Paradox?

Originally posted by bobbob1056th
Is 0.9 = 1? no. is 0.99 = 1? no. How about 0.999? None of these numbers equal one, although the more nines you add the closer you get to 1. Your first conjecture doesn't prove anything, it just assumes that 1 - 0.999... = 0. Your second one: it doesn't make any more sense to have a number with an infinite number of nines after the decimal than it ...[text shortened]... ly small distances.
Also, is it accepted as a fact that sum(1/2, 1/4, 1/8...) does not equal 2?

Fight the machine! I admire your spirit, bobbo. Unfortunately, in this case it's misplaced.

What davegage said is that there is no number "x" that satisfies this equation:

1 - x = 0.9999...

or equivalently:

1 - 0.9999... = x

Pick a number, any number for "x". Let's try x = 0.1. Well, all we have to do to show that 0.1 doesn't work is have two 9's after the decimal place:

1 - 0.99 = 0.01 < 0.1

OK, maybe 0.1 was too big. Let's try 0.01. Well, all we have to do to show that 0.01 doesn't work is have three 9's after the decimal place:

1 - 0.999 = 0.001 < 0.01

OK, maybe 0.01 was too big too. Let's try 0.000000001. Well, all we have to do to show that 0.000000001 doesn't work is have ten 9's after the decimal place:

1 - 0.9999999999 = 0.0000000001 < 0.000000001

Etc... So what this shows is that the idea that there is some small number "x" that fits between 1 and 0.9999... is false. The counter examples above can go on forever, because we have an infinite number of 9's to add after the decimal place. And like Sherlock Holmes said (and I paraphrase), after all the possibilities have been exhausted, whatever is left, no matter how improbable, must be the truth. This is the basis for the squeezing type solution above. Incindentally, this is also related to the delta-epsilon type solutions ussed as the foundations of calculus.

And for the record, sum(1/2, 1/4, 1/8...) is equal to 2!

Originally posted by davegage
I agree that all the series you are refering to diverge -- that is clear since the assumption we are working with is that the farmers' rows are infinite. But I don't think this answers the question, which deals primarily with cardinality.

Consider again the case where the farmers plant the first seed in 1 second, the second seed in 1/2 of a second, th ...[text shortened]... y said you agree with Premise 1. So if you reject Premise 3, then what is wrong with Premise 2?

Bird B will NOT eat all seeds.

The remaining seeds after 1 second is 1
The remaining seeds after 1,5 second is 2
The remaining seeds after 1,75 second is 3

Leaving some words out the number of remaining seeds is:

4, 5, 6, 7, 8, 9, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 18, ...

This diverges clearly. So after 2 seconds (assuming nor the farmer, nor the bird have exploded because of extreme friction) there are infinitely many seeds left.

The bird needs 10 times as much time as the farmer to eat all seeds when we talk about finite rows.

When dealing with infinity you have to realise that not all addition rules apply.

infinity - infinity can be anything, and is undefined. This 'paradox' only excists because you claim that infinity - infinity IS defined and equal to 0.

IN my eagerness to reply I missed alot of posts 🙂

To bobo:

1-0.999...=X

X has some nice properties.

Firstly: X is not negative
Secondly: X is smaller then any positive number.

ASSUMPTION: X is not 0

For real numbers we have a well-ordening principle (no idea what the correct english term is) wich means that a number is either smaller then 0, equal to 0 or greater than zero.

X is not less then 0 (first property)
X is not equal to 0 (Assumption)
So X must be greater than 0. Make Y = X/2. Y is postive since X was positive. Y is smaller than X, that's how we made Y. This is a contradiction with the second property of X!

This means our assumption must be wrong.

Therefore X is equal to 0.

So 1 - 0.999... = 0, wich is another way of saying 1 = 0.999...

Originally posted by TheMaster37
Bird B will NOT eat all seeds.

The remaining seeds after 1 second is 1
The remaining seeds after 1,5 second is 2
The remaining seeds after 1,75 second is 3

Leaving some words out the number of remaining seeds is:

4, 5, 6, 7, 8, 9, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 18, ...

This diverges clearly. So after 2 seconds (assuming nor the farm ...[text shortened]... s 'paradox' only excists because you claim that infinity - infinity IS defined and equal to 0.

No, Bird B will eat all the seeds. As davegage mentioned before, this has to do with the cardinality of the two sets (the number of seeds planted and the number of seeds eaten). In this case they are the same, so after an infinite number of iterations all the seeds planted will be accounted for by being eaten by the bird.

You say 1 - 0.999... = x where x = 0
I say 1 - 0.999... = x where x = 0.000...1

So if there are two points on a line and each point halfs the distance between them regularly then the points will eventually meet? No! They will not! In fact if you draw a graph and try this out for each time you half the ditance you can leave the graph the same and just say the proportions on the graph are different. But I guess that brings us back to square one because after it's been divided enough times [namely infinity(which it never can be)] the proportions would be 1/infinite (and to be consistent you must say this equals 0). But to me it is senseless because proportionately the points do not come any closer than the were when the started.

EDIT I just searched "1/infinity" and the first result that came up is helpful. It supports everything I've repeatedly been saying. And who can argue with Dr. Math? heres the link: http://mathforum.org/library/drmath/view/62486.html

Originally posted by PBE6
No, Bird B will eat all the seeds. As davegage mentioned before, this has to do with the cardinality of the two sets (the number of seeds planted and the number of seeds eaten). In this case they are the same, so after an infinite number of ...[text shortened]... he seeds planted will be accounted for by being eaten by the bird.

I'm still confused but thinking about it. Please explain what is wrong with my reasoning here:

Imagine that after the man plants the tenth seed and then, with EVERY seed the man plants, the crow eats the first seed which remains in the row.

Then in every iteration the number of seeds left is constant and equal to 10. If this is true, is makes no sense to me that after the two minutes, there is no seed left. There should be 10, as the value is constant.

If this is true, then it must be true that the number of seeds left is infinite in the case where with every tenth seed the man plants, the crow eats one seed. Because seeds in the previous case after the 12th seed are always less than in this case.

When you ask "how many seeds are left" you have to consider the series "Seeds left" not the series "eaten seeds".

🙄

Originally posted by TheMaster37
Bird B will NOT eat all seeds.

The remaining seeds after 1 second is 1
The remaining seeds after 1,5 second is 2
The remaining seeds after 1,75 second is 3

Leaving some words out the number of remaining seeds is:

4, 5, 6, 7, ...[text shortened]... ause you claim that infinity - infinity IS defined and equal to 0.

No, your analysis is wrong. Bird B eats all of the seeds.

This 'paradox' only excists because you claim that infinity - infinity IS defined and equal to 0.

Please be good enough to point out precisely where I made the statement that 'infinity - infinity is defined and equal to 0.' I am well aware that infinity - infinity is undefined because I happen to understand the concepts associated with cardinality.

I never stated that any paradox exists -- it was a poser, as in a trap many people fall into. Bird B eats all of the seeds in his row; Bird A leaves infinitely many seeds. Is this a paradox? No, because of cardinality -- you can create a 1-1 mapping between an infinite set and a subset of itself.

Originally posted by bobbob1056th
You say 1 - 0.999... = x where x = 0
I say 1 - 0.999... = x where x = 0.000...1

So if there are two points on a line and each point halfs the distance between them regularly then the points will eventually meet? No! They will not! In fact if you draw a graph and try this out for each time you half the ditance you can leave the graph the same and j ...[text shortened]... who can argue with Dr. Math? heres the link: http://mathforum.org/library/drmath/view/62486.html

I would like to respond to your posts, but I need some time to understand what you are arguing. I honestly find your arguments very confusing and hard to follow logically. 🙄

1 - 0.9 = 0.1
1- 0.99 = 0.01 etc.
So 1 - 0.999... = 0.000...1.
The only thing is you say 0.999... = 1 and 0.000...1 = 0.

If you have a graph of two points the graph can be any size and mean the same thing. If necessary you could give the proportion of the relative sizes of the two graphs to clear things up. As for what I meant when I said the proportions limit is 1/infinity... If the points are twice as close but you draw the graph the same, you'd have to say the proportions (difference between the two graph scales) are 1:2 (1/2, same thing), next time 1:4, 1:8 etc.

One more thing. I guess it would be correct to say the limit of 0.999... = 1.

Originally posted by TheMaster37
Bird B will NOT eat all seeds.

The remaining seeds after 1 second is 1
The remaining seeds after 1,5 second is 2
The remaining seeds after 1,75 second is 3

Leaving some words out the number of remaining seeds is:

4, 5, 6, 7, 8, 9, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 18, ...

This diverges clearly. So after 2 seconds (assuming nor the farm ...[text shortened]... s 'paradox' only excists because you claim that infinity - infinity IS defined and equal to 0.

The entire process ends after two seconds. By that time bird B has eaten all the seeds.

Originally posted by Palynka
I'm still confused but thinking about it. Please explain what is wrong with my reasoning here:

Imagine that after the man plants the tenth seed and then, with EVERY seed the man plants, the crow eats the first seed which remains in the row.

Then in every iteration the number of seeds left is constant and equal to 10. If this is true, is makes no sense ...[text shortened]... are left" you have to consider the series "Seeds left" not the series "eaten seeds".

🙄

I'll do my best to explain, but explaining infinite processes is difficult when you look at a finite section of the process. If you put a bound on the process, that is to say, "after X seeds are planted", then the Bird has not eaten all the seeds, no matter how large X is. We are talking about a limit though. As others have said, this problem has to do with the cardinality of two infinite sets. If you were to number the multiples of ten, would you run out of multiples of ten before you ran out of natural numbers? No, there are as many multiples of 10 as there are natural numbers. That's what this problem tries to illustrate, but it is deliberately counter intuitive. The farmer takes 2 seconds to go through all the natural numbers, (Mapping 1 seed to 1 slice of time) and the bird takes 2 seconds to eat them, even though it eats 1 after 10 are planted (Mapping 1 seed to 10 slices of time). There are infinite slices of time, even though they are getting smaller, and smaller, so the bird has as many slices of time as it needs to eat all of the seeds. I don't think this clears things up very well... Sorry, it's not easy to extrapolate infinite processes from finite ones.

I think the word infinity is meaningless, someone just made it up to confuse mathmeticians after they thought they "solved" mathematics poor mathematicians.:'( Just kidding. But seriously, anyone who has a question about this should check the link I posted, here I'll post it again: http://mathforum.org/library/drmath/view/62486.html Also I thought it rather amusing that Palynka presumed the bird was a crow (not to say you're wrong, make fun of you or anything) It just sounded funny. Am I too easily amused? And has anyone noticed my tendency to use parenthetical notes?

Originally posted by Palynka
I'm still confused but thinking about it. Please explain what is wrong with my reasoning here:

Imagine that after the man plants the tenth seed and then, with EVERY seed the man plants, the crow eats the first seed which remains in the ...[text shortened]... er the series "Seeds left" not the series "eaten seeds".

🙄

I think the problem here is that you have defined a function to be the difference between two numbers of indefinite and infinite magnitude. When you do that, strange things happen.

For example, as you have framed the question the number of seeds left after some finite number of plantings n is given as the difference between the following two sums:

seeds planted after n plantings = S(n) = 10 + 1+ 1 + 1... with (n-1) 1's
seeds eated after n plantings = E(n) = 1 + 1 + 1... with (n-1) 1's

We know intuitively that S(n) > E(n) for all finite n, and indeed that is borne out through subtraction:

S(n) - E(n) = [10 + 1 + 1 + 1... with (n-1) 1's] - [1 + 1 + 1...with (n-1) 1's] = 10

However, for infinite n, E(n) can be rearragned to appear greater than S(n) as follows:

lim[E(n)]|n-->inf = 1 + 1 + 1 + 1... = (1 + 1) + (1 + 1) + ... = 2 + 2 + 2 + 2...

Now when compared to S(n), E(n) seems much bigger:

lim[S(n)]|x-->inf - lim[E(n)]|x-->inf = [10 + 1 + 1 + 1...] - [2 + 2 + 2 + 2...]

It is no longer clear that S(n) > E(n) for infinite n. It is this is a strange property of diverging infinite sums that makes substraction of two sums unreliable, and in fact indeterminate.

Now if we define the functions above in a slightly different manner, I believe we can use L'Hopital's Rule to show the two functions are of the same magnitude for infinite n. Let's define S(n) as follows:

S(1) = 10;
S(2) = 10 + 1;
S(n) = 10 + (n-1);

for all natural numbers n, and for all other real n:

S(n) = S(FLOOR(n,1))

I slipped back into Excel notation for the above definition - it's supposed to mean that S(n) for real n is equal to S(n) for the closest natural n les than the real n, so S(10.5) = S(10). This lets us define the functions for all real n. We define E(n) similarly.

Taking the ratio of these functions, we get:

R(n) = S(n)/E(s) = [10 + (n-1)]/[(n-1)] = [9+n]/[n-1] (for all n > 1)

Clearly, for any finite n greater than 1 this ratio is a positive real number. However, if we allow n to approach infinity we can use L'Hopital's Rule to show that:

lim[R(n)]|n-->inf = lim[S(n)]|n-->inf / lim[E(s)]|n-->inf
.........................= lim[S'(n)]|n-->inf / lim[E'(s)]|n-->inf
.........................= lim[n]|n-->inf / lim[n]|n-->inf
.........................= 1

Since the ratio is equal to 1 for infinite n, this implies that S(n) and E(n) become equal in magnitude for infinite n. Now that we have a definite determination of their relative size, it makes sense to say:

{lim[S(n)]|n-->inf} - {lim[E(n)]|n-->inf} = 0

And since the difference between the two functions is 0 for infinite n, there will be no seeds left uneaten. The farmer's finite head start evaporates in the face of infinite plantings and corresponding infinite bird snacks.

I know this explaination was a little long winded, but I hope it helps.

Originally posted by bobbob1056th
I think the word infinity is meaningless, someone just made it up to confuse mathmeticians after they thought they "solved" mathematics poor mathematicians.:'( Just kidding. But seriously, anyone who has a question about this shoul ...[text shortened]... And has anyone noticed my tendency to use parenthetical notes?

I just checked out that link, and it's not very good. I will agree that the expression "1/infinity" doesn't have concrete meaning, but the way they explain it is pretty club-footed.

Here's a quote:

So, to finish up, you are perfectly correct in saying that "1/infinity
= infinitesimally small." But only if you realize that you REALLY mean
"1 divided by a REALLY big number is a REALLY small number."

I agree with this part:

So, to finish up, you are perfectly correct in saying that "1/infinity
= infinitesimally small."

But this is wrong:

But only if you realize that you REALLY mean "1 divided by a REALLY big number is a REALLY small number."

Infinity (as used in this context) isn't a big number. It's not even a very big number, or a very, VERY big number. It's infinitely big. It's beyond the realm of the biggest number you can think of. It's beyond the realm of the biggest number Dr. Math can think of. And that's big.

More importantly, when you divide 1 by this colossus, you don't get a very small number, or a very, VERY small number, you get something vanishingly small. So small that if you ever looked at a number and said "that's small enough!" you'd be wrong. In fact, it's so close to 0 that there is no number you can squeeze between it and 0. Try it! Didn't work, did it? Nope. It's so close that you could say it is 0. And we do. And it is.

I think the confusion lies in trying to substitue a very big number in place of infinity. None of those results will point you in the right direction, because they're in a different class. "Infinite" comes from the Latin word "infinitum" meaning "without bound", which is a property no real number can claim.

What Dr. Math fails to mention is that numbers are concepts, too. They are not conrete. I can find examples of 1's and 2's in the real world, groups of things that have that certain 1-ness or 2-ness that let me identify them correctly as 1's or 2's, but again this is a concept (incidentally, we call this property cardinality - it has been discussed many times in this forum). However, that doesn't mean "1/2" is nonsense. Our mathematical syntax is our shorthand for dealing with these concepts. When Dr. Math says "1/inf = 0" is nonsense, he may be technically right because the syntax is lagging the ideas, but the idea described by "1/inf = 0" makes perfect sense to me.

This is getting very philosophical. I think I need a cold compress.

Originally posted by PBE6
I just checked out that link, and it's not very good. I will agree that the expression "1/infinity" doesn't have concrete meaning, but the way they explain it is pretty club-footed.

Here's a quote:

[i]So, to finish up, you are perfectly correct in saying that "1/infinity
= infinitesimally small." But only if you realize that you REALLY mean
" ...[text shortened]... rfect sense to me.

This is getting very philosophical. I think I need a cold compress.

You say infinity is not a big number (but an "infinitely big" number), I say that is a matter of semantics. You say infinity is a number, I say infinity is not a number. You say "when you divide 1 by this colossus, you don't get a very small number, or a very, VERY small number, you get something vanishingly small." I say that is semantics (not to mention it is impossbile to divide 1 by infinity). you say"So small that if you ever looked at a number and said "it's so close to 0 that there is no number you can squeeze between it and 0. Try it! Didn't work, did it? Nope. It's so close that you could say it is 0. And we do. And it is." I say that the number you are looking for is 0.000...1 or 0.000...2, 0.000...3 all the way "up" to 0.000...999... for that matter. Of course you say these numbers are equal to each other (0) You say that "What Dr. Math fails to mention is that numbers are concepts, too." I say that, again, this is a matter of semantics. (ie you could say anything is a concept) After stating that infinity is not a number, he goes on to imply that it best defined as a concept, and nothing more, whereas numbers are numbers, they have a very clear definition and representation. I admit maybe the way he explains is club-footed as you say. Also you seem to think I highly esteem Dr. Math because him proving/agreeing that 1/infinity = invalid function. However he is only a human like us and yes, I've found mathematical blunders on his site, but I found the link I posted to be a good way to show people why 1/infinity is invalid/showthat a guy on the internet agrees with me. I'd also like to appologize for any offence taken by either you or davegage, as I have (attempted to) invalidate just about everything either of you said. This is the nature of debate, it doesn't mean I think u guys (I presume) are idiots. Also I'd like to tell davegage that at first I took offence at your responce, only to later realize why (all those statements about your question being invalid/rediculous). At least no one thought what I was attempting to explain made any sense. And things naturally lightened up at the end. This could be a good conclusion to the thred although I'm sure some people may want to post a responce. Good night 😴