Author Topic: 0.999... ≠ 1  (Read 446 times)

Offline Daniel

  • Feldwebel
  • ***
  • Posts: 1708
  • Thanked: 269 times
  • Religion: Roman Catholic
0.999... ≠ 1
« on: June 26, 2017, 08:39:22 PM »
All right, time for round 2.

First proof:

I walk nine tenths of the way across the room. Then I walk nine tenths of the remaining distance. Then I walk nine tenths of the remaining distance. And I keep doing that, forever. Yet I will never make it to the other side of the room.

Second proof:

I have two empty bottles. I fill the first bottle nine tenths of the way to the top while I simultaneously fill the second bottle ten elevenths of the way to the top.  Then, I fill the first bottle's remaining space nine tenths while simultaneously filling the second bottle's remaining space ten elevenths. And I keep doing that. Yet even if I do that forever, the first bottle will always have less in it than the second bottle, and the second bottle itself will never be completely full.
 

Offline dolores

  • Korporal
  • **
  • Posts: 287
  • Thanked: 158 times
Re: 0.999... ≠ 1
« Reply #1 on: June 26, 2017, 09:12:01 PM »
The problem with your "proofs" is that it is, in fact, possible to walk to the other side of the room and fill the bottles.  You've stumbled upon Zeno's paradox:

Quote
In the paradox of Achilles and the tortoise, Achilles is in a footrace with the tortoise. Achilles allows the tortoise a head start of 100 meters, for example. If we suppose that each racer starts running at some constant speed (one very fast and one very slow), then after some finite time, Achilles will have run 100 meters, bringing him to the tortoise's starting point. During this time, the tortoise has run a much shorter distance, say, 10 meters. It will then take Achilles some further time to run that distance, by which time the tortoise will have advanced farther; and then more time still to reach this third point, while the tortoise moves ahead. Thus, whenever Achilles arrives somewhere the tortoise has been, he still has some distance to go before he can even reach the tortoise. Therefore, because there are an infinite number of points Achilles must reach where the tortoise has already been, he can never overtake the tortoise.

https://en.wikipedia.org/wiki/Zeno%27s_paradoxes#Paradoxes_of_motion

However, despite this "proof" that Achilles will never reach the tortoise, our senses tell us that this is obviously false, and Achilles will overtake the tortoise.

Here's an interesting video about it.
 
The following users thanked this post: Daniel

Offline Daniel

  • Feldwebel
  • ***
  • Posts: 1708
  • Thanked: 269 times
  • Religion: Roman Catholic
Re: 0.999... ≠ 1
« Reply #2 on: June 26, 2017, 09:28:20 PM »
I've seen videos from that channel before. They're pretty entertaining :)

Anyway, I'm aware of the problem with Zeno's paradox, but the reason that Zeno's paradoxes don't work is because he ignores that fact that Achilles is moving at a constant speed. Constant speed means that when you decrease the distance, you also need to decrease the time by the same ratio.

Mathematically, Zeno's model is like this:
(9/10)/1 + (9/100)/1 + (9/1000)/1 + ... < 1, so Achilles will never reach the tortoise

But that doesn't reflect reality. Reality would be represented:
(9/10)/1 + (9/100)/(1/10) + (9/1000)/(1/100) + ...
= .9 + .9 + .9 + ... > 1, and so Achilles can easily outrun the tortoise

My proof is more of a hypothetical situation. "You're in a room, and the rule of this room is that you are only ever allowed to move nine tenths of the way between yourself and the other side of the room."
« Last Edit: June 26, 2017, 09:33:42 PM by Daniel »
 

Offline dolores

  • Korporal
  • **
  • Posts: 287
  • Thanked: 158 times
Re: 0.999... ≠ 1
« Reply #3 on: June 26, 2017, 09:39:46 PM »
My proof is more of a hypothetical situation. "You're in a room, and the rule of this room is that you are only ever allowed to move nine tenths of the way between yourself and the other side of the room."

The problem with your hypothetical is that the only reason a person would not reach the other side of the room is that it would take an infinite number of steps to reach it, and humans, of course, cannot do something an infinite number of times.  If you could somehow perform all of the infinite steps, you would reach the other side of the room.  That is precisely what is expressed by "0.999...", which is why 0.999... = 1.  If the number is cut off any any point, it would be less than 1, but the expression "0.999..." means that it is never cut off and goes on forever.
 

Offline Daniel

  • Feldwebel
  • ***
  • Posts: 1708
  • Thanked: 269 times
  • Religion: Roman Catholic
Re: 0.999... ≠ 1
« Reply #4 on: June 26, 2017, 11:59:46 PM »
The problem with your hypothetical is that the only reason a person would not reach the other side of the room is that it would take an infinite number of steps to reach it, and humans, of course, cannot do something an infinite number of times.  If you could somehow perform all of the infinite steps, you would reach the other side of the room.  That is precisely what is expressed by "0.999...", which is why 0.999... = 1.  If the number is cut off any any point, it would be less than 1, but the expression "0.999..." means that it is never cut off and goes on forever.

That's kind of my point. If "0.999..." signifies the sum of all the parts, and if we assume that 0.999... equals 1, then that means all the parts can be summed to 1. But they can't. No man can sum all the parts, and even if he could, that missing piece wouldn't just magically fill itself in.

Look at it this way:
9/10 = .9
9/10 + 9/100 = 99/100 = .99
9/10 + 9/100 + 9/1000 = 999/1000 = .999
[. . .]
9/10 + 9/100 + ... + 9/1000000000 = 999999999/1000000000 = .999999999
[. . .]
9/10 + 9/100 + ... + 9/googol = (googol-1)/googol = "a decimal point followed by a hundred 9s"
[. . .]
9/10 + 9/100 + ... + 9/googolplex = (googolplex-1)/googolplex = "a decimal point followed by a googol 9s"

So long as you can manage to sum up everything in the series up to a certain point, you get an actual value. And the value can be expressed as ((10^n)-1)/(10^n), the limit of which, as n gets increasingly bigger, is 1.

However, that doesn't mean that you can just go plugging in ∞ for n and saying that when the whole thing goes on forever then its value is 1. That's not valid math and it leads to absurdities such as this:
((10^∞)-1)/(10^∞) = 1
((10^∞)-1) = (10^∞)
-1 = 0

The fact is, the expression simply cannot be evaluated if it goes on forever. Its value can only be actualized upon the sequence's termination, in a finite number of steps. Yet any potential value will always be less than 1. Thus, this "0.999..." figure isn't even really a number, since it has no actual value. And it's certainly not the same thing as "1".
« Last Edit: June 27, 2017, 09:28:44 AM by Daniel »
 
The following users thanked this post: PerEvangelicaDicta

Offline Quaremerepulisti

  • Feldwebel
  • ***
  • Posts: 3288
  • Thanked: 979 times
  • Religion: Catholic
Re: 0.999... ≠ 1
« Reply #5 on: June 27, 2017, 11:21:06 AM »
The fallacy of this is that you are taking a limit as something goes to infinity (which is an abstraction), insisting that it be directly applicable to something in real life (which is something concrete), noticing that actual infinities do not exist in real life, and then claiming the limit is invalid.

Saying that the limit of ((10^n)-1)/(10^n) is 1 as n goes to infinity does not mean that you can just plug in "infinity" for n.  That is not the mathematical definition of limit.  It means you can get arbitrarily close to 1 as long as n is large enough.  Your other counterexamples only show that if you instead take the limit ((10^n)-1)/(10^m) as both m and n go to infinity, the limit is undefined (can be anything) because it depends on how you take the limits.

The Zeno's paradox and empty bottle examples fail if space is quantized (as I think must be the case) and if matter is quantized (which we know to be the case).  At some point you can't walk 9/10 of the remaining distance.  There's only one more point between you and the destination.  At some point you can't fill in 9/10 of the rest of the bottle.  There's room for one more molecule and that's it.

So it is true that .9999... ≠ 1 if by this you mean that are going to keep adding another 9 at the end.  Your chain is potentially infinite, but it will never be actually infinite.

But the limit, as it is defined mathematically, is true and correct.
The real purpose of traditionalist polemics is not to find truth, but to attempt to construct an epistemological fortress rendering one's worldview impervious to attack.
 
The following users thanked this post: Daniel

Offline Daniel

  • Feldwebel
  • ***
  • Posts: 1708
  • Thanked: 269 times
  • Religion: Roman Catholic
Re: 0.999... ≠ 1
« Reply #6 on: June 27, 2017, 01:27:47 PM »
The Zeno's paradox and empty bottle examples fail if space is quantized (as I think must be the case) and if matter is quantized (which we know to be the case).  At some point you can't walk 9/10 of the remaining distance.  There's only one more point between you and the destination.  At some point you can't fill in 9/10 of the rest of the bottle.  There's room for one more molecule and that's it.
Agree'd. But ignoring physics, you can divide some geometric shape or distance and keep diving it and you will never hit a "smallest" or "indivisible" part. This is my point: there is always going to be some space between 0.999... and 1, meaning, 0.999... is not the same thing as 1. (And in the bottle example, I was showing that there are other infinite series which, if they could be summed, would sum to something between 0.999...'s hypothetical value and 1. For example, they hypothetical sum of 10/11+10/121+10/1331+... is larger than the hypotheitcal sum of 9/10+9/100+9/1000+... yet it's still smaller than 1. And that would not be possible if 0.999... was 1, since you cannot stick additional numbers between 1 and itself.)
« Last Edit: June 27, 2017, 08:05:57 PM by Daniel »
 

Offline cgraye

  • Hellebardier
  • *
  • Posts: 5
  • Thanked: 5 times
Re: 0.999... ≠ 1
« Reply #7 on: June 30, 2017, 02:28:00 PM »
Agree'd. But ignoring physics, you can divide some geometric shape or distance and keep diving it and you will never hit a "smallest" or "indivisible" part. This is my point: there is always going to be some space between 0.999... and 1, meaning, 0.999... is not the same thing as 1.

Daniel, I think what you are missing is that 0.999... captures all of that infinite number of divisions.  You can't stop it at some point and just examine a finite number of elements in the division.
 That will never be 1, nor will it be infinitely close to 1.  It will just be some finite number less than 1.

It's easier to think about it starting from 1 and dividing it up, as you describe above with dividing up a geometric shape or distance.  You have a line of length 1.  You divide it in half and you have two segments of length 1/2 and 1/2.  Then you divide one of those segments in half and you have three segments of length 1/2, 1/4, and 1/4.  And so on.  You started with 1 and described its division into an infinite number of pieces, so of course when you add all those pieces back up, you are going to get what you started with, which was 1.

Quote
(And in the bottle example, I was showing that there are other infinite series which, if they could be summed, would sum to something between 0.999...'s hypothetical value and 1. For example, they hypothetical sum of 10/11+10/121+10/1331+... is larger than the hypotheitcal sum of 9/10+9/100+9/1000+... yet it's still smaller than 1. And that would not be possible if 0.999... was 1, since you cannot stick additional numbers between 1 and itself.)

Why do you think the sum 10/11+10/121+10/1331+... is greater than 9/10+9/100+9/1000+..., even hypothetically?
« Last Edit: June 30, 2017, 11:24:21 PM by cgraye »
 
The following users thanked this post: Daniel

Offline Daniel

  • Feldwebel
  • ***
  • Posts: 1708
  • Thanked: 269 times
  • Religion: Roman Catholic
Re: 0.999... ≠ 1
« Reply #8 on: June 30, 2017, 04:49:12 PM »
Why do you think the sum 10/11+10/121+10/1331+... is greater than 9/10+9/100+9/1000+..., even hypothetically?
Well, it's because 10/11+10/121+10/1331 approaches 1 more quickly than 9/10+9/100+9/1000.

Suppose we were to graph y11=((11x)-1)/(11x) and y10=((10x)-1)/(10x). Regardless of which x you choose, y11 is always greater than y10. So it stands to reason that hypothetically, if y11 and y10 were single values rather than functions, y11 would be greater than y10.

In fact, every base has a similar function/number. But no matter which base you choose, none of them will ever equal 1.

Daniel, I think what you are missing is that 0.999... captures all of that infinite number of divisions.  You can't stop it at some point and just examine a finite number of elements in the division.
 That will never be 1, nor will it be infinitely close to 1.  It will just be some finite number less than 1.

It's easier to think about it starting from 1 and dividing it up, as you describe above with dividing up a geometric shape or distance.  You have a line of length 1.  You divide it in half and you have two segments of length 1/2 and 1/2.  Then you divide one of those segments in half and you have a three segments of length 1/2, 1/4, and 1/4.  And so on.  You started with 1 and described its division into an infinite number of pieces, so of course when you add all those pieces back up, you are going to get what you started with, which was 1.
I'll have to think about that. (No time at the moment.)
 

Offline Daniel

  • Feldwebel
  • ***
  • Posts: 1708
  • Thanked: 269 times
  • Religion: Roman Catholic
Re: 0.999... ≠ 1
« Reply #9 on: June 30, 2017, 07:57:46 PM »
Why do you think the sum 10/11+10/121+10/1331+... is greater than 9/10+9/100+9/1000+..., even hypothetically?
Well, it's because 10/11+10/121+10/1331 approaches 1 more quickly than 9/10+9/100+9/1000.

Suppose we were to graph y11=((11x)-1)/(11x) and y10=((10x)-1)/(10x). Regardless of which x you choose, y11 is always greater than y10. So it stands to reason that hypothetically, if y11 and y10 were single values rather than functions, y11 would be greater than y10.

In fact, every base has a similar function/number. But no matter which base you choose, none of them will ever equal 1.
Objection! There is a major hole in my theory.

y10=((10x)-1)/(10x)
y11=((11x)-1)/(11x)
y10 < y11

True so far. However, this number "0.999..." is ambiguous. At first glance it appears to represent 9/10 + 9/100 + 9/1000 + ... (after all, there is a 9 in the tenths place, a 9 in the hundredths place, a 9 in the thousandths place, and so on). And if that be the case, it's the same thing as y10.

But who says that that has to be the case? Couldn't it just as well represent 99/100 + 99/10000 + 99/1000000+...? (After all, ignoring the carry, there is a 99 in the hundredths place, a 99 in the ten-thousandths place, and so on). And if that be true then 0.999... represents not y10 but y100:

y100=((100x)-1)/(100x), which approaches 1 much more quickly than y10 or even y11 for that matter.

(Come to think of it, 0.999... could represent any of functions y10n.)

Same for base-eleven's "0.AAA...". No way to know whether it represents y11 or y121 or y11n.

And y121 > y100 > y11 > y10. So it's like, yeah, y11 is greater than y10, but it's sandwiched between y10 and y100, which itself is sandwiched between y11 and y121, making it impossible to know for sure that 0.999... is lesser than 0.AAA..., unless of course we ignore all bases which are powers of other bases. But why anyone would feel so inclined to do that remains a mystery.
« Last Edit: June 30, 2017, 08:10:44 PM by Daniel »
 

Offline Daniel

  • Feldwebel
  • ***
  • Posts: 1708
  • Thanked: 269 times
  • Religion: Roman Catholic
Re: 0.999... ≠ 1
« Reply #10 on: June 30, 2017, 09:38:10 PM »
Agree'd. But ignoring physics, you can divide some geometric shape or distance and keep diving it and you will never hit a "smallest" or "indivisible" part. This is my point: there is always going to be some space between 0.999... and 1, meaning, 0.999... is not the same thing as 1.

Daniel, I think what you are missing is that 0.999... captures all of that infinite number of divisions.  You can't stop it at some point and just examine a finite number of elements in the division.
 That will never be 1, nor will it be infinitely close to 1.  It will just be some finite number less than 1.

It's easier to think about it starting from 1 and dividing it up, as you describe above with dividing up a geometric shape or distance.  You have a line of length 1.  You divide it in half and you have two segments of length 1/2 and 1/2.  Then you divide one of those segments in half and you have a three segments of length 1/2, 1/4, and 1/4.  And so on.  You started with 1 and described its division into an infinite number of pieces, so of course when you add all those pieces back up, you are going to get what you started with, which was 1.
Ok, I get what you're saying. But what I don't get is why that final piece is being counted among the rest.

Suppose it goes down like this:
Some guy has a line segment of length 1. He splits it into two pieces and gives one of those pieces to you. So at this point there are two pieces: the 1/2 in your possession and the 1/2 in his possession. The total length of these two pieces together adds up to 1. I don't deny that much. However, we are only concerned with the pieces in your possession, which so far is only 1/2.
Now the guy splits his piece into two again, and gives one to you again. So now there are three pieces: the 1/2 and the 1/4 in your possession, and the 1/4 in his possession. Again, the total of all three pieces makes 1, and I don't deny that. But the total of your two pieces only makes 3/4.
Now the guy does it again. He splits his piece in two and gives you one. So now there are four pieces: the 1/2 and the 1/4 and the 1/8 in your possession, and the 1/8 in his possession. The total of all four pieces still adds up to 1. But the total of your three pieces is only 7/8.

The guy keeps doing this and doing this ad infinitum, always giving you half but keeping the other half for himself. So clearly you will never end up with all the pieces, thus the sum of the lengths of the pieces in your possession will always fall just short of 1. But since the process never terminates, the actual sum remains forever undefined.
« Last Edit: June 30, 2017, 09:51:28 PM by Daniel »
 

Offline cgraye

  • Hellebardier
  • *
  • Posts: 5
  • Thanked: 5 times
Re: 0.999... ≠ 1
« Reply #11 on: June 30, 2017, 10:36:46 PM »
True so far. However, this number "0.999..." is ambiguous. At first glance it appears to represent 9/10 + 9/100 + 9/1000 + ... (after all, there is a 9 in the tenths place, a 9 in the hundredths place, a 9 in the thousandths place, and so on). And if that be the case, it's the same thing as y10.

But who says that that has to be the case? Couldn't it just as well represent 99/100 + 99/10000 + 99/1000000+...? (After all, ignoring the carry, there is a 99 in the hundredths place, a 99 in the ten-thousandths place, and so on). And if that be true then 0.999... represents not y10 but y100:

y100=((100x)-1)/(100x), which approaches 1 much more quickly than y10 or even y11 for that matter.

(Come to think of it, 0.999... could represent any of functions y10n.)

Yes, it does represent all of those series.  And all of those series are equal to 1.  That's not to say that the series are all the same, just that they all have the same sum.  1 + 4 and 2 + 3 are both equal to 5.  That's not saying that 1 + 4 and 2 + 3 are exactly the same thing - they clearly are not when you compare their elements one by one.  But they have the same sum.  Likewise, all the series you are describing are different series, and they are different when you compare their elements one by one, but they have the same sum.

It's exactly the same thing as noticing that there are infinitely many ways to divide a line of length 1 into infinitely many segments.  All the segmentations are different when you compare them segment by segment, but they all combine to form the same line.
 
The following users thanked this post: Daniel

Offline cgraye

  • Hellebardier
  • *
  • Posts: 5
  • Thanked: 5 times
Re: 0.999... ≠ 1
« Reply #12 on: June 30, 2017, 10:59:00 PM »
The guy keeps doing this and doing this ad infinitum, always giving you half but keeping the other half for himself. So clearly you will never end up with all the pieces, thus the sum of the lengths of the pieces in your possession will always fall just short of 1. But since the process never terminates, the actual sum remains forever undefined.

The pieces in your possession do not represent an infinite series.  If you stop and sum up all your pieces after any finite number of steps, of course you will just get some finite number less than 1.  What the infinite series represents is a division of the line into infinitely many pieces, and unless you consider all of the infinitely many pieces together, you cannot reconstruct the entire line.  This division is not some process you can actually do in real life, as in your example of the man dividing up the line, but if you want to try to think about it that way, the right question to ask is not, "How much of the line do I have after he has given me a million or a billion or a trillion segments?", it is, "How long is the complete line, considering that he is giving me segments in this manner?"
« Last Edit: July 01, 2017, 01:06:29 AM by cgraye »
 
The following users thanked this post: Daniel