.999... = 1

[ser stiefel]

I have a theory.

You (TheProwler) obviously understood Jonesthecurls post to be a bit of fun using Achilles and the Tortoise because you presented Aristotle as your rebuttal.

Hmm.... So I was thinking about giving another fun example using dropping an egg on the floor and discussing the dichotomy paradox (which perplexed me to no end when I was a kid). But then I had an epiphane!

I think you are a student who came across this thread, and mentioned it to your teacher/professor/instructor and he/she encouraged you to post arguments in opposition to the idea that 0.999... = 1 as an intellectual exercise.

This is the quote that has given you away:

I am arguing against the idea that N will ever reach infinity.

Are your really making this argument?

[AAFitz]

could it be that mans mind cant quite grasp infinity?

could we be failible?

could there be a greater power that has estabilished all this

could it be that somethings we will never quite understand because we were not created with the ability nor purpose to understand.

If this really is such an easy and simple question as many have said then how come there is so much controversy....infinity means just that there is no closure no final equation. Man cannot grasp that something might not have an expliantion because we live in a a finite world at least on earth everything has a solution and final ending...but that is not so in eternity and also in the infinity of space. Hahaha for all our great accomplishments man really is a pathetic creature and very small in true knowledge

Oh yeah, tell that to the crocodile. Its been on this planet for millions of years. He still sleeps in the water, and we wear him as boots, belts and carry our money around in wallets made of him. We are discussing a creature that has existed for millions of years, on a planet that we completely dominated and took over. ((Please do not think I do not understand the fact that this same intelligence, may very well be our undoing. Furthermore, and technically speaking, such undoing only really matters to those around when said undoing unfolds. The rest of us still get to play with our toys and internet until it happens...))

The very fact that we can ask such questions, and ponder them shows your statement to be pathetic, and small in knowledge. The human imagination actually knows no bounds and no limits. It does take time, however. Further, you are assuming that people dont actually understand infinity, simply because everyone doesnt agree what infinity is... the fact is, some may know and understand it perfectly and be perfectly correct in their opinions. The fact that others disagree does not make them incorrect, only not fully accepted. Those that realized the earth orbited the moon, were thought of as madmen in their day. They of course were right, even while the rest of the population disagreed.

The great thing about being correct, is that it does not require acceptance of all, many or even one other person. If you are right, you are right, and it simply does not matter who disagrees, so, to say no one here understands infinity, or lacks the imagination to fully understand it, only shows that you do not understand the infinite possibilities of the human mind.

[jonesthecurl]

How Doc_Brown and others can read something this idiotic and not correct your mistakes is a testament to the lack of integrity people have when they've "picked a side".

This is a perfect example of someone who is patting himself on the back for "understanding" something when he doesn't have a clue.

You didn't disprove him...

Who? jonesthecurl?

How about he comes up with his proof to show how he came up with 0.9999...? You know, with a somewhat proper solution; not just using vague wording.

Let's call Fred's speed F.

Joe's speed is J.

F = .9J

To simplify things, pick actual speeds that these two travel that will satisfy the 90% condition.


I can go on naming variables and solving the equation for the time that Joe catches up to Fred.

But I'd like to see jones show how he is getting 0.999...

"He goes 90% of the way, then he goes 90% of the next distance, and so on."

And he comes up with 0.999...!!! Hahahaha!!! Plug in some real numbers and you can see there is no 0.999... forming. First snapshot, Fred went 9 yards. The second snapshot, Fred went 8.1 yards. The next, he went 7.29 yards. So where's the 0.999... coming from?


We should all know that if we plot this on a graph, time on the x-axis and distance on the y-axis, we are going to just have two lines with different slopes. And they are obviously going to intersect if you start the slower slope at a greater distance than the faster slope.

His explanation is bloody ridiculous!

"This is of course, nonsense, and it turns out he does pass him.
Because, as it happens, .999... is equal to 1. It's just a different way of expressing it.
If it weren't, overtaking anything would be physically impossible."

This has nothing to do with 0.999...!!!!

Does he not realize that his explanation of the "problem", he is only allowing smaller and smaller periods of time to elapse between taking snapshots? The first segment, Joe ran 10 yards. The next, 9 yards. The next, 8.1 yards. And so on. But he runs a consistent speed. So obviously we are dealing with smaller and smaller periods of time. And the periods of time will continue to get smaller and smaller.

We have two numbers that are of interest. The TIME that has to expire before Joe catches Fred (call that T). And the DISTANCE that Joe has to run before he catches Fred (call that D). As long as we continue to allow smaller and smaller segments of time to elapse...as long as we keep stopping to take measurements every time Joe goes 90% as far as the last time we stopped for a measurement...the limit of the time that will expire is T and the limit of the distance Joe has run is D. (As long as we are allowed to pause time to take our snapshots...ha!!)

So if we were mapping out these two lines on a graph, we'd just be slowing down...slower....slower....slower....slower....never allowing the two lines to intersect. Because we have to stop to take that snapshot.

But unfortunately for the artsy types that read this little "problem" and found it interesting, you can't slow down time. Oh no!!! Did I just say that?? I take it back!!! All of you can slow down time!!! You are so awesome!!!


So, john9blue, did you also think "Logically, joe can never catch Fred, he can only get .999999.... of the way."?

Your image of making a graph of the example is exactly the right approach.

It is why, in the original Xeno's paradox of Achilles and the tortoise. Achilles does pass the tortoise. The reason that there is an apparent paradox is that the Greeks didn't have graphs to enable them to envisage this. In the original paradox, Achilles runs ten times as fast as the tortoise, and Xeno has him always catching up the distance the tortoise has run since the last calculation, by which time the tortoise has moved one tenth of that diatance again. The solution (hold on to your hat) is that the distance the tortoise has moved before Achilles overtakes him is 1/9 of the original lead that the tortoise had. measuring it like Xeno does and putting it in modern numerals we get 0.11111111... of the original lead.

It is an infinite series and it does equal 1/9. or Achilles will never pass the tortoise.

And 0.99999... does equal 1 or Joe would never pass Fred in my previous example.

[natty dread]

Concept of infinity is easy to comprehend.

If you think of a waveform, the wavelength of the wave correlates inversely to the frequency. Thus an infinitely large frequency gives an infinitely small wavelength, and vice versa.

[TheProwler]

could it be that mans mind cant quite grasp infinity?

could we be failible?

could there be a greater power that has estabilished all this

could it be that somethings we will never quite understand because we were not created with the ability nor purpose to understand.

If this really is such an easy and simple question as many have said then how come there is so much controversy....infinity means just that there is no closure no final equation. Man cannot grasp that something might not have an expliantion because we live in a a finite world at least on earth everything has a solution and final ending...but that is not so in eternity and also in the infinity of space. Hahaha for all our great accomplishments man really is a pathetic creature and very small in true knowledge

Oh yeah, tell that to the crocodile. Its been on this planet for millions of years. He still sleeps in the water, and we wear him as boots, belts and carry our money around in wallets made of him. We are discussing a creature that has existed for millions of years, on a planet that we completely dominated and took over. ((Please do not think I do not understand the fact that this same intelligence, may very well be our undoing. Furthermore, and technically speaking, such undoing only really matters to those around when said undoing unfolds. The rest of us still get to play with our toys and internet until it happens...))

The very fact that we can ask such questions, and ponder them shows your statement to be pathetic, and small in knowledge. The human imagination actually knows no bounds and no limits. It does take time, however. Further, you are assuming that people dont actually understand infinity, simply because everyone doesnt agree what infinity is... the fact is, some may know and understand it perfectly and be perfectly correct in their opinions. The fact that others disagree does not make them incorrect, only not fully accepted. Those that realized the earth orbited the moon, were thought of as madmen in their day. They of course were right, even while the rest of the population disagreed.

The great thing about being correct, is that it does not require acceptance of all, many or even one other person. If you are right, you are right, and it simply does not matter who disagrees, so, to say no one here understands infinity, or lacks the imagination to fully understand it, only shows that you do not understand the infinite possibilities of the human mind.

Rah rah rah!!!

Three cheers for us!!!!

C'mon AA...with all our "great intelligence", we still haven't scratched the surface. Yeah, we might "think" we have a lot of things nailed-down, and we might "think" that our imaginations are unlimited. But look at some of the facts. As an example: Space Travel. Great explorers that we are...we've gone....to our own moon!! Wow.

And we have no idea how powerful our imagination really is....or how accurate.

I agree with THORNHEART. No offense.

[jonesthecurl]

You mean we're failible?

[anonymus]

but if we drop the numbers and do it like this? (i only browsed this thread and am by no means a mathematician) but if we compare 1 and 0.999... with identical twins. indentical twin A reprecents 1 and identical twin B reprecents 0.999.. noone would argue that they are the same but they ARE identical (twins)

nah forget it i was going somewhere with this but you know.. kind of got bored half way through..
/

[Timminz]

Seriously, why would the pattern all of a sudden end at infinity?

Because it is impossible to arrive "at infinity".

He is trying to make a point against me....but he is actually defining the core problem when dealing with the concept of infinity.

If 0.999... never reaches an infinite number of decimal places, it will never equal 1.

I don't know how many times you can ignore a simple fact, but let's try again.

0.999... is not reaching towards anything. It is a static number, with an exact value of 1.

In other words, it does indeed have an infinite number of decimal places. That's what the '...' represents.

[jiminski]

yes i think the concept of infinity is beyond human comprehension. like the infinite universe it gets a little bigger as we reach its limitless limit.
One day (if we do not die out as a species) through biochemical 'computerised brains, genetics and collective consciousness we will increase our intellects to a level which we now consider to be godlike, we will achieve our 'heaven on earth' .... and i assume we will only learn to prove that we genuinely, actually and in reality do know nothing instead of it being a trick which anticipates our flawed intellects....

[TheProwler]

Your image of making a graph of the example is exactly the right approach.

It is why, in the original Xeno's paradox of Achilles and the tortoise. Achilles does pass the tortoise. The reason that there is an apparent paradox is that the Greeks didn't have graphs to enable them to envisage this. In the original paradox, Achilles runs ten times as fast as the tortoise, and Xeno has him always catching up the distance the tortoise has run since the last calculation, by which time the tortoise has moved one tenth of that diatance again. The solution (hold on to your hat) is that the distance the tortoise has moved before Achilles overtakes him is 1/9 of the original lead that the tortoise had. measuring it like Xeno does and putting it in modern numerals we get 0.11111111... of the original lead.

It is an infinite series and it does equal 1/9. or Achilles will never pass the tortoise.

And 0.99999... does equal 1 or Joe would never pass Fred in my previous example.

It has nothing to do with 0.999...

Somebody else should jump in because you are unable to grasp what I'm telling you.

You don't seem to understand that all you are doing is taking smaller and smaller periods of time. By only allowing Achilles to catch up to the spot that the tortoise was "at last measurement", we are taking a smaller amount of time, because Achilles' speed is consistent and the distance is smaller. And smaller. And smaller. Given this "stop and start" method or recording their positions, you will never get to the point in time where Achilles reaches the tortoise. Never.

This is an example of reaching a limit:

The limit of time passing before the next measurement (starting from the previous measurement) is zero. It never reaches zero, but it gets closer and closer.

The limit of the distance between the Achilles and the tortoise is zero. It never reaches zero, but it gets closer and closer.

The limit of the total amount of time that has passed, as each measurement is taken, is the time that Achilles will actually reach the tortoise (if we'd just stop taking those measurements every split-second!). It will approach that limit, but it will get closer and closer.

Again, this is all because you are taking your measurements as smaller and smaller periods of time are passing. You will never take a "last measurement". Because you are always "freeze framing" when Achilles reaches the last position of the tortoise. So this is also an example of infinity. Because there will be an infinite number of measurements taken. And you still won't be done. See that? And you still won't be done.

This has nothing to do with 0.999...

jones, ask Doc_Brown if I am right or wrong about this. Maybe he can convince you where you are making your mistakes. And maybe you will start to understand your limits*.



* See last post to AAFitz.

[TheProwler]

yes i think the concept of infinity is beyond human comprehension. like the infinite universe it gets a little bigger as we reach its limitless limit.

Timminz has it all sorted out. It's really quite simple. Just ask him.

[Snorri1234]

yes i think the concept of infinity is beyond human comprehension. like the infinite universe it gets a little bigger as we reach its limitless limit.

Timminz has it all sorted out. It's really quite simple. Just ask him.

He's not saying we can grasp the concept, he's saying that we can use it in math.

0.999... is defined as 1 by mathematics. Your objection to it is philosophical rather than mathematical. Your post to jones, while certainly true, should have clued you in to what you are missing. 0.999... is the limit of the sequence (0.9, 0.99, 0.999, 0.9999, ...), therefore it is 1.

Your objection is philosophical because you are arguing that one can never reach, or "finish", an infinte number of tasks (calculating the last digit of 0.999...). While that's certainly a viable objection, Math does not care about that. Math is not the real world, the axioms and definitions it's based upon are not neccesarily true for the real world, they are however certainly true for mathematics. So if you use infinity in mathematics, you end up with using Archimedean properties and therefore conclude that 0.999... and 1 are the same number.


That is fundamentally true. There are no infinitesimal positive numbers because that would be logically contradictory. That means logically that there is nothing between 0.999... and 1. When you compare two things which aren't different in anyway, they are the same.

[Neoteny]

But one has nines and the other has a one...

[jonesthecurl]

Also: there is a difference between counting numbers and measuring numbers.

You can have exactly six coins, but you can't have exactly six yards.

Go back to your graph. Imagine it. The exacct place at which one runner overtakes the other, or Achilles overtakes the tortoise, is shown on the graph.

But the two lines on your graph have the thickness of the pencil you've drawn them with, or the number of pixels you've got your computer to display. The point at which the two lines meet has a thickness.

You can refine this, and theoretically the lines should each have no thickness, being onely one-dimensional. They meet at a point which has no dimensions at all, where the two one-dimension lines intersect.

You can refine the "point" you are actually imagining to the thickness of a pencil line, a single pixel, the width of an atom, or an electron.

You can refine its position as exactly as you like, but the point itself is infinitely smaller than that, having no dimensions at all. the "difference" between the point which you can measure and the "ideal" point is exactly the same as the difference between .99... and 1:

It's not just that it's so small you can't measure it. It doesn't exist. They are the same.

[TheProwler]

You guys are simply lost.

Jones, you don't understand your own problem. Or you are just playing dumb.

You can have exactly six coins, but you can't have exactly six yards.

This is a whole different discussion.

Go back to your graph. Imagine it. The exacct place at which one runner overtakes the other, or Achilles overtakes the tortoise, is shown on the graph.

Yes jones, the purpose of the graph is to help visualize the problem. Drawing the lines to show a point of intersection would help some people understand the problem and the solution.

What you are failing to understand is that by your definition of the problem, if you were to draw segments of the lines and stop each time the fast runner gets to the slow runner's last position, the lines would never intersect. But only because you are "freezing" time.

As I already explained, you are just taking shorter and shorter periods of time; the real problem is with anyone who hears the explanation of the situation and isn't able to realize this immediately.

This has nothing to do with 0.999... You chose "90% as fast". But all you have to do is chose two runners of different speeds. Start the faster one behind the slower one and he'll catch up after a period of time. The "paradox" remains the same. 0.999... has nothing to do with it.

This is simple high school math. You are trying to make it something more, but it's not.

But the two lines on your graph have the thickness of the pencil you've drawn them with, or the number of pixels you've got your computer to display. The point at which the two lines meet has a thickness.

Yeah dude. I know that graphs are visual representations and they are not exact. Now we're getting into early high school stuff.

You can refine this, and theoretically the lines should each have no thickness, being onely one-dimensional. They meet at a point which has no dimensions at all, where the two one-dimension lines intersect.

That is correct. And that is the precision that we have to deal with when looking at problems. We can't draw them as they really are. Again, this is common knowledge.

But it doesn't change the fact that you don't seem to understand your "paradox".

You can refine the "point" you are actually imagining to the thickness of a pencil line, a single pixel, the width of an atom, or an electron.

No, go smaller.

You can refine its position as exactly as you like, but the point itself is infinitely smaller than that, having no dimensions at all. the "difference" between the point which you can measure and the "ideal" point is exactly the same as the difference between .99... and 1:

It's not just that it's so small you can't measure it. It doesn't exist. They are the same.

Oh, so that's what you were getting at. I'm pretty sure that must have been stated on page 1 or 2 of the thread.

jones, you can't understand a simple linear equation involving a couple of runners. Don't fool yourself into thinking you understand infinity.

[frankiebee]

.333+.333+.333=.999

1/3+1/3+1/3= 1

1/3 is not .333

So .999 is not 1

[MeDeFe]

.333+.333+.333=.999

1/3+1/3+1/3= 1

1/3 is not .333

So .999 is not 1

True.
But on the other hand

0.3... + 0.3... + 0.3... = 0.9...

1/3 + 1/3 + 1/3 = 1

1/3 is 0.3...

So 0.9... is 1

[the.killing.44]

yes i think the concept of infinity is beyond human comprehension. like the infinite universe it gets a little bigger as we reach its limitless limit.

Timminz has it all sorted out. It's really quite simple. Just ask him.

He's not saying we can grasp the concept, he's saying that we can use it in math.

0.999... is defined as 1 by mathematics. Your objection to it is philosophical rather than mathematical. Your post to jones, while certainly true, should have clued you in to what you are missing. 0.999... is the limit of the sequence (0.9, 0.99, 0.999, 0.9999, ...), therefore it is 1.

Your objection is philosophical because you are arguing that one can never reach, or "finish", an infinte number of tasks (calculating the last digit of 0.999...). While that's certainly a viable objection, Math does not care about that. Math is not the real world, the axioms and definitions it's based upon are not neccesarily true for the real world, they are however certainly true for mathematics. So if you use infinity in mathematics, you end up with using Archimedean properties and therefore conclude that 0.999... and 1 are the same number.


That is fundamentally true. There are no infinitesimal positive numbers because that would be logically contradictory. That means logically that there is nothing between 0.999... and 1. When you compare two things which aren't different in anyway, they are the same.

Well I was going to say /thread, but now that I see Prowler didn't reply to it, it's more like /typicalofthisthread.

[frankiebee]

.333+.333+.333=.999

1/3+1/3+1/3= 1

1/3 is not .333

So .999 is not 1

True.
But on the other hand

0.3... + 0.3... + 0.3... = 0.9...

1/3 + 1/3 + 1/3 = 1

1/3 is 0.3...

So 0.9... is 1

To simplify things, if you share 100 pizza slices with 3 people. Than 3 persons would get 1/3 of the total. With the logic 1/3=0.3 that would mean that they would all get 30 slices. That cannot be right ?
Therefore 1/3=0.33 is also wrong, cause you would still miss one slice.
In the end, youll have 0.33^(end-limit), but that still wouldn't be the same as 1/3

So 1/3 is not 0.3, is not 0.33, is not 0,333 etc.
SO
0.333^(limit) is not 1/3

1/3 = just 1/3, in some cases we can't write it in a different way.

And sice 0.3333^(limit) is not 1/3

0.3333+0.3333+.033333=0.99999

1/3+1/3+1/3= 3/3 = 1

0.3333^(limit)= not 1/3

0.9999^(limit)=not 1

[Snorri1234]

1/3 = 0.3 ???
To simplify things, if you share 100 pizza slices with 3 people. Than 3 persons would get 1/3 of the total. With the logic 1/3=0.3 that would mean that they would all get 30 slices. That cannot be right ?
Therefore 1/3=0.33 is also wrong, cause you would still miss one slice.

He said 0.3..., not 0.3

[frankiebee]

1/3 = 0.3 ???
To simplify things, if you share 100 pizza slices with 3 people. Than 3 persons would get 1/3 of the total. With the logic 1/3=0.3 that would mean that they would all get 30 slices. That cannot be right ?
Therefore 1/3=0.33 is also wrong, cause you would still miss one slice.

He said 0.3..., not 0.3

Sorry, but 0.333^(limit) is still not 1/3.

[natty dread]

0.33... = 1/3, get over it.

Divide 1 with 3 and represent the answer in decimal. What do you get?

0.33333333333333...

In other words, 0.33..

This is simple if you use trinary numbers:

1 / 10 = 0.1

[nippersean]

yeah bur then there's 0.000000.....1 bit of pizza left.

Is this limit / approaching thing like that stuff you use in calculus?

[Snorri1234]

yeah bur then there's 0.000000.....1 bit of pizza left.

Is this limit / approaching thing like that stuff you use in calculus?

Nope. What you are describing is a finite sequence. It's not 0.999.....9, that's a finite string of nines.

[nippersean]

Well I have exactly the same non finite string of zero's