with objects of zero width in day to day living this situation never comes up.

This is math, people! Everyone with common sense please stop posting!

Grey
2005-Mar-13, 12:50 AM
This is math, people! Everyone with common sense please stop posting!
:D

Fram
2005-Mar-14, 02:22 PM
This must have been said before, but I have noticed that it helps some people understand it better (or makes it tougher to defend their opposite cause).

If you think 1 &lt;> 0.9999999......, answer this:

1 - 0.99999999...... = 0.0000000..... Agreed?

Now, what's the difference between 0 and 0.00000000...... ?

Lance
2005-Mar-14, 02:28 PM
1 - 0.99999999...... = 0.0000000..... Agreed?

I know I'm going to regret this...

As a layman, it would be clear to me that is wrong...

1 - 0.99999... = 0.00000...1

azazul
2005-Mar-14, 02:33 PM
1 - 0.99999... = 0.00000...1
Where did you get that one from, you can only get that one if you find the last nine. Go through all the nines and post a reply to me when you get to that last nine.

Disinfo Agent
2005-Mar-14, 03:29 PM
1 - 0.99999999...... = 0.0000000..... Agreed?

I know I'm going to regret this...

As a layman, it would be clear to me that is wrong...

1 - 0.99999... = 0.00000...1

Lance
2005-Mar-14, 03:36 PM

No, but I can continue to be stupid.

Again, from a layman's perspective...

0.99999... is 9's going on for infinity, right?
Never ending?
Never quite getting there?

And before you pick that apart, isn't it self-evident that if you "got there" you wouldn't have to keep going?

A Thousand Pardons
2005-Mar-14, 03:46 PM
0.99999... is 9's going on for infinity, right?
Never ending?
Never quite getting there?

And before you pick that apart, isn't it self-evident that if you "got there" you wouldn't have to keep going?
"get where"? :)

"There is no there there"

Disinfo Agent
2005-Mar-14, 03:46 PM
Again, from a layman's perspective...

0.99999... is 9's going on for infinity, right?

Never ending?

Never quite getting there? And before you pick that apart, isn't it self-evident that if you "got there" you wouldn't have to keep going?
Allow me to reply with another question: isn't it self-evident that if you didn't "get there" it's precisely because you stopped, rather than keep going?

Lance
2005-Mar-14, 06:08 PM
0.99999... is 9's going on for infinity, right?
Never ending?
Never quite getting there?

And before you pick that apart, isn't it self-evident that if you "got there" you wouldn't have to keep going?

"get where"?

"There is no there there"

Exactly. 1 isn't ambiguous like that.

Allow me to reply with another question: isn't it self-evident that if you didn't "get there" it's precisely because you stopped, rather than keep going?

That seems a bit side-ways. I though by definition it was never ending.

fosley
2005-May-22, 01:30 AM
So, I've been thinking, and discussed the idea with a couple people, and thinking, and busy with other stuff, and thinking, decided that I would post one last post.

First, I think I may have made a mistake,and I can't find the post I may have made the mistake in, so just in case:
1/3 = 0.1 in ternary, not 0.3 .

Second, my final line of debate:
Add 9/10ths of the difference between X and 1:
X += 9 * (1 - X) / 10
Repeat this process forever. Every time you do this, X gets ten times closer to 1 than it was one iteration previous to the current iteration:
X(N) = X(N - 1) / 10
Let's assume, for a moment, there's a maximum point right before 1. We'll call it 0.999999 . So, after 6 iterations of getting ten times closer to 1, we are as close to 1 as we can get:
N(6) = 0.999999
So, the 7th iteration is closer to 1 than you can possibly get:
N(7) = 0.9999999
Since N(7) is closer to 1 than N(6), we could say N(>6) = 1, by virtue of 1 being the closest real number to it:
1 - N(7) &lt; N(7) - N(6)
I still say that would make N(>6) a non-existent (non-real) number, not 1. But I can see the logic behind it. As was decided earlier though, if a number is less than 1, there are an infinite number of numbers between it and 1. With that, no matter how far out you go, there would still be an infinite number of numbers between X and 1. Even if you go to infinity. Unless the infinity that represents the number of iterations of X is greater than the infinity that represents the number of numbers between X and 1 at some point:
Infinity1 > Infinity2
If there is a limit to how many 9's we can put at the end of a decimal, then our numbering system loses and 0.9... &lt; 1. If there is a limit to how many numbers are between 0 and 1 then the count of real numbers loses and 0.9... = 1. But if there is no limit to either, they tie. However, since the difference started in the lead, the difference will always lead the number of 9's. 0.9... simply cannot reach 1 unless there is a limit to how close you can get to 1. And even then it hasn't reached 1, it's just ceased to be a real number, and we rounded it to 1 because it's closer to 1 than almost 1.

Third, some notes on the relevance of the answer to real life:
When I'm measuring the width of a desk to see if it will fit in my house, I'm measuring to about a sixteenth of an inch, so I need the ruler or tape-measure to be accurate to within several hundredths of an inch. When I'm measuring the clearance between the rod bearings and the crankshaft in my car, I'm measuring in thousands of an inch, so I need the plastiguage to be accurate to ten thousandths. When scientists measure the size of bacteria, they are measuring in millionths of an inch, so they need to be accurate to the ten millionth. Even the smallest things we can measure are only on the order of billionths, trillionths and quadrillionths of an inch. I was actually reading something the other day about the various sizes of all the things we've measured, and the difference between the size of the known universe and the size of the smallest sub-atomic particle can be expressed with 60 or 70 digits (it's based on meters, but that doesn't change much). As far as the real world is concerned 0.99999999999 is practically the same as 1, and 0.9 followed by a few thousand 9's is indistinguishable from 1. So it's really quite irrelevant to anything useful whether 0.9... = 1 or is just almost 1.

Being an engineer and not a mathmatician, if somebody can cut me a piece pipe to 26.999999999999999999999... inches and keep it from being 27 inches or 26.99999999...999990 I'll vote no.
If I cut a pipe to 26.999999999999999999999 inches (note that's exact, unlike the original intent of 26.999999999999999999999... inches), it would obviously not equal 27 inches or 26.999999999999999999990 inches, but nobody would know the difference. So, I would expect an engineer to say 26.9... inches is exactly 27 in, because they behave exactly the same way to an engineer. Or so we assume - it's actually rather unlikely that anyone has ever made something that was exactly 27 inches long, because that means the sub-atomic influences at one end extend to exactly 27 inches away from the sub-atomic influences at the other end. Being as such forces aren't exactly defined, that leads to a bit of ambiguity, which we would say means that if 27 inches fell anywhere in the grey area it counts (so we're down to a few picometers or so in the accuracy department). Since the atoms are vibrating constantly (they are above absolute zero, after all), if you can set them to 27 inches +- the amount of vibration, the pipe would have to equal exactly 27 inches at some point. But I would think that we would measure the average length over a period of time, since we can't measure the exact length of time any more than we can measure the exact length of space.

So basically, many people have made the argument: Prove that 0.9... is different from 1 by showing the difference on a measuring tape.

I propose the counter argument: Prove that 0.9 followed by 7 trillion 9's is different from 1 by showing the difference on a measuring tape.

The result of any attempt to physically prove either will be the same resoundingly negative result. Since I think we would all agree that 0.9 followed by any finite number of 9's is different than 1, one shouldn't attempt to use the lack of physical verification as "proof" for 0.9... and 1 being the same.

And, one last example of the real-world: half-life. Take any given amount of radioactive stuff, and it will convert half of the unstable atoms to more stable atoms in some specific amount of time (we're going to pretend that this is an exact truth). Now, a very long time after the batch is created, we will have two unstable atoms left (possibly - let's assume this was a perfect batch). Then, you're going to have one. Then, the last atom will convert. You can't convert half the atom, then a quarter all the way to infinitesimal, so it will just convert. That doesn't mean 1 / 2 = 0. It just means physical reality doesn't convert one plutonium atom into two radioactive silver atoms, one of which then gets converted to a more stable atom. But if it did, that would only work down to having 1 hydrogen atom, at which point it would have to start splitting into quarks and leptons and things. Then you would run out of those, at which point you would have to either leave 1, or leave 0.

mickal555
2005-May-22, 01:34 AM
So, I've been thinking, and discussed the idea with a couple people, and thinking, and busy with other stuff, and thinking, decided that I would post one last post.

First, I think I may have made a mistake,and I can't find the post I may have made the mistake in, so just in case:
1/3 = 0.1 in ternary, not 0.3 .

Second, my final line of debate:
Add 9/10ths of the difference between X and 1:
X += 9 * (1 - X) / 10
Repeat this process forever. Every time you do this, X gets ten times closer to 1 than it was one iteration previous to the current iteration:
X(N) = X(N - 1) / 10
Let's assume, for a moment, there's a maximum point right before 1. We'll call it 0.999999 . So, after 6 iterations of getting ten times closer to 1, we are as close to 1 as we can get:
N(6) = 0.999999
So, the 7th iteration is closer to 1 than you can possibly get:
N(7) = 0.9999999
Since N(7) is closer to 1 than N(6), we could say N(>6) = 1, by virtue of 1 being the closest real number to it:
1 - N(7) &lt; N(7) - N(6)
I still say that would make N(>6) a non-existent (non-real) number, not 1. But I can see the logic behind it. As was decided earlier though, if a number is less than 1, there are an infinite number of numbers between it and 1. With that, no matter how far out you go, there would still be an infinite number of numbers between X and 1. Even if you go to infinity. Unless the infinity that represents the number of iterations of X is greater than the infinity that represents the number of numbers between X and 1 at some point:
Infinity1 > Infinity2
If there is a limit to how many 9's we can put at the end of a decimal, then our numbering system loses and 0.9... &lt; 1. If there is a limit to how many numbers are between 0 and 1 then the count of real numbers loses and 0.9... = 1. But if there is no limit to either, they tie. However, since the difference started in the lead, the difference will always lead the number of 9's. 0.9... simply cannot reach 1 unless there is a limit to how close you can get to 1. And even then it hasn't reached 1, it's just ceased to be a real number, and we rounded it to 1 because it's closer to 1 than almost 1.

Third, some notes on the relevance of the answer to real life:
When I'm measuring the width of a desk to see if it will fit in my house, I'm measuring to about a sixteenth of an inch, so I need the ruler or tape-measure to be accurate to within several hundredths of an inch. When I'm measuring the clearance between the rod bearings and the crankshaft in my car, I'm measuring in thousands of an inch, so I need the plastiguage to be accurate to ten thousandths. When scientists measure the size of bacteria, they are measuring in millionths of an inch, so they need to be accurate to the ten millionth. Even the smallest things we can measure are only on the order of billionths, trillionths and quadrillionths of an inch. I was actually reading something the other day about the various sizes of all the things we've measured, and the difference between the size of the known universe and the size of the smallest sub-atomic particle can be expressed with 60 or 70 digits (it's based on meters, but that doesn't change much). As far as the real world is concerned 0.99999999999 is practically the same as 1, and 0.9 followed by a few thousand 9's is indistinguishable from 1. So it's really quite irrelevant to anything useful whether 0.9... = 1 or is just almost 1.

Being an engineer and not a mathmatician, if somebody can cut me a piece pipe to 26.999999999999999999999... inches and keep it from being 27 inches or 26.99999999...999990 I'll vote no.
If I cut a pipe to 26.999999999999999999999 inches (note that's exact, unlike the original intent of 26.999999999999999999999... inches), it would obviously not equal 27 inches or 26.999999999999999999990 inches, but nobody would know the difference. So, I would expect an engineer to say 26.9... inches is exactly 27 in, because they behave exactly the same way to an engineer. Or so we assume - it's actually rather unlikely that anyone has ever made something that was exactly 27 inches long, because that means the sub-atomic influences at one end extend to exactly 27 inches away from the sub-atomic influences at the other end. Being as such forces aren't exactly defined, that leads to a bit of ambiguity, which we would say means that if 27 inches fell anywhere in the grey area it counts (so we're down to a few picometers or so in the accuracy department). Since the atoms are vibrating constantly (they are above absolute zero, after all), if you can set them to 27 inches +- the amount of vibration, the pipe would have to equal exactly 27 inches at some point. But I would think that we would measure the average length over a period of time, since we can't measure the exact length of time any more than we can measure the exact length of space.

So basically, many people have made the argument: Prove that 0.9... is different from 1 by showing the difference on a measuring tape.

I propose the counter argument: Prove that 0.9 followed by 7 trillion 9's is different from 1 by showing the difference on a measuring tape.

The result of any attempt to physically prove either will be the same resoundingly negative result. Since I think we would all agree that 0.9 followed by any finite number of 9's is different than 1, one shouldn't attempt to use the lack of physical verification as "proof" for 0.9... and 1 being the same.

And, one last example of the real-world: half-life. Take any given amount of radioactive stuff, and it will convert half of the unstable atoms to more stable atoms in some specific amount of time (we're going to pretend that this is an exact truth). Now, a very long time after the batch is created, we will have two unstable atoms left (possibly - let's assume this was a perfect batch). Then, you're going to have one. Then, the last atom will convert. You can't convert half the atom, then a quarter all the way to infinitesimal, so it will just convert. That doesn't mean 1 / 2 = 0. It just means physical reality doesn't convert one plutonium atom into two radioactive silver atoms, one of which then gets converted to a more stable atom. But if it did, that would only work down to having 1 hydrogen atom, at which point it would have to start splitting into quarks and leptons and things. Then you would run out of those, at which point you would have to either leave 1, or leave 0.

Your going to be in sooooooooo much trouble for bumping this thread :lol:

A Thousand Pardons
2005-May-22, 02:19 AM
First, I think I may have made a mistake,and I can't find the post I may have made the mistake in, so just in case:
1/3 = 0.1 in ternary, not 0.3 .
No, you got it right (http://www.badastronomy.com/phpBB/viewtopic.php?p=430713&amp;highlight=ternary#430713). Someone would have caught that.

Lurker
2005-May-22, 06:06 AM
NNNNNNNNNNNNNNNNNNNOOOOOOOOOOOOOOOOOOOOOOOO!!!!!!! ! http://loresinger.com/FWIS/images/smiles/shocked.gif

Must put right what was never meant to be!!!
http://loresinger.com/FWIS/images/smiles/blowingup.gif

worzel
2005-May-22, 08:46 AM
Sorry, couldn't resist :oops:

So, I've been thinking, and discussed the idea with a couple people, and thinking, and busy with other stuff, and thinking, decided that I would post one last post....
First off, this about maths, not physical limitations on measurements.

Secondly, you could say that 0.999.... is not a real number, but then you could argue that 0.888.... isn't either by the same logic. But then what is the decimal representation of 8/9? Now we even can't represent all rationals in decimal. Alternatively, you could decide that 0.999... is a real (and rational) number like any other recurring decimal, and make the only consistent choice that it equals 1.

mickal555
2005-May-22, 08:59 AM
I was thinking

IF 0.999999999.....
=1

then 0.555555555555555......
=
.66666.......

which would equal

.7777777........

then

.888.........

then

.999999999........

then

1.00........

so

:o

worzel
2005-May-22, 09:18 AM
I was thinking

IF 0.999999999.....
=1

then 0.555555555555555......
=
.66666.......

How did you figure that out?

mickal555
2005-May-22, 09:33 AM
well if

0.999999....
tips over into

1.00000 or 1

then 0.55555..... (or 0.anything)

would tip over 0.666666666666......

then that would tip over into

0.7777777777777....

etc...

PS. when I say tip over I don't mean rounding- I mean in the same way 0.999999...=1

worzel
2005-May-22, 09:41 AM
well if

0.999999....
tips over into

1.00000 or 1

then 0.55555..... (or 0.anything)

would tip over 0.666666666666......

then that would tip over into

0.7777777777777....

etc...

PS. when I say tip over I don't mean rounding- I mean in the same way 0.999999...=1
Something like:

0.7 - 0.6 = 0.1
0.6 - 0.5 = 0.1
1.0 - 0.9 = 0.1

0.77 - 0.66 = 0.11
0.66 - 0.55 = 0.11
1.00 - 0.99 = 0.01 [ oops ]

0.777 - 0.666 = 0.111
0.666 - 0.555 = 0.111
1.000 - 0.999 = 0.001 [ oops ]

.
.
.

0.777... - 0.666... = 0.111...
0.666... - 0.555... = 0.111...
1.000... - 0.999... = 0.0...01 [ oops ]

01101001
2005-May-22, 09:43 AM
well if

0.999999....
tips over into

1.00000 or 1

then 0.55555..... (or 0.anything)

would tip over 0.666666666666......

Tips over? I must have skipped math class that day.

The difference between .555... and .666... is .111...

6/9 - 5/9 = 1/9

So what .999... "tips over" to is .999... + .111... = 1.111...

mickal555
2005-May-22, 09:56 AM
I do belive that- .999... =1.0000000...

well if .9999999999... = 1.000000000000...
then I thought 0.55555.... would go up to 0.6666666666....

I've clearly gotten something wrong

what does 0.555... = then :-?

Lycus
2005-May-22, 09:59 AM
what does 0.555... = then :-?

5/9

mickal555
2005-May-22, 10:04 AM
in decimal- like .99999.... eqaling 1.0000.....

*looks around- stop laughing at me-They laughed at Galileo, too!8-[

frogesque
2005-May-22, 10:07 AM
in decimal- like .99999.... eqaling 1.0000.....

*looks around- stop laughing at me-They laughed at Galileo, too!8-[

It would depend on where you took a cut-off - to 4 decimal places it would be 0.5556

mickal555
2005-May-22, 10:09 AM
why do you need a cut off?

assume no cut off

skwirlinator
2005-May-22, 10:24 AM
The way it was explained to me is

everything this side of the decimal is over 1. everthing this side is under 1

mickal555
2005-May-22, 10:30 AM
hmmm

frogesque
2005-May-22, 10:40 AM
why do you need a cut off?

assume no cut off

Then it's 0.5555 ... or 5/9.

I can't think of a simple way to explain it but (0.6666 ...) -(0.5555 ...) =(0.1111 ...)

Rounding 0.5555555555555555 ... would give 0.5555555555555556 It's only the last figure that would become a 6, not all of them

mickal555
2005-May-22, 11:28 AM
why do you need a cut off?

assume no cut off

Then it's 0.5555 ... or 5/9.

I can't think of a simple way to explain it but (0.6666 ...) -(0.5555 ...) =(0.1111 ...)

Rounding 0.5555555555555555 ... would give 0.5555555555555556 It's only the last figure that would become a 6, not all of them

But It's impossable to have a last # in an infinite series.

Why is .999999999999.... so special

does 0.55555555 simple =.6?

no wait...

OK I'm confused again #-o

worzel
2005-May-22, 11:42 AM
Why is .999999999999.... so special
Anything ending with 999... is special in that the same real number has another distinct decimal representation. This isn't the case for 0.555...

mickal555
2005-May-22, 11:49 AM
why not?

Lycus
2005-May-22, 11:57 AM
Because there is a difference between, for example, 0.6 and 0.5555...

0.6 - 0.5555... = 0.0444...

In other words, there is a distance between them on a number line. Within that distance, you can find other numbers; such as 0.58.
0.6 and 0.5555... have different values.

The difference between 0.9999... and 1 is zero. There are no other numbers in between them. The is no space in between them on a number line. Therefore, they both represent the same value.

jfribrg
2005-May-22, 01:53 PM
I say we take up a collection and pay the BA to lock all these threads.

A Thousand Pardons
2005-May-22, 02:17 PM
well if

0.999999....
tips over into

1.00000 or 1

then 0.55555..... (or 0.anything)

would tip over 0.666666666666......

then that would tip over into

0.7777777777777....

etc...

PS. when I say tip over I don't mean rounding- I mean in the same way 0.999999...=1
Well, if you mean that, then in the same way that 0.999999... tips over into 1,

0.55555..... tips over into 0.55555.....

Not 0.666666666666......

Sam5
2005-May-22, 02:25 PM
Well, if you mean that, then in the same way that 0.999999... tips over into 1,

0.55555..... tips over into 0.55555.....

Not 0.666666666666......

Maybe he was thinking of his car’s odometer that has a limited number of digits.

A Thousand Pardons
2005-May-22, 02:53 PM
Well, if you mean that, then in the same way that 0.999999... tips over into 1,

0.55555..... tips over into 0.55555.....

Not 0.666666666666......

Maybe he was thinking of his car’s odometer that has a limited number of digits.
No, that can't be it, right? On an odometer,

555555 still tips over into 555556 not 666666

whereas 999999 tips over into 000000

Lance
2005-May-22, 03:39 PM
I've avoided these threads for the most part, but that's similar to a question I've had too.

If .99999... = 1 then why doesn't

.55555... = .6? (Not .66666..., just .6)

Wouldn't the same comments about "space on the number line" be true here as well?

A Thousand Pardons
2005-May-22, 03:45 PM
I've avoided these threads for the most part, but that's similar to a question I've had too.

If .99999... = 1 then why doesn't

.55555... = .6? (Not .66666..., just .6)

Wouldn't the same comments about "space on the number line" be true here as well?
Instead of the BABB, try a bloody mary

.55555... is less than or equal to .5600000... right? Just like 0.9999... is less than or equal to 1.0000...., everybody seems to agree with that--no one I've ever seen has suggested that .999... might be greater than 1.0

So, is .56 less than .60? Of course it is. So, .5555.... has to be even less than that.

Lance
2005-May-22, 03:50 PM
So, is .56 less than .60? Of course it is. So, .5555.... has to be even less than that.

Okay, so I get a big fat hairy DUH! sign to wear around my neck for the day.

I guess the questiong should really be if:

.59999... = .6

Grey
2005-May-22, 03:57 PM
I guess the questiong should really be if:

.59999... = .6
Yes, that's true.

Sam5
2005-May-22, 04:48 PM
Hmm, what if we had an electronic odometer with an infinite number of numbers. Could it ever click over from .55555….. to .55556….?

frogesque
2005-May-22, 04:54 PM
Hmm, what if we had an electronic odometer with an infinite number of numbers. Could it ever click over from .55555….. to .55556….?

Wouldn't work, there is a finite (though very large) number of electrons in the universe so you could never reach an infinite number.

skwirlinator
2005-May-22, 05:01 PM
This is SO FUNNY!

Numbers are not FUZZY.
They are not up for interpretation

They are solid and Reality

Just because you can 'Round up' a number doesn't change what the reality really is.

.9999 to infinity will never be ONE

After you pass out of reality anything is possible

If I have .9 apples and I look really hard it appears to be 1 apple but really there is a piece missing. As you add places .99 there is still not 1 apple but its harder to figure out whats missing. After a while that missing part is too small to calculate but being as below the point {.} means it's still not 1 apple.
If you are figuring Plank energy it still holds true. when you pass beyond the realm of your measurement it doesn't matter but it still holds true

Makes me think you want to tell me something is colder than frozen.

Or lighter than zero

or less massive than empty

or a 'partial' vacuum

or zero gravity

well That how this mechanic thinks of this stuff anyway
LoL

Sam5
2005-May-22, 05:10 PM
This is SO FUNNY!

Numbers are not FUZZY.
They are not up for interpretation

They are solid and Reality

Just because you can 'Round up' a number doesn't change what the reality really is.

.9999 to infinity will never be ONE

Well, suppose we say this… mathematically, .99999999…. will never equal 1, but in reality, any device we try to use to measure the difference between a long number of .9999999….s and 1 will eventually not be able to determine a difference between a long list of .9999999…..s and 1. Like frogesque says, there is a finite number of electrons in the universe.

skwirlinator
2005-May-22, 05:25 PM
I only say this

What would an advanced civilization say to that?

remember our age- we be babys yet - 14 billion years is a little more than 200,000 or so

perhaps the secrets we have not found depend on PRECISE calculations to the number of gluons in the universe or something like that.

Rounding off in mathematics is just designer numbering and voids all true calculations.

WoW.......LOL

frogesque
2005-May-22, 05:34 PM
I only say this

What would an advanced civilization say to that?

remember our age- we be babys yet - 14 billion years is a little more than 200,000 or so

perhaps the secrets we have not found depend on PRECISE calculations to the number of gluons in the universe or something like that.

Rounding off in mathematics is just designer numbering and voids all true calculations.

WoW.......LOL

Meanwhile, engineers get the job done and mathematicians contemplate the number of angels dancing on a pin head (or the meaning of fluff in the belly button - which probably amounts to the same thing 8) )

Grey
2005-May-22, 06:08 PM
This is SO FUNNY!

Numbers are not FUZZY.
They are not up for interpretation

They are solid and Reality

Just because you can 'Round up' a number doesn't change what the reality really is.

.9999 to infinity will never be ONE

well That how this mechanic thinks of this stuff anyway
LoL
Strange, I'd say that numbers aren't up for interpretation either, but come to the opposite conclusion you have. They have a rigorous definition mathematically, and using those mathematical definitions, 0.999... can be shown to be equal to 1. It's not a matter of "rounding it up", or approximating it in any way. The real number system would not be consistent if they were not equal.

A Thousand Pardons
2005-May-22, 06:59 PM
Makes me think you want to tell me something is colder than frozen.

Or lighter than zero

or less massive than empty

or a 'partial' vacuum

or zero gravity

well That how this mechanic thinks of this stuff anyway
LoL
What's between 0.9999.... and 1.0? How're you going to wrench something in there?

If they are two different numbers, there must be another number in between them, right? Just take their average. What number would that be?

If A > B then

their sum A + B > B + B
and their average (A + B)/2 > (B + B)/2
so their average (A+B)/2 is greater than B

1.0 plus 0.999... is 1.99999....

Divide that by 2, what do you get?

Lycus
2005-May-22, 07:02 PM
skwirlinator

You would agree that if two numbers are not equal, then their difference would have to equal something other than zero, right?

So, what's the answer to this:

1 - 0.9999.... = ?

TrAI
2005-May-22, 09:28 PM
Heh, so this thread is around still... I have been thinking about this a little since the last time, but I still cant accept that 0 is not nothing, even something infinitely small is something...

skwirlinator

You would agree that if two numbers are not equal, then their difference would have to equal something other than zero, right?

So, what's the answer to this:

1 - 0.9999.... = ?

I guess that I might have a crack at this, even if I am not the one you are asking this of.

1 - 0.9999...= one of two things, that is 0 with infinite resolution(though this is catastrophic cancellation(IMHO), so it can't be considered accurate, remember everyone, always make sure you have the needed amount of resolution if you need accurate answers), and 0,000...0001 with overinfinite resolution...

The idea behind this(Yea, I have been thinking a little more about this part since the last time) is that even if you never can get to the end of something infinite, you do have one end, there is not any reason for the end to be 0,999.... It might as well be ...999, so, I can predict how an infinite string of the same number may look if I started writing it from the other end(I can do this since the question clearly states that there is an infinite string of 9s after the "0,"). Now I can make two copies of the same number just writing it from one side in one, and the other side in the other. Then I can use the combination to predict the 0,999...999.

Of course, I guess this may be considered bad math, but then I guess that could be said about, not calculating the subtraction to the end is bad too, even if it is infinite, you can start in any end when doing a calculation, but if you do not take in to account that you have to keep going to the end, it is to be considered an approximation.

You know, it is a bit strange that we humans find it entertaining to discuss something like this that really has no significance at all, and is never likely to end with everyone agreeing...

mickal555
2005-May-22, 09:34 PM
I've avoided these threads for the most part, but that's similar to a question I've had too.

If .99999... = 1 then why doesn't

.55555... = .6? (Not .66666..., just .6)

Wouldn't the same comments about "space on the number line" be true here as well?
Instead of the BABB, try a bloody mary

Well maybe I will [-(

A Thousand Pardons
2005-May-22, 09:40 PM
I still cant accept that 0 is not nothing
I think this sets a new BABB record for negatives (OK, non-positives) per inch

PS: are you sure that's what you meant?

::snip::

Then I can use the combination to predict the 0,999...999.

Of course, I guess this may be considered bad math, but then I guess that could be said about, not calculating the subtraction to the end is bad too, even if it is infinite, you can start in any end when doing a calculation, but if you do not take in to account that you have to keep going to the end, it is to be considered an approximation.
So, how would you go about subtracting 1.000...000 minus 0.999...999 ?

TrAI
2005-May-22, 10:05 PM
I still cant accept that 0 is not nothing
I think this sets a new BABB record for negatives (OK, non-positives) per inch

PS: are you sure that's what you meant?

::snip::

Then I can use the combination to predict the 0,999...999.

Of course, I guess this may be considered bad math, but then I guess that could be said about, not calculating the subtraction to the end is bad too, even if it is infinite, you can start in any end when doing a calculation, but if you do not take in to account that you have to keep going to the end, it is to be considered an approximation.
So, how would you go about subtracting 1.000...000 minus 0.999...999 ?

Indeed, that is my answer. Sure it is a bit of a stretch to do it like that, it is kind of me making a sort of philosophical idea of the answer one would get if one could apply higher resolution than infinite for the numbers we are computing with. I think that can be considered, when 1-0,999... would get cancellation problems if not, I mean, we cant really get to the end of that subtraction, so at some point we stop and just say its infinitely small...

Anyway, all our attempts at prooving one or the other may perhaps be beside the point, as what we are really discussing is the definition of zero and nothing, isn't it?. One side feels anything infinitely small is to be considered zero while the other feels that zero is nothing at all, and that even an infinitely small amount is something(perhaps it is a bit like the difference between the absence of anything(even the universe) and just vacuum)...

Captain Kidd
2005-May-22, 11:16 PM
This is the thread that never ends.
It goes on and on my friends.
Some people started debating it not knowing what it was,
and they'll continue debating it forever just because,

This is the thread that never ends.
It goes on and on my friends.
Some people started debating it not knowing what it was,
and they'll continue debating it forever just because,

Fortis
2005-May-23, 12:43 AM
What's between 0.9999.... and 1.0? How're you going to wrench something in there?

And if there is no number that lies between them they are contiguous on the number line, which given that a number on that line is a point of zero extent is a tricky thing to do if they are supposed to be different numbers. :)

A Thousand Pardons
2005-May-23, 12:47 AM
Anyway, all our attempts at prooving one or the other may perhaps be beside the point, as what we are really discussing is the definition of zero and nothing, isn't it?.
No, we're just trying to make sure everybody understands the definitions. I'm pretty sure mine's the same as yours.

So, is 0.000...001 less than 0.000...01?

mopc
2005-May-23, 02:21 AM
this thread still exists? Wasnt this thread that triggered that banning-unbanning episode of sci-fi chick and some guy?

skwirlinator
2005-May-23, 02:24 AM
I'm done
its either one or its not
if its less than one then its not one

no mater how you look at it

bye #-o

Fortis
2005-May-23, 02:39 AM
I'm done
its either one or its not
if its less than one then its not one

no mater how you look at it
Agree 99.999999...% ;)
And 0.9999... is not less than 1. :)

Lycus
2005-May-23, 02:41 AM
this thread still exists? Wasnt this thread that triggered that banning-unbanning episode of sci-fi chick and some guy?
Not exactly, that was one of the other half-dozen or so 0.9999... threads. :P

I suppose these threads will be around as long as there are people having some trouble understanding the concepts behind them.

Lance
2005-May-23, 02:48 AM
If 0.99999... truly equals 1, why don't you just write "1"?

Fortis
2005-May-23, 02:57 AM
If 0.99999... truly equals 1, why don't you just write "1"?
There's nothing to stop you doing that, but there are a lot of people who would get upset if we did, as they don't believe that they are equal.

01101001
2005-May-23, 03:15 AM
If 0.99999... truly equals 1, why don't you just write "1"?
... or 1.0 or 1.00 or 1.000 or 1.0000 or...?

They are all among the infinitely many names of the one thing.

mopc
2005-May-23, 03:17 AM

Chuck
2005-May-23, 03:41 AM
It has a final digit. 0.999999... has no final digit. To which nine do you add that far right one to make the total come out to 1? It can't be the last nine because there is no last nine.

01101001
2005-May-23, 04:48 AM

What exactly does that mean? An infinite number of 0's followed by a 1? If so, there is no such thing.

If it is a finite number, n, of 0's followed by a 1, then the difference between it and 1.0 is a small but finite quantity, 1*10^-(n+1), making them different.

mopc
2005-May-23, 05:40 AM

What exactly does that mean? An infinite number of 0's followed by a 1? If so, there is no such thing.

If it is a finite number, n, of 0's followed by a 1, then the difference between it and 1.0 is a small but finite quantity, 1*10^-(n+1), making them different.

So what is the smallest thing after 1? How do you denote it?

Lycus
2005-May-23, 05:46 AM
So what is the smallest thing after 1? How do you denote it?
There's no such thing as "smallest thing after 1." Choose any number that's greater than 1, and there'll be an infinite number of others in between them.

01101001
2005-May-23, 06:37 AM
So what is the smallest thing after 1? How do you denote it?
There's no such thing as "smallest thing after 1." Choose any number that's greater than 1, and there'll be an infinite number of others in between them.
Yep. And... there is no largest number less than 1.

mopc
2005-May-23, 06:38 AM
So what is the smallest thing after 1? How do you denote it?
There's no such thing as "smallest thing after 1." Choose any number that's greater than 1, and there'll be an infinite number of others in between them.

So there is no smallest thing before 1, thus any number smaller than one has infinite numbers between it and 1, so 0,999... is smaller than one.

Lycus
2005-May-23, 06:50 AM
So there is no smallest thing before 1, thus any number smaller than one has infinite numbers between it and 1, so 0,999... is smaller than one.
If that's true, then you should be able to give us an example of a number between 0.9999... and 1.

mopc
2005-May-23, 06:54 AM
So there is no smallest thing before 1, thus any number smaller than one has infinite numbers between it and 1, so 0,999... is smaller than one.
If that's true, then you should be able to give an example of a number between 0.9999... and 1.

And if that is not true, than you shouldn't be able to give an example of a number between 1 and 1,0000......1

Lycus
2005-May-23, 07:06 AM
1,0000......1
That's not a number, it doesn't exist. That last digit has no definition.

01101001
2005-May-23, 07:12 AM
1,0000......1
Would you please define that notation. It is not part of the math I learned. What does it mean? (And, if you think that it means "the smallest number greater than 1", then prove that such a thing exists.)

mickal555
2005-May-23, 07:35 AM
I had something to say but as it answed the question once and for all I deleted it

Captain Kidd
2005-May-23, 12:15 PM
... or 1.0 or 1.00 or 1.000 or 1.0000 or...?

They are all among the infinitely many names of the one thing.
Ah but now you're in the realm of significant digits.
0.9754*1 = 1
0.9754*1.0 = 0.98
0.9754*1.00 = 0.975
0.9754*1.000 = 0.9754

:wink:

jfribrg
2005-May-23, 01:39 PM
Why is it that when a phyicist asserts something that is counterintuitive, people tend to agree without necessarily understanding. Examples are the wave-particle duality, Heisenberg Uncertainty principle, dark matter, dark energy. I know there is experimental evidence to make all of these ideas at least plausible, but in the end, many people believe these phenomena to be real because the consensus of world famous scientists is that these things are real. But when you discuss the thread topic, the fact that all (not just a consensus) world famous mathematicians agree that 0.99999.... = 1.00000...., a good number of people on this board ( and a greater percentage of others in the general public) simply assert the all these math geeks don't know what they are talking about.

Captain Kidd
2005-May-23, 01:56 PM
Probably because a lot of that is obviously so over their heads and they know it that they'll take the physicist's word for it. However, 0.9999...=1 has an apparent simplicity to it. To many, if 0.9999... were to equal 1 then it'd be 1 not 0.9999... which has an implied less than one (otherwise you'd write "1".)

Heisenberg Uncertainty = complex
numbers = simple grade school stuff

Thus, trust the physicists and question the mathematicians.

There's probably a lot more to it, but that's my 2 minute take.

Sam5
2005-May-23, 02:28 PM
Probably because a lot of that is obviously so over their heads and they know it that they'll take the physicist's word for it. However, 0.9999...=1 has an apparent simplicity to it. To many, if 0.9999... were to equal 1 then it'd be 1 not 0.9999... which has an implied less than one (otherwise you'd write "1".).

I think my solution solves the basic puzzle.

Using pure mathematics only, .99999999…, with an infinite number of 9s, will never equal 1, but in physics, any device we try to use to measure the difference between a long number of .9999999…s and 1 will eventually not be able to determine a difference between a long list of .9999999…s and 1.

For example, if we try to divide a mm into an infinite number of parts, we discover that we have only a finite number of divisions we can make before our instruments can no longer detect a difference between .999999999999999999999999 and 1. The same goes for weight measurements, time measurements, and other kinds of physics measurements.

Disinfo Agent
2005-May-23, 02:31 PM
So there is no smallest thing before 1, thus any number smaller than one has infinite numbers between it and 1, so 0,999... is smaller than one.
How's that? I don't think I follow you.
How does the fact that:

1) Any number smaller than one has infinite numbers between it and 1

imply the hypothesis that

2) 0,999... is smaller than one? :-?

Disinfo Agent
2005-May-23, 02:35 PM
Anyway, all our attempts at prooving one or the other may perhaps be beside the point, as what we are really discussing is the definition of zero and nothing, isn't it?.
No, what we're really discussing is the definition of decimal numerals in standard math, which is what people learn in college and high school.

This is why any appeals to nonzero infinitesimals, or to numerical computation, or to physical measurements, are irrelevant and completely miss the point.

worzel
2005-May-23, 02:41 PM
I think my solution solves the basic puzzle.
Wow, another seemingly insoluble puzzle solved first by Sam5!

Using pure mathematics only, .99999999…, with an infinite number of 9s, will never equal 1
Except that it does.

No dbout when you find a page explaining this to your satisfaction by a doctor of mathematics you'll post a link and increase this thread by another page or two by arguing that it was you who first posted the solution. Then, if you follow form, you'll forget that you once supposedly understood the answer and go back to aguing the inequality again.

Disinfo Agent
2005-May-23, 02:43 PM
I can't think of a simple way to explain it but (0.6666 ...) -(0.5555 ...) =(0.1111 ...)

Rounding 0.5555555555555555 ... would give 0.5555555555555556 It's only the last figure that would become a 6, not all of them
You can check that 0.6666 ... - 0.5555 ... = 0.1111 ... by adding 0.5555 ... and 0.1111... together. Each 1 in the numeral expansion will add to 5 to give 6. Since there is an infinity of both, we get 0.5555 ... + 0.1111... = 0.6666...

Lance
2005-May-23, 02:49 PM
Yep. And... there is no largest number less than 1.

But if nothing comes between 0.99999... and 1, wouldn't 0.99999... be the largest number less than one?

Disinfo Agent
2005-May-23, 02:51 PM
But if nothing comes between 0.99999... and 1, wouldn't 0.99999... be the largest number less than one?
Only if they are different.

Lance
2005-May-23, 02:54 PM
But if nothing comes between 0.99999... and 1, wouldn't 0.99999... be the largest number less than one?
See above. Only if they are different.

ACK! I missed all of page 34 when I posted that.

Disinfo Agent
2005-May-23, 02:55 PM
Well, suppose we say this… mathematically, .99999999…. will never equal 1, but in reality, any device we try to use to measure the difference between a long number of .9999999….s and 1 will eventually not be able to determine a difference between a long list of .9999999…..s and 1.
That's exactly backwards. It is mathematics which shows that 1 and .99999999… are the same, and our limited physical experience which misleads us into doubting their equality.

Disinfo Agent
2005-May-23, 03:01 PM
Deleted.

Sam5
2005-May-23, 03:02 PM
I think my solution solves the basic puzzle.
Wow, another seemingly insoluble puzzle solved first by Sam5!

Many seemingly “unsolvable” puzzles are usually solvable if people would analyze them rationally and figure out how to separate the different hidden components of the puzzle and also realize that many puzzles have more than one type of solution.

If we are speaking in purely mathematical terms, .9999...∞ never = 1. But if we are speaking in practical terms, even .9 can = 1 and .009 can = .01.

For example I know of a county that has a sales tax of = .059375¢ on every dollar.

For a bill of \$5,234.86, the tax would be \$310.8198125

Since we can’t divide a penny, we must round off the last numbers, and the actual tax bill would be \$310.82

If the tax were officially .999...∞¢ on every dollar, then the actual tax would have to be rounded off to 1 dollar on every 1 dollar.

But this fact of practical life regarding US money will never make the mathematical number .999...∞ = 1.

Disinfo Agent
2005-May-23, 03:07 PM
Can't really comment on that, as I've never seen the symbol ".9999...infinity" before in my life, and have no idea what it means. Anyway, it's not what you had written earlier, Sam5.

Sam5
2005-May-23, 03:10 PM
Can't really comment on that, as I've never seen the symbol ".9999...infinity" before in my life, and have no idea what it means. Anyway, it's not what you had written earlier, Sam5.

That is my shorthand way of saying an infinite number of 9s after .999999...

doh

Disinfo Agent
2005-May-23, 03:12 PM
That is my shorthand way of saying an infinite number of 9s after .999999...

doh
Another one?
There already is an infinite number of nines in ".999999..." Hence the ellipsis.

JohnW
2005-May-23, 03:17 PM
Aargh! It's back! The Thread That Would Not Die! Where's my stake?

So, has anyone who thinks 0.999... ^= 1 come up with a non-zero value for the difference yet?

Thought not.

Disinfo Agent
2005-May-23, 03:20 PM
Second, my final line of debate:
Add 9/10ths of the difference between X and 1:
X += 9 * (1 - X) / 10
Repeat this process forever.
What's X supposed to be, there?

Sam5
2005-May-23, 03:20 PM
That is my shorthand way of saying an infinite number of 9s after .999999...

doh
Another one?
There already is an infinite number of nines in ".999999..." Hence the ellipsis.

If you think my ∞ symbol is being redundant, then why did you earlier write “0.999... is not smaller than 1.”

Why didn’t you just write “0.9... is not smaller than 1.”?

Disinfo Agent
2005-May-23, 03:23 PM
If you think my ? symbol is being redundant, then why did you earlier write “0.999... is not smaller than 1.”

Why didn’t you just write “0.9... is not smaller than 1.”?
0.999... and 0.9... mean exactly the same thing in standard mathematical notation. This has been explained before in these threads. I wrote down three nines earlier because mpoc had written four of them, too, and I was replying to his post. (I actually thought I had written four nines, too, but I see that I misremembered. It makes no difference, regardless.)

Sam5
2005-May-23, 03:25 PM
If you think my ? symbol is being redundant, then why did you earlier write “0.999... is not smaller than 1.”

Why didn’t you just write “0.9... is not smaller than 1.”?
0.999... and 0.9... mean exactly the same thing in standard mathematical notation. This has been explained before in these threads. I used four nines earlier because that's what mpoc had done, too, and I was replying to his post.

Then you must use an ∞ symbol when responding to my posts. :D

Disinfo Agent
2005-May-23, 03:28 PM
That's not standard mathematical notation. The one used by mopc is.

Disinfo Agent
2005-May-23, 03:31 PM
So, has anyone who thinks 0.999... ^= 1 come up with a non-zero value for the difference yet?

Thought not.
You can't say that they're discouraged by failure. :D

Sam5
2005-May-23, 03:35 PM
That's not standard mathematical notation. The one used by mopc is.

My notation is more in line with the title of this thread, which is:

“Do you think 0.9999999~ infinite 9s is exactly the equal to 1.” :D

Disinfo Agent
2005-May-23, 03:42 PM
My notation is more in line with the title of this thread, which is:

“Do you think 0.9999999~ infinite 9s is exactly the equal to 1.” :D
Actually what VTBoy wrote was

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

Edited because I took your word for it, initially. I should have known better than to trust you to accurately quote someone else...

A Thousand Pardons
2005-May-23, 04:24 PM
I think my solution solves the basic puzzle.

There is no real puzzle, it's just that a lot of people misunderstand the notation. .999... means 9/10+9/100+9/1000+..., and there are a lot of webpages out there that will give you the simple formula for the sum of an infinite geometric series (http://mathworld.wolfram.com/Series.html). In this case, it is 1.0

Using pure mathematics only, .99999999…, with an infinite number of 9s, will never equal 1,

No, it does equal 1. .999... with a finite number of 9s will not equal 1.

PS: fixed series

Sam5
2005-May-23, 04:25 PM
My notation is more in line with the title of this thread, which is:

“Do you think 0.9999999~ infinite 9s is exactly the equal to 1.” :D
Actually what VTBoy wrote was

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

Edited because I took your word for it, initially. I should have known better than to trust you to accurately quote someone else...

Actually, he wrote two different versions of the sentence.

I copied and pasted the title of his poll that is shown up at the top of this page, which is:

“Do you think 0.9999999~ infinite 9s is exactly the equal to 1.”

I now see that the actual title of the thread is a little different:

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

But this doesn’t really matter.

What matters is that he seems to have been talking about .999... equaling 1 in the field of calculus, so I won’t dispute that at all. If that’s a rule of calculus, then that’s a rule of calculus.

But if he was talking about the question only in relation to calculus, then he should have said so in the title, otherwise many people will come on this thread and give their opinions about different variations of the puzzle as applied to different circumstances that are not related specifically to the field of calculus.

Sam5
2005-May-23, 04:29 PM
I think my solution solves the basic puzzle.

There is no real puzzle, it's just that a lot of people misunderstand the notation. .999... means 9/10+9/100+9/1000+..., and there are a lot of webpages out there that will give you the simple formula for the sum of an infinite geometric series (http://mathworld.wolfram.com/Series.html). In this case, it is 1.0

Using pure mathematics only, .99999999…, with an infinite number of 9s, will never equal 1,

No, it does equal 1. .999... with a finite number of 9s will not equal 1.

PS: fixed series

I went back and read some of VTBoy’s original posts on page 1. He apparently was talking specifically about “calculus”. If there is a fixed law in the field of calculus that says .999... (with an infinite number of 9s) = 1, then so be it. :D

A Thousand Pardons
2005-May-23, 04:30 PM
But if he was talking about the question only in relation to calculus, then he should have said so in the title, otherwise many people will come on this thread and give their opinions about different variations of the puzzle as applied to different circumstances that are not related specifically to the field of calculus.
The only field where the professionals might dispute it, is mathematics, so that doesn't make a difference.

Grey
2005-May-23, 04:31 PM
But if he was talking about the question only in relation to calculus, then he should have said so in the title, otherwise many people will come on this thread and give their opinions about different variations of the puzzle as applied to different circumstances that are not related specifically to the field of calculus.
It's not just in the realm of calculus. It's true using the formal definitions of the real number system. I'm not sure why it would be necessary to specify that he meant "within the real number system", since what other system of numbers would you assume he meant by default?

(Oh, and here (http://mathforum.org/dr.math/faq/faq.0.9999.html)'s a page on the subject from the Dr. Math site, if you're more inclined to take their word on the matter than that of anyone here.)

Sam5
2005-May-23, 04:38 PM
If VTBoy was specifically talking about “calculus”, and if .999... = 1 in calculus, then there is nothing to debate. The thread is over, finished, kaput.

A Thousand Pardons
2005-May-23, 04:44 PM
If VTBoy was specifically talking about “calculus”, and if .999... = 1 in calculus, then there is nothing to debate. The thread is over, finished, kaput.
It's not just you that we have to convince.

PS: I see where VTBoy mentions calculus on page 1, but it's not in the setup of the OP, it's in one of the solutions.

Disinfo Agent
2005-May-23, 04:48 PM
I went back and read some of VTBoy’s original posts on page 1. He apparently was talking specifically about “calculus”.
Not that it makes any difference, but why do you think he was talking specifically about calculus?
He never rejected the algebraic proofs of the equality which were given in the first few pages.

Candy
2005-May-23, 04:49 PM
http://home.att.net/~candy.stair/graph.jpg

Lurker
2005-May-23, 05:14 PM
http://home.att.net/~candy.stair/graph.jpg
Exactly... except that you should consider this about the sequence... :o

"Listen and understand! [It's] out there. It can't be bargained with, it can't be reasoned with. It doesn't feel pity or remorse or fear, and it absolutely will not stop-EVER, until..."
-- Reese (The Terminator)

Well... you get the idea... 8)

Candy
2005-May-23, 05:19 PM
Moose made me do it. 8-[

Donnie B.
2005-May-23, 07:41 PM
[Ricky Ricardo voice] Ay ay ay!!!... [/Ricky Ricardo voice]
That's infinite !s after the last "ay"...

worzel
2005-May-23, 09:42 PM
Damnation! I could have predictated Sam5's eventual dispute with the wording of the opening post. Oh well, there goes my chance at Psychic Idol.

fosley
2005-May-23, 11:49 PM
Second, my final line of debate:
Add 9/10ths of the difference between X and 1:
X += 9 * (1 - X) / 10
Repeat this process forever.
What's X supposed to be, there?
X started as 0. I meant to write that, but I'm used to wussy Visual Basic programming, in which any numeric variable automatically starts at 0.

And the "+=" means you add what's on the right side to the variable on the left side. Again, I should have specified.

So:
0 + 9 * (1 - 0) / 10 = 9 / 10 = 0.9
0.9 + 9 * (1 - 0.9) / 10 = 0.9 + 9 * 0.1 / 10 = 0.9 + 0.9 / 10 = 0.9 + 0.09 = 0.99
etc.

No, what we're really discussing is the definition of decimal numerals in standard math, which is what people learn in college and high school.
No. If that were true, then we'd just break out the definition proposed by these books, and be done with it. What we are discussing is whether that definition describes reality. Now, we obviously can't look at something that is 0.9... inches long and something that is 1 inch long and see if they are the same or different, unless we can see to infinite resolution (I like that term, so I'm stealing it). So, we have to debate this on logic.

The &lt;people who say 0.9... is not 1> can't come up with a result for 1 - 0.9... that is not 0.
This goes back to the original problem that we say infinitesimal is not 0. Therefore, we *have* come up with an answer that isn't 0. Only if infinitesimal = 0 do we lose. Which is a problem I kept having earlier of people bringing things like "1/3 = 0.3... and 3 * 1/3 = 1 and 3 * 0.3... = 0.9... so 0.9... = 1" into play, even though I had already noted that 1/3-0.3... = something like infinitesimal / 3 . That number is not defined, because it is super-infinite resolution, but that doesn't mean it doesn't exist. sqrt(-1) isn't defined, but we still assume that it has meaning.

There is zero distance between 0.9... and 1, so they must be equal.
No. In the realm of integers, there is zero distance between 0 and 1. That is, you cannot go between 0 and 1. But that does not mean they are the same number. There is a difference of 1. Just like there is a difference of infinitesimal between 0.9... and 1. Which is not 0.

What is the average of 0.9... and 1?
In the set of integers, what is the average of 0 and 1? Or are they equal?

Donnie B.
2005-May-24, 12:17 AM
There is zero distance between 0.9... and 1, so they must be equal.
No. In the realm of integers, there is zero distance between 0 and 1. That is, you cannot go between 0 and 1. But that does not mean they are the same number. There is a difference of 1. Just like there is a difference of infinitesimal between 0.9... and 1. Which is not 0.
That's incorrect, on both counts. In integer arithmetic, the distance between 0 and 1 is the same as it is in real arithmetic: 1. And formal mathematics does not recognize infinitesimals.

The latter is the real clue to the misconception. In the real number domain, there is an infinite number of values that fall between any two values that are not equal, no matter how small. You cannot construct an infinitesimal that's small enough to violate this rule.

Way, way back on an early page of this thread, I challenged the group to define such a number (that is, a value that is infinitely close to 1 but not one) and work out a consistent mathematics that includes that number. I suggested treating it like i (or j, to us electrical engineers), the square root of -1 -- a value that doesn't exist but is treated as though it did. But so far, no one has taken up the gauntlet. How about you? Start with some simple things: pick a symbol for this infinitesimal, and show how it behaves under simple arithmetic operations. What happens when you add it to itself, or to a real number? Or subtract it? How do you do multiplication and division with it? What is its value squared, or to another power, or to its own power? If you can work that out you can move on to the more complex things like algebraic manipulations, trig, calculus, differential equations...

What is the average of 0.9... and 1?
In the set of integers, what is the average of 0 and 1? Or are they equal?
In integer arithmetic, the average of 1 and 0 is (1 + 0) / 2 = 0.

Grey
2005-May-24, 01:38 AM
No. If that were true, then we'd just break out the definition proposed by these books, and be done with it. What we are discussing is whether that definition describes reality.
We're talking here about pure mathematics. Those definitions are reality, since the real number system is something created by us. Whether the real numbers correspond to something external in the physical universe is another question (I'd certainly say the real numbers are a useful tool in modeling numerous aspects of the universe), not related to the issue at hand.

Lurker
2005-May-24, 02:26 AM
No. If that were true, then we'd just break out the definition proposed by these books, and be done with it. What we are discussing is whether that definition describes reality.
fosley -- limit theory is the underpinning of calculus. The idea that an infinite series converges to a finite value is the basis of the Fundamental Theorem of Calculus. You may not think that limit theory and therefore calculus represents reality, but before you head too far down that road, you might want to consider how much of science is described by calculus.

Michael Faraday pretty much laid out the laws of electromagnetism with out calculus and James Clerk Maxwell did it with calculus. So....
:wink:

This is what makes science sooooo fascist 42% of those who answered the poll in this thread say that these two quantities are not equal. Does science listen and do the "big tent thing"... nope

It simply says, "OK... then 42% of those polled are wrong!! [-("

That is soooooo fascist!!! [-X

Sam5
2005-May-24, 03:04 AM

This is what makes science sooooo fascist 42% of those who answered the poll in this thread say that these two quantities are not equal. Does science listen and do the "big tent thing"... nope

It simply says, "OK... then 42% of those polled are wrong!! [-("

That is soooooo fascist!!! [-X

It's just a simple internet message-board trick.

Lurker
2005-May-24, 03:16 AM

This is what makes science sooooo fascist 42% of those who answered the poll in this thread say that these two quantities are not equal. Does science listen and do the "big tent thing"... nope

It simply says, "OK... then 42% of those polled are wrong!! [-("

That is soooooo fascist!!! [-X

It's just a simple internet message-board trick.
Internet message board trick?? :o

Disinfo Agent
2005-May-24, 12:33 PM
Second, my final line of debate:
Add 9/10ths of the difference between X and 1:
X += 9 * (1 - X) / 10
Repeat this process forever.
According to your explanation (http://www.badastronomy.com/phpBB/viewtopic.php?p=475394#475394), the various values of X are the terms of the sequence:

0.0...
0.90...
0.990...
0.9990...
...

These are given by the formula 1 - 0.1^n, where n = 0, 1, 2, ...

Every time you do this, X gets ten times closer to 1 than it was one iteration previous to the current iteration:
X(N) = X(N - 1) / 10
But now there's also an X(N), which is something different. Your notation is a bit inconsistent. What is this X(N) supposed to be? You haven't defined it, although it seems you're thinking of the distance between the N-th term of the 'X' sequence and 1. Am I right?

Sylas
2005-May-24, 01:36 PM
0.999999~is just a weird way to write 1.

This is the key. "1.34e7" is not a number. It denotes a number, in a certain notation. 15 (in base 8 ) denotes the same number as 13 (in base 10).

0.9999999~~ as an infinite string of characters, is a different string of characters to 1.000000000~~ But they both denote the same number, if taken in the conventional base 10 notation. There is no other number for 0.999999999~ to denote in base 10, other than the number also denoted by 1.000000~

Thinking they are different is fundamentally a confusion between a notation, and the abstraction denoted by the abstraction.

[[ Edit to kill unwanted smilie with "8 )". ]]

Disinfo Agent
2005-May-24, 02:40 PM
No, what we're really discussing is the definition of decimal numerals in standard math, which is what people learn in college and high school.
No. If that were true, then we'd just break out the definition proposed by these books, and be done with it. What we are discussing is whether that definition describes reality.
I find the idea of a number standing just before 1 very unphysical, myself.

The &lt;people who say 0.9... is not 1> can't come up with a result for 1 - 0.9... that is not 0.
This goes back to the original problem that we say infinitesimal is not 0. Therefore, we *have* come up with an answer that isn't 0.
Except that you aren't able to produce it...

Which is a problem I kept having earlier of people bringing things like "1/3 = 0.3... and 3 * 1/3 = 1 and 3 * 0.3... = 0.9... so 0.9... = 1" into play, even though I had already noted that 1/3-0.3... = something like infinitesimal / 3 . That number is not defined, because it is super-infinite resolution, but that doesn't mean it doesn't exist. sqrt(-1) isn't defined, but we still assume that it has meaning.
The square root of -1 is a well defined concept. Infinitesimals, however, are never defined in high school or college, I believe.

What is the average of 0.9... and 1?
In the set of integers, what is the average of 0 and 1? Or are they equal?
The average of 0 and 1 is 0.5. If you can't work with fractional numbers, then you either round the result, as Donnie B. did, or leave it undefined.

Way, way back on an early page of this thread, I challenged the group to define such a number (that is, a value that is infinitely close to 1 but not one) and work out a consistent mathematics that includes that number. I suggested treating it like i (or j, to us electrical engineers), the square root of -1 -- a value that doesn't exist but is treated as though it did. But so far, no one has taken up the gauntlet.
Nitpick: j does exist. It's just not real. (Ain't mathematical terminology fun? :D)

We're talking here about pure mathematics. Those definitions are reality, since the real number system is something created by us. Whether the real numbers correspond to something external in the physical universe is another question (I'd certainly say the real numbers are a useful tool in modeling numerous aspects of the universe), not related to the issue at hand.
I would go so far as to say that 0.(9) and 1 being equal is what makes the most sense, physically. Just consider Zeno's paradoxes.

Grey
2005-May-24, 02:48 PM
Thanks for correcting the attribution before I could even respond!

I would go so far as to say that 0.(9) and 1 being equal is what makes the most sense, physically. Just consider Zeno's paradoxes.
I'd agree, actually, but when people on this thread start discussing how this problem relates to the physical world, they start talking about limits of measurement accuracy, or whether 0.999... is "close enough" to be considered equal to 1 from an engineering standpoint, or other things of that sort. I was trying to remind fosley (and anyone else still listening after 35 pages :)) that we're dealing here with the abstract realm of pure numbers.

Lurker
2005-May-24, 05:14 PM
Thanks for correcting the attribution before I could even respond!

I would go so far as to say that 0.(9) and 1 being equal is what makes the most sense, physically. Just consider Zeno's paradoxes.
I'd agree, actually, but when people on this thread start discussing how this problem relates to the physical world, they start talking about limits of measurement accuracy, or whether 0.999... is "close enough" to be considered equal to 1 from an engineering standpoint, or other things of that sort. I was trying to remind fosley (and anyone else still listening after 35 pages :)) that we're dealing here with the abstract realm of pure numbers.
But this is what gets me... it may be an abstract realm, but with very real physical world applications. If you drop a pencil the behavior of gravity acts according to laws develped with calculus and therefore limit theory. When you walk into a room and turn on a light or your PC, it obeys the laws of electromagnetic theory which are also expressed by calculus.

This is not about measurement; it's not about precision or accuracy. Its about reasoning in a logical fashion about the world around us. The mathematics is logical and consistent and observation demonstrates that the world behaves according to the laws derived from the mathematics.

fosley
2005-May-24, 08:06 PM
Infinitesimals would behave the same way with each other as integers work with each other. Because an infinitesimal is infinitely small, it would never reach the realm of finite numbers unless you multiplied it by infinity. Basically, you would work with the last number, something like 0.9...9 - 0.0...1 = 0.9...8 . Now, I keep getting told that you can't have a "last digit", because infinity means "never-ending". However, as I've stated before, you can have two ends, and a never-endingly infinite number of digits in between.

An example: let's say you take a line between two physical points, and each point on that line represents a digit. There are a never-endingly infinite number of points between the ends of the line (I suppose that would be a segment, since it has two ends), but there are still two ends. So the first point would represent 0, the second point would represent the decimal point, the last points (as needed) would represent the numbers you are playing with, and all of the other infinite points would represent one of the infinite 9's.

I think 0.9... would have to be the closest decimal number to 1 without equaling 1. Now, there may be numbers between 1 and 0.9..., but they can't be defined (just like the average of 0 and 1 wouldn't be defined). Or you could round it up to 1 (since 0.5 rounds up, by common standards--and that's not a definition, just something people agreed to do to make life simpler), but only as an approximation for dealing with reality (just like rounding 0.9... as a perfectly reasonable approximation to 1).

Again, I will state my thought that saying infinitesimal and 0 are the same is no different than calling 0 and 1 the same. On an infinitely long number line, 0 and 1 are right next to each other. They might as well be equal, from the perspective of anyone who could actually see the entire number line. But that doesn't change the fact that, to all us finite beings, 0 and 1 are very different. To us finite beings, 0 and infinitesimal might as well be the same, but to an infinitesimal being, they would be very different. Also, to an infinitesimal being, things like 1 and 2 would be infinitely large, so they would be no different.

If infinitesimal = 0, then why is division by 0 undefined?
(note: I couldn't make a strikeout, so terms with a backslash were "struck-out" to show cancellation)

N 0
---------- = ---
infinity

N * infinity 0 * infinity
-------------- = --------------
infinity

N * infinity 0 * infinity
-------------- = --------------
infinity * 0 0

N * \infinity\ \0\ * infinity
---------------- = ----------------
\infinity\ * 0 \0\

N infinity
--- = ----------
0

Now, if that (or something else I've said) doesn't either convince all of you that I'm right, or prompt you to say something that convinces me you are right, I'm going to withdraw from this discussion until I get done with all my college math, algebra, calculus, etc. I think that would probably have a much higher chance of showing me what I need to see, or give me a much more stable base to debate from. And I feel I've used enough (probably too much, but I'll never admit to that 8) ) of our collective time. Of course, you will all probably hunt me down and murder me if I actually bump this post yet again several years from now, but hey! it's all good.

Thanks for the debate, sorry if I offended anyone unduly, and see you on some other thread (I'm sure).

Disinfo Agent
2005-May-24, 08:40 PM
Infinitesimals would behave the same way with each other as integers work with each other. Because an infinitesimal is infinitely small, it would never reach the realm of finite numbers unless you multiplied it by infinity. Basically, you would work with the last number, something like 0.9...9 - 0.0...1 = 0.9...8 . Now, I keep getting told that you can't have a "last digit", because infinity means "never-ending".
Not exactly.
It's just that you don't need that last digit to cover all the real numbers. All you need are infinite strings with a first digit to the left, a decimal point, and no last digit to the right. What we call "sequences", in math.

An example: let's say you take a line between two physical points, and each point on that line represents a digit. There are a never-endingly infinite number of points between the ends of the line (I suppose that would be a segment, since it has two ends), but there are still two ends. So the first point would represent 0, the second point would represent the decimal point, the last points (as needed) would represent the numbers you are playing with, and all of the other infinite points would represent one of the infinite 9's.
Bad representation. Different kind of infinity. There are many more points in that line than the digits you'd ever need to write down a real number in the decimal notation. :)

Again, I will state my thought that saying infinitesimal and 0 are the same is no different than calling 0 and 1 the same. On an infinitely long number line, 0 and 1 are right next to each other. They might as well be equal, from the perspective of anyone who could actually see the entire number line.
The real line is always infinitely long, but there's still a distance of one unit between 0 and 1. It wouldn't make much sense to call it 'the number 1' otherwise, don't you think?

If infinitesimal = 0, then why is division by 0 undefined?
In some contexts we do define division by zero, but it's often too messy to be of practical use. Many of the properties that other fractions have don't work for divisions by zero.

N 0
---------- = ---
infinity

N * infinity 0 * infinity
-------------- = --------------
infinity

N * infinity 0 * infinity
-------------- = --------------
infinity * 0 0

N * \infinity\ \0\ * infinity
---------------- = ----------------
\infinity\ * 0 \0\

N infinity
--- = ----------
0

Just as it isn't 'legal' to divide each side of an equation by zero, it isn't legal to multiply it by infinity, either.

Now, if that (or something else I've said) doesn't either convince all of you that I'm right, or prompt you to say something that convinces me you are right, I'm going to withdraw from this discussion until I get done with all my college math, algebra, calculus, etc. I think that would probably have a much higher chance of showing me what I need to see, or give me a much more stable base to debate from.
An excellent idea. 8)

Donnie B.
2005-May-24, 08:56 PM
What is the average of 0.9... and 1?
In the set of integers, what is the average of 0 and 1? Or are they equal?
The average of 0 and 1 is 0.5. If you can't work with fractional numbers, then you either round the result, as Donnie B. did, or leave it undefined.
Nitpick right back at ya... I didn't round. If I'd done what you suggest, the result would have been 1 (the rule is, .5 and up round upward).

But in integer arithmetic there is no such thing as 0.5. Ask any microprocessor (without a floating-point unit). Add the integers 0 and 1, and you get 1. Divide by 2, and you get zero -- not 1/2 or 0.5. So, 0 is the average of 0 and 1 in the integer domain.

Disinfo Agent
2005-May-24, 09:00 PM
Nitpick right back at ya... I didn't round. If I'd done what you suggest, the result would have been 1 (the rule is, .5 and up round upward).
It depends on whether we're rounding up or down... :wink:

mopc
2005-May-25, 01:51 AM
So there is no smallest thing before 1, thus any number smaller than one has infinite numbers between it and 1, so 0,999... is smaller than one.
How's that? I don't think I follow you.
How does the fact that:

1) Any number smaller than one has infinite numbers between it and 1

imply the hypothesis that

2) 0,999... is smaller than one? :-?

The problem is that infinity does not exist. It would take forever for anyone to calculate infinite decimal 9s. So 0,9999..... is less than 1.

Mathematicaly, how do you denote the smallest fraction after 1???

Lycus
2005-May-25, 02:03 AM
The problem is that infinity does not exist.
So how many distinct points is there on a number line between 1 and 2?

It would take forever for anyone to calculate infinite decimal 9s. So 0,9999..... is less than 1.
If you don't think that infinity exists, then what do you think you're representing with that ellipsis?

Mathematicaly, how do you denote the smallest fraction after 1???
As it's been said, such a number does not exist.

Lurker
2005-May-25, 02:09 AM
So there is no smallest thing before 1, thus any number smaller than one has infinite numbers between it and 1, so 0,999... is smaller than one.
How's that? I don't think I follow you.
How does the fact that:

1) Any number smaller than one has infinite numbers between it and 1

imply the hypothesis that

2) 0,999... is smaller than one? :-?

The problem is that infinity does not exist. It would take forever for anyone to calculate infinite decimal 9s. So 0,9999..... is less than 1.

Mathematicaly, how do you denote the smallest fraction after 1???

Um... so... PI is an infinite series of digits and yet no one seems to have a problem with the idea that PI represents C/D for a circle. So, we let Sigma represent the value defined by the digit '0' followed by a decimal point followed by an infinite series of '9's. So now Sigma represents this value just as PI represents the irrational number defined by the Circumference of a circle divided by its Diameter.

Tell me... why are we still discussing this, the concepts are only covered on about 1000 first semester calculus text books. Have those who still believe the values are not equal bothered to learn a bit of limit theory??

01101001
2005-May-25, 02:17 AM
The problem is that infinity does not exist. It would take forever for anyone to calculate infinite decimal 9s. So 0,9999..... is less than 1.

0.999... is not a calculation, any more than 1/9 is a calcluation. It is a symbol.

It symbolizes the same entity that "1", "1.0" and "1.000000..." symbolize.

Does it take you forever to figure out what 1/9 is? How do you function is this world?

mopc
2005-May-25, 02:38 AM
All of mathematics is symbols in our mind. It's pertty close to reality, but not reality. Theoretically, 0,9999.... is 1, but since in reality no one will wait until those 9s reach infinity, it'll just be something very close to 1. In reality, however, the difference will be negligible so it's just easier to say 1.

But what is 1? Numbers are for measuring and counting, right? You have 1 pencil, than you sharpen it, is it still one pencil or 0,9999999 pencil?

01101001
2005-May-25, 02:44 AM
All of mathematics is symbols in our mind.

So, what is the sum of 9 instances of 1/9? Will this inifinte calculation take you forever to answer? Is it not equal, less than 1?

skwirlinator
2005-May-25, 02:45 AM
All of mathematics is symbols in our mind. It's pertty close to reality, but not reality. Theoretically, 0,9999.... is 1, but since in reality no one will wait until those 9s reach infinity, it'll just be something very close to 1. In reality, however, the difference will be negligible so it's just easier to say 1.

But what is 1? Numbers are for measuring and counting, right? You have 1 pencil, than you sharpen it, is it still one pencil or 0,9999999 pencil?

It is now less than 1 pencil

if you sharpened it to 1/2 it would be a 1/2 pencil

if you have a pen and a pencil and sharpen the pencil you still have a pen and a pencil= even if you only have 1/2 a pencil.

Lurker
2005-May-25, 02:50 AM
All of mathematics is symbols in our mind. It's pertty close to reality, but not reality. Theoretically, 0,9999.... is 1, but since in reality no one will wait until those 9s reach infinity, it'll just be something very close to 1. In reality, however, the difference will be negligible so it's just easier to say 1.

But what is 1? Numbers are for measuring and counting, right? You have 1 pencil, than you sharpen it, is it still one pencil or 0,9999999 pencil?
But this is where you are mistaken... Everytime you turn on your computer, the electromagnetic forces that you unleash obey Maxwell's Equations. Maxwell's Equations are a representation of these forces that are based on Calculus and therefore on limit theory. As a result the concept of infinity and the properties of infinite series are a part of your every day life. Even if you lived 200 years ago, infinite series would still have been part of your every day life. The laws of gravitational attraction are also described through calculus. So every time you dropped something and it fell, you would still have been witnessing forces that obey laws that are described by the concepts and properties of infinite series.

Applications of limit theory are all around us... Limit theory may be abstract, but it's application is the physical world in which we live!!

mopc
2005-May-25, 03:01 AM
All of mathematics is symbols in our mind. It's pertty close to reality, but not reality. Theoretically, 0,9999.... is 1, but since in reality no one will wait until those 9s reach infinity, it'll just be something very close to 1. In reality, however, the difference will be negligible so it's just easier to say 1.

But what is 1? Numbers are for measuring and counting, right? You have 1 pencil, than you sharpen it, is it still one pencil or 0,9999999 pencil?

It is now less than 1 pencil

if you sharpened it to 1/2 it would be a 1/2 pencil

if you have a pen and a pencil and sharpen the pencil you still have a pen and a pencil= even if you only have 1/2 a pencil.

What if you didnt know it had been sharpened? By the way, have you ever borrowed 0,9999999 pencil?

Fortis
2005-May-25, 03:04 AM
All of mathematics is symbols in our mind. It's pertty close to reality, but not reality. Theoretically, 0,9999.... is 1, but since in reality no one will wait until those 9s reach infinity, it'll just be something very close to 1. In reality, however, the difference will be negligible so it's just easier to say 1.
You can imagine doing it in a finite time.

Write "0."
Wait 16 minutes and write down the first "9"
Wait 8 minutes and write another "9"
Wait 4 minutes and write another "9"
Wait 2 minutes and write another "9"
Wait 1 minute and write another "9"
etc.

If you can keep up, I guarantee that you will have written out the entire series in no more than 32 minutes. ;) :)

mopc
2005-May-25, 03:07 AM
All of mathematics is symbols in our mind. It's pertty close to reality, but not reality. Theoretically, 0,9999.... is 1, but since in reality no one will wait until those 9s reach infinity, it'll just be something very close to 1. In reality, however, the difference will be negligible so it's just easier to say 1.

But what is 1? Numbers are for measuring and counting, right? You have 1 pencil, than you sharpen it, is it still one pencil or 0,9999999 pencil?
But this is where you are mistaken... Everytime you turn on your computer, the electromagnetic forces that you unleash obey Maxwell's Equations. Maxwell's Equations are a representation of these forces that are based on Calculus and therefore on limit theory. As a result the concept of infinity and the properties of infinite series are a part of your every day life. Even if you lived 200 years ago, infinite series would still have been part of your every day life. The laws of gravitational attraction are also described through calculus. So every time you dropped something and it fell, you would still have been witnessing forces that obey laws that are described by the concepts and properties of infinite series.

Applications of limit theory are all around us... Limit theory may be abstract, but it's application is the physical world in which we live!!

What if you add those infinte decimal 9s, wouldnt it be an infinite number? No. But why?

I'm still puzzled by the lack of a notation for the smallest thing after 1, which doesnt exist, but since we can write the smallest thing before 1, which doesnt exist either, we could write its 'opposite'.

mopc
2005-May-25, 03:11 AM
All of mathematics is symbols in our mind. It's pertty close to reality, but not reality. Theoretically, 0,9999.... is 1, but since in reality no one will wait until those 9s reach infinity, it'll just be something very close to 1. In reality, however, the difference will be negligible so it's just easier to say 1.
You can imagine doing it in a finite time.

Write "0."
Wait 16 minutes and write down the first "9"
Wait 8 minutes and write another "9"
Wait 4 minutes and write another "9"
Wait 2 minutes and write another "9"
Wait 1 minute and write another "9"
etc.

If you can keep up, I guarantee that you will have written out the entire series in no more than 32 minutes. ;) :)

I don't think so. The kinetic energy of my frantic fingers would surpass light speed soon after the 20 minute and create a black hole, engulfing our galaxy. So infinity does not exist!!!!!!!!!

Lycus
2005-May-25, 03:11 AM
I'm still puzzled by the lack of a notation for the smallest thing after 1, which doesnt exist, but since we can write the smallest thing before 1, which doesnt exist either, we could write its 'opposite'.
We can't write the smallest thing before 1 either, because it doesn't exist. 0.9999... is not less than 1.

mopc
2005-May-25, 03:15 AM
I'm still puzzled by the lack of a notation for the smallest thing after 1, which doesnt exist, but since we can write the smallest thing before 1, which doesnt exist either, we could write its 'opposite'.
We can't write the smallest thing before 1 either, because it doesn't exist. 0.9999... is not less than 1.

But at least you can write it!

Fortis
2005-May-25, 03:22 AM
All of mathematics is symbols in our mind. It's pertty close to reality, but not reality. Theoretically, 0,9999.... is 1, but since in reality no one will wait until those 9s reach infinity, it'll just be something very close to 1. In reality, however, the difference will be negligible so it's just easier to say 1.
You can imagine doing it in a finite time.

Write "0."
Wait 16 minutes and write down the first "9"
Wait 8 minutes and write another "9"
Wait 4 minutes and write another "9"
Wait 2 minutes and write another "9"
Wait 1 minute and write another "9"
etc.

If you can keep up, I guarantee that you will have written out the entire series in no more than 32 minutes. ;) :)

I don't think so. The kinetic energy of my frantic fingers would surpass light speed soon after the 20 minute and create a black hole, engulfing our galaxy. So infinity does not exist!!!!!!!!!
I never claimed that it wouldn't destroy the universe... ;)
(Now getting worried about being sued for liability... :o )

mopc
2005-May-25, 03:27 AM
All of mathematics is symbols in our mind. It's pertty close to reality, but not reality. Theoretically, 0,9999.... is 1, but since in reality no one will wait until those 9s reach infinity, it'll just be something very close to 1. In reality, however, the difference will be negligible so it's just easier to say 1.
You can imagine doing it in a finite time.

Write "0."
Wait 16 minutes and write down the first "9"
Wait 8 minutes and write another "9"
Wait 4 minutes and write another "9"
Wait 2 minutes and write another "9"
Wait 1 minute and write another "9"
etc.

If you can keep up, I guarantee that you will have written out the entire series in no more than 32 minutes. ;) :)

I don't think so. The kinetic energy of my frantic fingers would surpass light speed soon after the 20 minute and create a black hole, engulfing our galaxy. So infinity does not exist!!!!!!!!!
I never claimed that it wouldn't destroy the universe... ;)
(Now getting worried about being sued for liability... :o )

Besides, the exercise would end a few seconds before completion, so there would still be infinite 9s to write, and infinite universes to destroy 8-[

Chuck
2005-May-25, 03:54 AM
There are other ways of representing nines than by writing them. You could use a domino sliding along a ruler. When the domino has moved 0.9 inches it represents 1 nine. 0.99 inches represents 2 nines. 0.999 inches, 3 nines. Etc, Etc. When you arrive at one inch you've moved the domino through an infinity of such points so one inch represents an infinity of nines. Since you can easily push a domino one inch you can represent 0.99999... this way. Stopping the domino short of one inch, no matter how small the difference, won't do it since you'd have pushed it through a finite number of such points. The whole inch is sufficient and required.

Lurker
2005-May-25, 04:01 AM
mopc -- I would suggest that if you are interested you should go get a first semester calculus book and do a little study on limit theory. If it can get through my fool bone head, it can't be that difficult. It might clear up a lot of questions for you.

Grey
2005-May-25, 04:04 AM
All of mathematics is symbols in our mind. It's pertty close to reality, but not reality.
I again take a different tack than Lurker when addressing this objection. We're talking about pure mathematics. The only reality that these symbols have is the one in our mind. When we write the numbers 1 or 0.999..., we aren't referring to something scribed on a tablet in space somewhere, we're referring to nothing other than the abstract concepts that we've invented: the real number system. It might in principle be possible to develop a consistent number system where 0.999... and 1 represent different numbers, but I'm unaware of anyone that has done so, and even if you did, that system would be something other than the real numbers.

mopc
2005-May-25, 04:47 AM
mopc -- I would suggest that if you are interested you should go get a first semester calculus book and do a little study on limit theory. If it can get through my fool bone head, it can't be that difficult. It might clear up a lot of questions for you.

No I prefer to bug you guys until you wanna kill yourselves! \:D/

Lurker
2005-May-25, 05:02 AM
mopc -- I would suggest that if you are interested you should go get a first semester calculus book and do a little study on limit theory. If it can get through my fool bone head, it can't be that difficult. It might clear up a lot of questions for you.

No I prefer to bug you guys until you wanna kill yourselves! \:D/
Well I think you're pretty much there... you might want to think about declaring victory before one of us decides to act!! [-(

:P

mopc
2005-May-25, 05:09 AM
mopc -- I would suggest that if you are interested you should go get a first semester calculus book and do a little study on limit theory. If it can get through my fool bone head, it can't be that difficult. It might clear up a lot of questions for you.

No I prefer to bug you guys until you wanna kill yourselves! \:D/
Well I think you're pretty much there... you might want to think about declaring victory before one of us decides to act!! [-(

:P

No I still have one more itsy bitsy objection... I'm kidding...

Lurker
2005-May-25, 05:17 AM
Michael Corleone: Just when I thought that I was out they pull me back inWhy do I let this happen to me in these threads!! #-o

A Thousand Pardons
2005-May-25, 07:02 AM
I'm still puzzled by the lack of a notation for the smallest thing after 1, which doesnt exist, but since we can write the smallest thing before 1, which doesnt exist either, we could write its 'opposite'.
We can't write the smallest thing before 1 either, because it doesn't exist. 0.9999... is not less than 1.
You meant the largest thing before 1, right?

Lycus
2005-May-25, 07:12 AM
I'm still puzzled by the lack of a notation for the smallest thing after 1, which doesnt exist, but since we can write the smallest thing before 1, which doesnt exist either, we could write its 'opposite'.
We can't write the smallest thing before 1 either, because it doesn't exist. 0.9999... is not less than 1.
You meant the largest thing before 1, right?
Yeah. I noticed it later, but didn't bother fixing it 'cause mopc made the same mistake and I figured he'd know what I meant. :)

A Thousand Pardons
2005-May-25, 07:16 AM
Michael Corleone: Just when I thought that I was out they pull me back inWhy do I let this happen to me in these threads!! #-o
Gotcha

worzel
2005-May-25, 08:45 AM
Assuming those dots represent a finite number of digits, that'd equal 0,9999999999999999999.....6.......7. If they represent infinite digits then the number is not defined - what is the place value of the first 6, for instance?

Fram
2005-May-25, 09:42 AM
I guess you mean 0.999999999999999....7..... ? You had a 6.... too much there.

worzel
2005-May-25, 09:48 AM
I guess you mean 0.999999999999999....7..... ? You had a 6.... too much there.
It depends on whether the dots always represent 9s or always represent more of the same digit to the left of the dots. Either way, there's no ... after the 7. It's all pretty moot though, if finite then we'd just write the numbers in anyway - I think mpoc intending for the dots to represent infinite digits.

Fram
2005-May-25, 10:08 AM
Right, my mistake. What I meant was t oreplace 0.99999...6999... by 0.99999...7, but it could just as well be read as 0.99999... 66666...9999..., and then your 0.99999...6666....7 is correct. Either way, it stops at the 7, so again we have two ways of writing the same number (though both wrong in the ... in the middle), just like in the original question.

worzel
2005-May-25, 12:10 PM
We're allowing ourselves to be forced into the ill-defined teritory of the inequality believers! :)

Disinfo Agent
2005-May-25, 01:18 PM
Your notation is ambiguous. You have to explain what each ellipsis stands for. Do you mean:

a) 0,9999999999999999999 followed by a couple of nines, followed by 6, followed by a couple of nines, followed by 99999, followed by a denumerable infinity of nines?

b) 0,9999999999999999999 followed by a couple of sixes, followed by 6, followed by a couple of sixes, followed by 99999, followed by a denumerable infinity of nines?

c) 0,9999999999999999999 followed by a couple of nines, followed by 6, followed by a couple of sixes, followed by 99999, followed by a denumerable infinity of nines?

d) Something else?

What if you add those infinte decimal 9s, wouldnt it be an infinite number? No. But why?
For much the same reason that 1/2 + 1/4 + 1/8 + ... is not an infinite number, that an infinite sum can add up to a finite number.

Lance
2005-May-25, 05:46 PM
They are now debating this on GLP (http://godlikeproductions.com/bbs/message.php?message=110969). If you can stand the occasional "[bad word deleted]", "[bad word deleted]" &amp; "[bad word deleted]" it's rather amusing.

Note:
I didn't actually type "bad words" that got filtered out. I intentionally typed it exactly as it appears for the humor value.

Lurker
2005-May-25, 05:49 PM
They are now debating this on GLP (http://godlikeproductions.com/bbs/message.php?message=110969). If you can stand the occasional "[bad word deleted]", "[bad word deleted]" &amp; "[bad word deleted]" it's rather amusing.

Note:
I didn't actually type "bad words" that got filtered out. I intentionally typed it exactly as it appears for the humor value.
Now who would have started a firefight like that over there?? :wink:

Lance
2005-May-25, 05:59 PM
Now who would have started a firefight like that over there?? :wink:

I can't imagine... Not a clue. \:D/

skwirlinator
2005-May-25, 06:18 PM
You have given them Mudd's Women, Ah, But Fitting...

Lance
2005-May-25, 06:21 PM
You have given them Mudd's Women, Ah, But Fitting...

I got tired of arguing with them about chemtrails so I thought I would try to turn them loose on themselves and see how they like it.

So far, it has worked out quite well.

Lurker
2005-May-25, 07:50 PM
You have given them Mudd's Women, Ah, But Fitting...

I got tired of arguing with them about chemtrails so I thought I would try to turn them loose on themselves and see how they like it.

So far, it has worked out quite well.
Its all a government plot!! :o

We have no idea what they are really hiding in all those '9''s... 8-[

THE TRUTH IS OUT THERE!!!

:o

Lance
2005-May-25, 08:44 PM
A few of them are coming up with some pretty good, albeit twisted arguments in support of the inequality camp.

skwirlinator
2005-May-26, 03:21 AM
http://voyager.jpl.nasa.gov/spacecraft/images/image003.gif

We told ET this is ONE

mickal555
2005-May-26, 06:48 AM
Cough

monster
2006-Jan-04, 04:54 PM
Furgeson Jenkins

Thomas(believer)
2006-Jan-04, 05:39 PM
I did not read all replies in this thread. But did someone think about the following question. What is greatest real value just not equal to 1?
I would say this equals: 0.999~

JohnW
2006-Jan-04, 06:07 PM
Oh no. It's back...

For new readers: the poll can be interpreted as:

Do you know anything about mathematics?
( ) Yes
( ) No

Moose
2006-Jan-04, 06:10 PM
There was a followup poll when this was settling down that was overwhelmingly "yes", except for a few holdouts.

But yeah, this is the thread that simply won't die.

Thomas, you probably should read this thread, or at least the first ten pages of it. Nearly all of the proofs are covered in detail by that point.

hhEb09'1
2006-Jan-04, 06:14 PM
I did not read all replies in this thread. But did someone think about the following question. What is greatest real value just not equal to 1?Yes. There is no such value.

I would say this equals: 0.999~But that equals 1! :)

Thomas(believer)
2006-Jan-04, 06:25 PM
I know a little mathematics.
I would say 1 belongs to the closed set e.g. [-1,1]
and 0.999~ belongs to the open set e.g. <-1,1>

I agree sum(0.9x10^-n) where n takes integer values from 0 to inifinity equals 1.

mickal555
2006-Jan-04, 06:58 PM
Who dares to awaken this thread, I sence a great disurbance...

Dragon Star
2006-Jan-04, 07:03 PM
Who dares to awaken this thread, I sence a great disurbance...

Whokay master Jedi....:rolleyes: :D

Taks
2006-Jan-04, 07:05 PM
I know a little mathematics.
I would say 1 belongs to the closed set e.g. [-1,1]
and 0.999~ belongs to the open set e.g. <-1,1>

I agree sum(0.9x10^-n) where n takes integer values from 0 to inifinity equals 1.
the open set <-1,1> does not include 1 therefore by your logic, 0.999~ != 1. yet you follow to say (0.9x10^-n) summed to infinity = 1. a contradiction since 0.999~ = (0.9x10^-n) summed to infinity.

taks

Candy
2006-Jan-04, 07:32 PM
Thank you, the percentage for the poll adds up to 100% now. :D

umop ap!sdn
2006-Jan-04, 07:38 PM
If .999~ = 1, then does it follow that the reciprocal of zero is infinity?

Apologies if this has already been covered - I'm still on page 3. :D

Thomas(believer)
2006-Jan-04, 07:43 PM
the open set <-1,1> does not include 1 therefore by your logic, 0.999~ != 1. yet you follow to say (0.9x10^-n) summed to infinity = 1. a contradiction since 0.999~ = (0.9x10^-n) summed to infinity.

taks

I think a nice way to notate 1> is 0.999~. But probably is not a mathematical convention.
And of course is not the same. Look at it. 1 contains 1 digit. 0.999~ contains infinite digits (and a dot).:)

Candy
2006-Jan-04, 07:48 PM
Must find old Moose diagram... classic! :clap:

http://home.att.net/~candy.stair/graph.jpg

Disinfo Agent
2006-Jan-04, 08:29 PM
There was a followup poll when this was settling down that was overwhelmingly "yes", except for a few holdouts.Not quite (http://www.bautforum.com/showthread.php?t=16806), I'm afraid. :)

Taks
2006-Jan-04, 08:46 PM
I think a nice way to notate 1> is 0.999~. But probably is not a mathematical convention.not only not mathematical convention, but not correct. 1> means "up to, but not inclusive of, 1." 1] means "up to and including 1." the latter is a set that includes 0.999~, the former is not. this has been shown many times (there are probably half a dozen or so unique proofs for this, probably all mentioned within this thread).

And of course is not the same. Look at it. 1 contains 1 digit. 0.999~ contains infinite digits (and a dot).:)1 contains infinite digits, too, we just don't write them all out: 1.000~ or 0.999~. i.e. the integer notation "1" is a sort of shorthand representation.

taks

Thomas(believer)
2006-Jan-04, 08:51 PM
But I do agree.
What is proven is that the limit of this sequence equals 1.
For that I know maths well enough.
My point is the following. Inifnity is a mathmatical abstraction. It is a very strange niumber, because infinity + something still equals infinity.
And with this strange property of the "number" infinity, all the proofs are very correct.
I like to think of 0.999~ as 1> which equals 1] when I replace > for ].

Or when I replace ~ with an infinite number of 9's
[edit 2] My last comment on this. "1" represents an integer value, 0.999~ a sequence. It's like telling the same story in a different way. End that's where it is different[/edit 2]

Hugh Jass
2006-Jan-04, 09:51 PM
But I do agree.
What is proven is that the limit of this sequence equals 1.
For that I know maths well enough.
My point is the following. Inifnity is a mathmatical abstraction. It is a very strange niumber, because infinity + something still equals infinity.
And with this strange property of the "number" infinity, all the proofs are very correct.
I like to think of 0.999~ as 1> which equals 1] when I replace > for ].

Or when I replace ~ with an infinite number of 9's
[edit 2] My last comment on this. "1" represents an integer value, 0.999~ a sequence. It's like telling the same story in a different way. End that's where it is different[/edit 2]

Thanks, Thomas, you've just convinced me to think like everybody else. I have known for years what I was supposed to answer just didn't like it. I've been thinking along the lines of this last post. I had never really thought much about it or else I would have given the correct answer years ago. Basically re-read what you just posted ~ (infinity) only equals infinity. As such if you want to write 1.00000~ != 0.999999~ as a sequence they are equal because whatever you put in front of the ~ is academic.

There for in order to define 0.99999~ in anyway you MUST (this is the part that has bothered me since I first heard of this problem my freshmen year of high school) define it as a number, (I see the ~ and I don't want to round it, but you have to), otherwise the 0.99999 part of ~ has no meaning. at that point it has a limit, then enter any number of the dozen proofs given thought this thread. I suppose you could say the answer is NO because
~!=1
I'm sure that is in this thread somewhere right? But for the question as soon as you want to use 0.999999 or whatever other symbol you desire, you are expressing something to be defined, in order to define it... see previous proofs.

Thanks again, everytime i've heard the question it has bothered me, and the simple proofs just didn't do it for me.

Taks
2006-Jan-04, 10:32 PM
What is proven is that the limit of this sequence equals 1.
uh, no. what is proven is that the two numbers are equal. there is no limit involved, i.e. 0.999~ does not approach 1, it is equal to 1.

i realize that you agree 0.999~ and 1 are equal. however, the result of this is that the former really is in the set defined 1]. the "closed" notation is very intentional.

My point is the following. Inifnity is a mathmatical abstraction. It is a very strange niumber, because infinity + something still equals infinity.depends upon your point of view. take a lot of higher level math classes and you may not see it as strangely.

And with this strange property of the "number" infinity, all the proofs are very correct.
i suppose. but many of these proofs never even use the concept of infinity (other than the tilde shorthand representation) or a limit.

I like to think of 0.999~ as 1> which equals 1] when I replace > for ].i see where you're coming from but you should shake this concept from your head. the distinction between open and closed sets is purposeful (though i usually use a parenthesis for open). they mean different things, and not just when thinking of limits.

It's like telling the same story in a different way. true, but there's a reason for convention. there's a reason the notation means what it means. it is so we can all communicate the same concepts using the same language without having to repeatedly explain ourselves.

btw, on this last topic, i should point out the very real difference in notation between mathematics and engineering disciplines. in general, they are similar, though often differ enough that you have to double-check the source to see who wrote it. it's bad even within various engineering circles, with some writers preferring certain notations over others (mostly dealing with matrices and vectors). ugh. very hard given that i bounce between the two in my recent classes.

taks

Hugh Jass
2006-Jan-04, 10:43 PM
i suppose. but many of these proofs never even use the concept of infinity (other than the tilde shorthand representation) or a limit.

See this is the part that has always bothered me. If infinity is part of the question how do you just not use it or convert it in the proof. Did my last post/explanation to myself make any sense?

Granted I got very lost toward the end of the second semester of calculus (as high as I went), and I haven't done anything resembling math since then, for oh about 10 or so years, it has always bothered me when I reach the point of not understanding. for me math is either incredibly simple or just impossible there is almost no middle ground.

Candy
2006-Jan-04, 10:57 PM
Can we remind viewers that there are several different math "theories" in question, and you guys are fighting the battle of one of those? :shifty:

Taks
2006-Jan-04, 11:05 PM
See this is the part that has always bothered me. If infinity is part of the question how do you just not use it or convert it in the proof.technically, it is not part of the proof. my fave is...

1/9 = 0.111~
9*(1/9) = 9*0.111~
1 = 0.999~

the concept of infinity is never used. the fact that the tilde is there (normally a bar over the last digit, but hard to do in here) is nothing more than shorthand notation. nowhere are you actually manipulating infinity.

Did my last post/explanation to myself make any sense?sort of confusing, actually.

for me math is either incredibly simple or just impossible there is almost no middle ground.for me it is life. :)

taks

Taks
2006-Jan-04, 11:05 PM
Can we remind viewers that there are several different math "theories" in question, and you guys are fighting the battle of one of those? :shifty:? which ones?

taks

Hugh Jass
2006-Jan-04, 11:18 PM
sort of confusing, actually.

Figured as much. Ah well, next time the question comes up i'll play ball and answer the way I'm supposed to. I guess I have no issue with the actual hard and fast math, it is the conceptual side of it that just doesn’t intuitively sit well with me. I can move on to other more important things like PMS stings now.

Candy
2006-Jan-04, 11:26 PM
? which ones?

taks
Don't make me beat you! Ask milli360, he is the math brain.

William_Thompson
2006-Jan-04, 11:59 PM
I am trying to see how well people at BABB understand this, because other forums seem to show most people have a poor grasp of this idea.

There is a correct answer to the question. When I posted this in another forum, it was really surpised at how most people answered. So now I want to see how people here accept the correct answer. The correct answer is they are exactly equal. There is no difference between the two.

Another bit of evidence that the human species is not intelligent.
:(

William_Thompson
2006-Jan-05, 12:01 AM
Candy,

one third plus one third plus one third equals exactly one

(1/3 + 1/3 + 1/3 = 1)

and the only way to show this in decimals is

0.333~ + 0.333~ + 0.333~ = 0.999~

Candy
2006-Jan-05, 12:04 AM
Grrrrrrrrrr!

Taks
2006-Jan-05, 12:18 AM
it's really all one theory... though the believer's questions are delving into the semantics of sets and real numbers (believe it or don't, infinity itself really does not mean anything here).

huge, i understand the conceptual issue for you, and it takes some work to get around. i've spent so many years dealing with it all, it has become second nature, really.

now, compare my background (probably 60 credits of pure math, 150 credits total counting "math related" subjects) to true mathemeticians and the conceptual gap is even larger than between you and i. my weirdest professors have always been the math guys. very weird.

taks

PS: beatings are welcome. after today, i feel like a masochist anyway... torturing myself over this darned processor that just can't fill its pipe properly... ugh. i digress. :)

Candy
2006-Jan-05, 12:26 AM
I still think there is something wrong with this theory. :D

Candy
2006-Jan-05, 12:27 AM
I accept it, but there is something seriously wrong. :shifty:

Dragon Star
2006-Jan-05, 12:28 AM
I still think there is something wrong with this theory. :D

Their is, the fact that it is going on it's 1,000th post!

Hugh Jass
2006-Jan-05, 12:32 AM
If it only makes it to 999 it'll be the same as 1000 right? ;)

kucharek
2006-Jan-05, 12:37 AM
Wasn't there somewhere a big "Don't touch this!" sign at this thread?
Gee, new kid on the block, I see...

*LOL*

Joff
2006-Jan-05, 12:46 AM
like this, kucharek?

or we could just get a mod to lock it.... ;) :lol:

Candy
2006-Jan-05, 12:50 AM
I think kucharek is dreamy. :razz:

umop ap!sdn
2006-Jan-05, 12:51 AM
Wait a minute. I never said I had any doubts. I asked a question about the implication of the statement that .999~ = 1, namely, if this statement is true then is infinity the result of dividing by zero, i.e. does infinitessimal equal zero. Nowhere in my post did I contradict the statement.

But now that you bring it up, I do have doubts that a number written as a decimal point followed by an infinity of nines can be exactly equal to a number written simply as a one, for the reason that each digit represents a quantity multiplied by its position (e.g. tenths, hundredths, thousandths, etc...) and no number of nines will "carry" that last minuscule quantity to the ones position. They are equal to the margin of error of any measurement that one could ever attempt, and so in that sense are practically equal, but to say they are exactly equal would seem to stretch the definition of "exactly".

On the other hand, this is contradicted by the simple proof that 1/3 = .333~, so what I see here is a paradox, and the arguement is meaningless. ;) The only way I can think of to resolve the paradox is to assume that infinitessimal equals zero, .999~ = 1, and anything divided by zero is infinity.

It will be time to eat supper (probably out someplace) shortly, and after that I have 2 solid days of work. I probably won't get a chance to read the entire thread, and I don't see a search thread function or else I'd check to see if the infinity thing has already been brought up.

snarkophilus
2006-Jan-05, 01:08 AM
Let's look at the definition of the real numbers.

Suppose that 0.999~ != 1. One might claim that 0.999~ is distinct from 1, the largest real less than but not equal to 1. But adding anything but 0.000~ will bring the sum to greater than 1 (the first non-zero element in the sequence will cause carrying to occur right back to the decimal point, leaving a number like 1.000...000999~ > 1). Therefore there is no real number x > 0 such that 0.999~ + x = 1. Therefore if 0.999~ is real and not equal to 1, the reals are not continuous (and, in fact, are not continuous at every non-transcendental point, and possibly discontinuous at the transcendentals, too -- I don't know enough to deal with those).

Actually, if you take the definition of the real numbers as a complete ordered field, then having no x such that 0.999~ + x = 1 means the additive group is not closed (which makes it not a field, a contradiction), meaning that either 0.999~ is not a real number or 0.999~ = 1.

The possibility that 0.999~ is not a real number still exists, though. It takes a great deal of work to get from the construction of the reals to the fact that decimals can represent them.

I like the idea that maybe 0.999~ isn't a real number (though I'm convinced it is, and is equal to 1, because I'm convinced that writing things in decimal notation is equivalent to writing them in set notation). That's often the solution to seemingly contradictory problems. It's like saying "this statement is false." The apparent logical contradiction is due to assuming that it really is a statement, that it really has semantics. Once you accept that it isn't a statement at all, then it's vacuously true.

snarkophilus
2006-Jan-05, 01:21 AM
i.e. does infinitessimal equal zero.

When dealing in real numbers only, yes. There are other numbers out there, like the surreals and hyperreals, where infinitesimals are definitely not equal to 0.

and anything divided by zero is infinity.
If we're going to start dividing by zero, we can have some fun with algebra. :)

Let's divide something by 0 and get 7 (not infinity), shall we? In fact, let's do one better, and at the same time divide 0 by something to get something that doesn't equal 0.

Let's say that
Eqn 1: 7 x a = b, for any a and b.
Assuming we can divide by a,
7 x a / a = b / a.
Eqn 3: 7 x (1) = b / a.
If we let b = 0, then
7 = 0 / a,
so a is the number by which we can divide 0 to get 7. But from Eqn 1,
a = b / 7 = 0 / 7 = 0.
Therefore 7 = 0 / 0.

You'll notice that in Eqn 3, we assumed that a / a = 1, the normal meaning of division, meaning that 0 / 0 also equals 1. It is a schizophrenic creature at best. Although, you can do similar calculations to get rid of that inconsistency (here, you could use 1 instead of 7, and 0/0 would be "proven" to equal 1).

montebianco
2006-Jan-05, 04:48 AM
I did not read all replies in this thread. But did someone think about the following question. What is greatest real value just not equal to 1?
I would say this equals: 0.999~

If there is a greatest real value less than one, let's call it x. Now, let's call y == (x+1)/2. Two questions:

a) Is y less than one?

b) Is y greater than x?

Ain't no such thing as the greatest real value less than one...

N

montebianco
2006-Jan-05, 04:51 AM
If we're going to start dividing by zero, we can have some fun with algebra. :)

Yep, no doubt about that :D

It's possible to extend the real number system to include infinity, but the price of the extension is that some of the laws obeyed by real numbers don't work anymore...

montebianco
2006-Jan-05, 04:56 AM
If .999~ = 1, then does it follow that the reciprocal of zero is infinity?

Apologies if this has already been covered - I'm still on page 3. :D

In what is sometimes called the extended real number system, the reciprocal of zero can be defined to be infinity. I'm not really sure how that relates to .999~ == 1 though. Do keep in mind, though, that infinity is not a real number, and does not obey all the same rules of arithmetic as the real numbers...

N

montebianco
2006-Jan-05, 04:59 AM
If it only makes it to 999 it'll be the same as 1000 right? ;)

No, if it makes it to 999.999~ it will be the same as 1000.

N

umop ap!sdn
2006-Jan-05, 05:01 AM
That makes a lot of sense - I hadn't considered the 0.000~ angle before. :)

montebianco
2006-Jan-05, 05:03 AM
Can we remind viewers that there are several different math "theories" in question, and you guys are fighting the battle of one of those? :shifty:

Well, a mathematical system is not a theory in the sense of, for example, a theory of gravity, it is a logical system created by assumption. I'm not aware of any system of math in which 0.999~ is not equal to one.

N

umop ap!sdn
2006-Jan-05, 05:06 AM
I should have previewed before quick replying to a page opened a while ago... :doh:

In what is sometimes called the extended real number system, the reciprocal of zero can be defined to be infinity. I'm not really sure how that relates to .999~ == 1 though.
I guess I was thinking of the difference between .999~ and 1 as infinitessimal, and infinitessimal as 1/∞. :lol:

Do keep in mind, though, that infinity is not a real number, and does not obey all the same rules of arithmetic as the real numbers...
That would seem to also resolve the paradox. If, for instance, one multiplies .999~ by 10, there wouldn't be an extra zero added to the end because adding or subtracting from infinity still gives infinity (or at least I think it does).

snarkophilus
2006-Jan-05, 06:06 AM
I guess I was thinking of the difference between .999~ and 1 as infinitessimal, and infinitessimal as 1/∞.

You can certainly treat an infinitesimal as 1/∞, and say that 1 - .999~ = 1/∞, but you could equally well say that 2 - 2 = 1/∞. All you're saying when you mix ∞ with real numbers is that you can pick an arbitrarily large number and get whatever logical statement you're making to be arbitrarily close to true.

That would seem to also resolve the paradox. If, for instance, one multiplies .999~ by 10, there wouldn't be an extra zero added to the end because adding or subtracting from infinity still gives infinity (or at least I think it does).
Adding a finite number to infinity still gives infinity, yes. But you can certainly add (or subtract) infinities and end up with finite numbers. (Feynman, Tomonaga and Schwinger got a Nobel prize for figuring out a nice way to do this with particle physics.)

For example, say you have two functions, f(x) = x + 3 and g(x) = x + 5. The limit as x->∞ = ∞ for both functions. However, for g(x) - f(x), the limit as x -> ∞ = 2.

LurchGS
2006-Jan-05, 06:09 AM
.9999~ is generic
1.000~ is brand name

Says so in my insurance paperwork

montebianco
2006-Jan-05, 06:19 AM
But you can certainly add (or subtract) infinities and end up with finite numbers.

We're speaking informally here, of course...

I have a book (I can't recall which, maybe I'll look for it tomorrow) that, when discussing Lebesgue integration, points out that some so-called improper integrals appearing in calculus textbooks are actually not defined, containing an infinite amount of area above and below the line...

montebianco
2006-Jan-05, 06:20 AM
That would seem to also resolve the paradox. If, for instance, one multiplies .999~ by 10, there wouldn't be an extra zero added to the end because adding or subtracting from infinity still gives infinity (or at least I think it does).

Infinity possibly being the amount of time until my next move in our chess game at Scotson's Shack...

N

Disinfo Agent
2006-Jan-05, 01:32 PM
See this is the part that has always bothered me. If infinity is part of the question how do you just not use it or convert it in the proof.technically, it is not part of the proof. my fave is...

1/9 = 0.111~
9*(1/9) = 9*0.111~
1 = 0.999~

the concept of infinity is never used. the fact that the tilde is there (normally a bar over the last digit, but hard to do in here) is nothing more than shorthand notation. nowhere are you actually manipulating infinity.I'm not sure I agree. You say the tilde is a shorthand -- shorthand for what?

montebianco
2006-Jan-05, 01:34 PM
I'm not sure I agree. You say the tilde is a shorthand -- shorthand for what?

I think we have a semantic issue here. Certainly the tilde represents an infinite sum, i.e., a sum with infinitely many terms in it. But the partial sums are bounded, and approach a finite limit...

N

Disinfo Agent
2006-Jan-05, 01:35 PM
It's like saying "this statement is false." The apparent logical contradiction is due to assuming that it really is a statement, that it really has semantics. Once you accept that it isn't a statement at all, then it's vacuously true.What is "vacuously true"?

Disinfo Agent
2006-Jan-05, 01:41 PM
Well, a mathematical system is not a theory in the sense of, for example, a theory of gravity, it is a logical system created by assumption. I'm not aware of any system of math in which 0.999~ is not equal to one.Here's one. (http://www.bautforum.com/showpost.php?p=371396&postcount=280)

(Or you can always cheat, and use a base greater than 10. ;))

Disinfo Agent
2006-Jan-05, 01:44 PM
If, for instance, one multiplies .999~ by 10, there wouldn't be an extra zero added to the end because adding or subtracting from infinity still gives infinity (or at least I think it does).But .999~ is not an infinite number, is it? I mean, it's right there between 0 and 1.

pghnative
2006-Jan-05, 01:49 PM
But .999~ is not an infinite number, is it? I mean, it's right there between 0 and 1.Inclusive.

galacsi
2006-Jan-05, 01:51 PM
Actually according to Calculus it is equal to 1. 0.99999... can be written as a limit. That limit converges onto 1. Thus 0.999999... equals 1.

Also the limit of x as a apporaches 1 is exactly equal to 1, not aproximatly 1.

I am interested : Can you write down this suite and show us how it converges to 1 ?

Something like that :

1/101 + 1/102 + 1/103 +......+ 1/10n ????

Thanks.

Disinfo Agent
2006-Jan-05, 01:54 PM
Galacsi, I did that in one of the threads I linked to on the previous page. The graph that Candy posted on that page is of the sequence 0.9, 0.99, 0.999, ...

P.S. As a matter of fact, here it is (http://www.bautforum.com/showpost.php?p=412429&postcount=877).

ToSeek
2006-Jan-05, 04:26 PM
or we could just get a mod to lock it.... ;) :lol:

I'm tempted, at least until it drops back to page 4 or so, where it belongs. ;)

hhEb09'1
2006-Jan-05, 06:07 PM
Don't make me beat you! Ask milli360, he is the math brain.Just call me Pace (http://www.bautforum.com/showthread.php?p=445234#post445234) then. :)

The theory that us guys are advocating is the one of common ordinary notions. Any theory that would have .999... not equal to 1.0 would probably also have .333... not equal to 1/3. Most people accept the latter. If someone doesn't, they probably should be posting in the Against The Mainstream.

If someone thinks .333... does not equal 1/3, I'd like to hear their reasoning. I've asked this before, and I don't believe anyone has ever stepped forward.

Taks
2006-Jan-05, 06:23 PM
I still think there is something wrong with this theory. :Dit is not theory if mathematically proved. theory is when something is tested and shown to be true. theory is allowed to be later revised when further testing (perhaps more advanced or simply more clever) finds a difference or a flaw. mathematical proof is it. fine. done. law, i suppose, unless you change the underlying logical assumptions (which would change the statements you use to make your proof, and perhaps your conclusions in the end).

taks

Taks
2006-Jan-05, 06:27 PM
I'm not sure I agree. You say the tilde is a shorthand -- shorthand for what?shorthand for writing out an infinite number of 9s, which is obviously impossible.

taks

Disinfo Agent
2006-Jan-05, 06:34 PM
But then how can you make the bolded claims below?

technically, it [infinity] is not part of the proof. my fave is...

1/9 = 0.111~
9*(1/9) = 9*0.111~
1 = 0.999~

the concept of infinity is never used. the fact that the tilde is there (normally a bar over the last digit, but hard to do in here) is nothing more than shorthand notation. nowhere are you actually manipulating infinity.Yes, you are. There is an implicit infinity in the tilde. Giving it another name doesn't make it any less infinite.

Thomas(believer)
2006-Jan-05, 09:06 PM
1/9 = 0.111~
9*(1/9) = 9*0.111~
1 = 0.999~

taks
Suppose I won't believe 1=0.999~, why should I believe that 1/9=0.1111~.
So this is no proof to the real question, I think.
The real question is: does 0 equals 0.0000~. Or am I wrong?
I suppose that this is very trivial to most people.
We talking about two ways to write the same thing.
0 represemts exactness, it is impossible to write down in decimal notation.
The same goes for 1 and PI.

HenrikOlsen
2006-Jan-05, 09:18 PM
I'm tempted, at least until it drops back to page 4 or so, where it belongs. ;)
See if you can close it with the 1000th post :)

Thomas(believer)
2006-Jan-05, 10:06 PM
Let's go for the thousend.

0 = 0.00~ = 0.(~0's)1
~ = 1(~0's) = 1(~0's)0000

0 x ~ = 1000

galacsi
2006-Jan-05, 10:24 PM
Galacsi, I did that in one of the threads I linked to on the previous page. The graph that Candy posted on that page is of the sequence 0.9, 0.99, 0.999, ...

P.S. As a matter of fact, here it is (http://www.bautforum.com/showpost.php?p=412429&postcount=877).

Hum hum :whistle: i believed it was a new thread ! Better go to bed !

Thanks for the link ! :)

Disinfo Agent
2006-Jan-05, 10:35 PM
I wrote it in another thread, too.

ToSeek
2006-Jan-05, 11:06 PM
See if you can close it with the 1000th post :)

Dragon Star
2006-Jan-05, 11:15 PM
Can we please close this thread for it's 1,000th? It is so pointless.

JMV
2006-Jan-05, 11:22 PM
why should I believe that 1/9=0.1111~.

Taks
2006-Jan-05, 11:49 PM
0 = 0.00~ = 0.(~0's)1no.
taks

Taks
2006-Jan-05, 11:55 PM
Yes, you are. There is an implicit infinity in the tilde. Giving it another name doesn't make it any less infinite.no, there is not. there's a difference between an infinite series and the number infinity. there are an infinite number of 9s in the number (an infinite series), but the number infinity is not used in any calculation. i.e. we are not manipulating infinity with any operation (multiply, divide, add, subtract, etc.).

the tilde is simply there because we are indicating a repeating series. typically you actually put a bar over the part that repeats.

taks

Disinfo Agent
2006-Jan-06, 12:07 AM
You still have an infinite series, which is what confuses people. Some also seem to confuse infinite series with infinite numbers, though, so it's a good thing you pointed out the distinction between the two.

Do I get to have the last word? :D

Taks
2006-Jan-06, 12:09 AM
Suppose I won't believe 1=0.999~, why should I believe that 1/9=0.1111~.again, it is proved. do the math yourself. long division, 9 into 1...

mathematics are based on a set of assumptions, known as axioms, which are universal. arithmetical operators are all developed based on such accepted rules, and division is one of them. that you would "choose" not to believe simply means you choose not communicate in the accepted mathematical language the rest of us are using.

assuming this is true, why are you in here asking questions which you are obviously unqualified to understand answers for?

So this is no proof to the real question, I think.false, based on the previous comment.

The real question is: does 0 equals 0.0000~.yes.

I suppose that this is very trivial to most people.
We talking about two ways to write the same thing.
0 represemts exactness, it is impossible to write down in decimal notation.what do you mean by exactness? the digits are there regardless of whether you write them out or not.

The same goes for 1 and PI.except that pi is not exact. at least, we cannot describe it exactly in numerical terms. the complete stream of digits of pi are unknown as it is a transcendental number.

taks