[SpiralEye]

In the movie I.Q. Zeno's dichotomy paradox is discussed briefly. If you have a distance, say 1 meter, and you go half that distance, and half the remaining distance, and half of that, and so on, you never reach the other side. You get REEEALLLLY close, but don't ever travel 1 meter. This is, I think the conceptual illustration of a calculus limit.

Well, one day I was sitting in class when I thought, 'Hey! What if instead of going half the distance, you go nine-tenths the distance?' So, the first hop you travel .9 meters. Then you hop nine-tenths of .1 meters, which is .09 meters. The next hop you would travel .009 meters, and so on. If you add up each fraction of a meter that you travel as you go, you get .9999. . . repeating.

And this is interesting because .999 repeating mathematically is EQUAL, in every respect, to 1 (Really! Look it up if you don't believe me!). So, if you travel nine-tenths of the remaining distance with each hop, you actually DO travel to the other side. But there is a catch. You can never stop hopping! If you ever do, there will be a minute distance remaining to cross and you just won't make it to the other side. BUT if you keep traveling (making the total distance you travel expressed as .9 repeating), you DO get to the other side! Makes no sense, I know, but it's fun to think about, eh? I call this one 'Royal's Dichotemy Paradox.' Too pretentious? Who cares. The name stays.

It's also a good analogy for prorgress in life. As long as you keep moving forward, you'll make it (to wherever you want to go), but if you stop moving, you're not where you want to be.

[ends long random thought for the day]

Royal

[Impaler]

It's actually more a model of a tangential curve, which never reaches the x or y axis, but it gets continually closer.

.9... does not equal one. If you round off, it does, but it will never equal one.

This theorem is also the basis for all theories surrounding the speed of light. Matter, theoretically, will travel at an exponentially decreasing acceleration rate as it reaches the speed of light (essentially, you accelerate half as slowly from 99.98% to 99.99% the speed of light). It's relative to the energy need to move the mass at such speeds, versus the available energy. You need tremendous amounts of energy to move that fast, and once you hit the speed of light, all mass disappears and the object is completely energy.

This is all theoretical, of course, but, eh.

[sacrelicious]

If .999... doesn't equal one as Impaler states, then I find it a rather more depressing metaphor for life- no matter how long you travel, you'll never reach your destination. It will always be just out of reach.

[SpiralEye]

Impaler--My idea isn't that .99 or .9 equals one, but that .99999999....etc ad infinitum equals one. Check out this link:
http://forum.swarthmore.edu/dr.math/problems/point_9_repeating.html

This should explain how .9 repeating IS mathematically equal, in every respect to one, but that .9999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999 is nowhere close.
Never heard that theory about matter turning to energy at the speed of light. Interesting. .. mm..I have another random thought brewing about that. [evil grin]

Sacrelicious--don't be depressed, the theorem and analogy still stands (see above).

[PandaX52]

I think .999xinfinity could equal one....it makes sense if you think about it, and it explains why we are able to get out of bed every morning and actually reach our door with out being stuck moving in exponentialy smaller distances.

Just think about it... .999 times infinity...an infinite line of nines...I say it could equal one.

How do you represent infinity with a formula, the only real way to communicate math concepts, without actually using an infinite amount of characters?

Now there is a thought....

(edit) I think the real problem lies not in the fact that we can't round .99999xinfinity to one, but the fact that we can't comprehend infinity itself.

[ May 22, 2001: Message edited by: PandaX52 ]

[Impaler]

Spiraleye: Dr. Math is somewhat fallacious. Number one, 1/3 IS .333... (remember that ... is the mathematical term for a repeating phrase). The logic on what he is going by is that 1/3 is .3..., 2/3 is .3... + .3..., so thusly 3/3 must be .6... + .3... It IS a paradox in and of itself, seeing as 3/3 really IS one. However, because fractions are not always true representations of decimals (they're more like, estimations or icons, sort of like how infinity doesn't actually represent infinity, or how pi isn't 3.14longlistofnumbers), the paradox falls.

Thusly, .9... does NOT equal 1. It's a somewhat hard concept to grasp, never reaching one. But does happen. It's almost the exact model of reaching infinity. You keep adding one to your number, and infinity keeps getting one number farther away.

1/3 = .3333334
2/3 = .6666667
3/3 = 1

[ May 22, 2001: Message edited by: Impaler ]

[Impaler]

And since you're leaving your bedroom at a steady rate, and not an increasing one, it's quite possible to walk around.

However, go to a gym, road, whatever, and run your hardest. Notice how much easier it is to go from 0 to 5 mph than it is to 15 to 20 mph? The amount of energy you need to push your 200 pounds of flesh increses sharply as your acceleration rate decreases sharply.

Physics is way fun.

And bah on the metric system.

[SpiralEye]

Impaler--Haha! Now we're getting into the accuracy of our representations that we use in math. Well, Trying to imagine the concept of infinity is beyond our mental capacity, but the use of a symbol to represent such an abtract concept does not mean the symbol is inaccurate. Thus, .999... et al is an accurate representation of an infinite number of nines following a decimal point.

Now, in my first post I said I thought it was an illustration of a limit, but you're right, it's not. Because I was referring to continually adding HALF of the distance and was thinking of my 9/10ths variation I got 'em mixed up. I understand the concept of getting REAAAALLLLY close and not crossing--Zeno's dichotomy paradox holds true as a paradox. But his paradox deals with half the distance (.5 + .25 + .125 + .0625 etc) and is represented better by the graph of, say, a logarithm. but my variation is, I think correct.

Try this link for the proof (and a whole argument) as to why point 9 repeating DOES equal one.
[url=http://everything2.com/index.pl?node=Why%20one%20doesn't%20equal%20point%20nine%20repeating]http://everything2.com/index.pl?node=Why%20one%20doesn't%20equal%20point%20nine%20repeating[/url]

royal

[Hlid Skjalff]

Come on, guys, you are loosing it here!!!
It is just about math, that you can move half a distance or 90% of a distance per unit time, have you ever tried to make a step that was 90% of .1m??? thats not right...Yes, when we are talking about functions, asymptotes
and convergence it is relevant, but the theory of the limit is HARDLY applicable to our everyday life... And for those who is interested in 'speed of light' read Stephen Hawking's - Brief History of Time (if you still didn't :-))) Btw, you know there is such a thing as a 'dissappearing table'problem ? look it up on the internet, kinda cool... it actually disappears when you remove only HALF of it :-)))
I like to fiddle around with the infinities much more than with near-zero values... Like bodies with finite volume (converging), but infinite surface area (diverging) hee-hee math is cool !!!

[Impaler]

That entire page just proves everything I've said here. Fractions are not true representations, because of remainders, etc.

[edit]And infinity can be grasped mathematically, if you like fractals. in the real world, infinity is well beyond our comprehension. It's also true that most people think infinity is just a little more than a million. However, if you grasp the notion, the actual thing is somewhat irrelevant.[/edit]

[ May 22, 2001: Message edited by: Impaler ]

[Giant Hamster]

How about this. Shut the fuck up.

err...oop. thats my subconcious speaking.

*clicks off subconcious*

ok, back to the thing. how about this:

Just cause you got to your door, aren't you still .9 or more away from your car? and when you get there, aren't you still not quite at the street? and after that, not quite at the next house?

just cause you got "somewhere" doesnt mean your at the end. so really, you still haven't gotten anywhere.

"Math never got anyone anywhere except .9% of the way."

[Giant Hamster]

oh, and another thing.......you can even see that working in this post.

after a remark. someone is going to come in here and add their retort. then someone else will repeat that....on and on and on.

[wayfinder]

1/9 = 0.111...

1/9 * 9 = 0.999....

1/9 * 9 = 9/9

9/9 = 1

whee

[Ragnarok]

I think fractions represent the real number. Decimals are there because we understand them easily, but the fraction will represent the real thing.

[Chris]

fractions aren't real numbers.

9/10 != 1
19/21 != 1

simple as that. But 1 == 1

So you .9 of a step (each step) you will never ever ever ever ever ever ever reach 1.

You will get SO close that it will seem like you are at the destination, but technically (if you could even do this) you wouldn't be at the destination, you would be .9999999999999999 there or whereever you ware in your steps. But you aren't there

[leat hacksaw]

actually, some scientists got a wavepacket
to travel faster than the speed of light through a bunch of specially prepared cesium atoms, i.e the waveform was detected completely leaving the prism BEFORE it had completely entered it. very interesting.

about the whole 0.9999.... = 1, well i dont think it does, really. Think about it. If you have exactly one table, then you have 1 table. If you have .99... of a table, then you dont have one whole table, however small the difference is between this incomplete table and the full monty of a table. (i know that the 0.99... of a table could still be 'called' 1 table, but just take the analogy at face value - i couldnt think up a better one. its late )

anyway, thats my 2 bob worth.

[Ragnarok]

If you take a knife and make a little cut in a table, is it a table?
Following your post, it isn't.

[Awetopsy]

I hate math...

[v1510nAry]

With the half way thoery, you would eventually reach the other side...

[Chris]

no you wouldn't, you would just get so damn close that it would seem you would reach, like atom-size length away.

[Chris]

hey, just thought of something, if you were a atom-length away and you went half that length would you not split the atom?

[Hlid Skjalff]

Ragnarok, stop talking If you didn't understand what I was talking about!!!
Its like that - first you remove 1/4 of the table-top (from the middle) then (1/8)/2 from the centers of the two segments left and so on, then when you let the lim of iterations to approach infinity - you will make the table 'dissapear' (SAW THAT QUOTE SIGNS?) i.e. NOT VISIBLE, while the ammount of the 'table removed' will only approach 1/2 of its original length...

[Ragnarok]

I wasn't answering you, Hlid Skjalff

[Hlid Skjalff]

Whatever, I'm lost here, if you weren't - I'm sorry - everybody's just posting some incomprehendable BS not saying to WHOM they relate, just saying 'You' duh! sorry

[Etict]

Ok, this is my opinion.

If time can be divided (not by as humans, but by Universe) into infinite amount of units (a unit of time, that is, which is impossible for us to record on any device, for we are stuck to our 3 dimensions), then we can, I quess, also say that space can be divided into infinite amount of units, units so small that we can't see them because blah-blah-blah and yadda-yadda-yaaah, read above.

Anyway, point being, everything in universe is actually built up by infinitely small units of space/time.



SO, that means that because we travel that infinitely small distance in infinitely small amount of time, it's obvious why we get there. Because UNDER and ABOVE of these dimensions which we travel in (hold on guys I'm trying to get this together... ), are dimensions of time and space that ACTUALLY RUN the whole universe.

Well, this obviously goes to discussion like "well show me the God and I believe in Him", but that's my theory.

Cheers.

[Impaler]

It's a difficult concept to grasp, not being able to get there.

Just think of going half the distance each step as slowing down twice as much each step, meaning that you will continually move slower and slower each step, comparatively. In reality, you're moving just as fast, you're just taking smaller and smaller steps.

If you think that's mind boggling, consider this. You're in a space ship traveling 99.9999999999% the speed of light. You turn on the headlights. What happens?

[SpiralEye]

I can't believe this post got so many responses!
Impaler--okay, did you read where you try to pick a number between .9 repeating and 1? The number between those two numbers is. . .zero. If there is no number between .9 repeating and 1 then. . . they are the same. The fact that .9 repeating and 1 are the same number is a fault (in Microsoft language--feature) of the decimal numbering system. Anyhow, I guess I can't explain this any better. If you don't believe it, nothing matters anymore and my whole world will cave in! Um. Maybe not, but I still had fun debating this, even if I am wrong.

[Chris]

There is always a number between .9 and 1

.99 is between .9 and 1

.999 is between .99 and 1

.9999 is between .999 and 1

and so on and so on.

Im: I like the headlights thinking. I would probably see the light exiting the headlight going very very slowly, but still traveling faster then me.

[SpiralEye]

Chris--not a number between .9 and 1, but .999999999999999999999, ad inifitum and 1.
.9 + .01 = 1
.99 + .001 = 1
etc
so if you have a decimal point followed by an infinite number of nines, the number between it and one would be: .000...ad infinitum...1 this infinity in the middle negates the one at the end--you never reach it, so the number between .999... and 1 is really .000... which is a milit that reaches Zero. Zero. So Zero is the number between .9 repeating and 1. Meaning, they are the same number.

[SpiralEye]

Ah, as for the Headlights thing. I think that like sound waves, the lightwaves would be all bunched up in front of the spacecraft, since light in a vacuum travels at a constant velocity. In space where there is dust, maybe the light is too slow and returns to the source (!) when a near-lgihtspeed is reached? Or maybe all the light stays bunched up, except where it scatters to the sides, until the vehicle slows down enough that it can move in front of the vehicle. Interesting quesiton, that.