I recall having seen somewhere a bit of fun via nerdy applications of nerdy theories… that indicated something like when looked at a certain way, a physics theory claims you can never walk out of a room — the idea being you can only decrease the distance between you and the door, you can only get infinitely closer to the door, but you’ll never reach a doorway because the distance can only get closer to zero.
Am I getting infinitely closer to anything here?
Sounds like Zeno’s paradoxes of motion.
It’s just like Zeno’s Paradox, although it’s math, not physics.
A topologist might say that if the door is open, there is no such thing as inside or outside the room.
An engineer and a mathematician (both male) were placed at one end of a hallway, and a naked female was placed at the other end. They were told, “You can make as many moves as you’d like. In each move, you can reduce the distance from you to the female by one-half.” The mathematician said, “That’s similar to Zeno’s Paradox. Even if I make an infinite number of moves, I’ll never reach her. I won’t even bother trying!” The engineer said, “I can get close enough for all practical purposes.” 
Using that reasoning, it’s a wonder you got into the room in the first place!
I wouldn’t call it 'nerdy applications of nerdy theories" though, as nerds know that calculus, helped by a rigorous definition of limits, has taken the teeth right out of this paradox by giving mathematicians and physicists a better understanding of infinity. It is known that one can traverse an infinite number of points in a finite amount of time. Furthermore, it is known how.
Now that I think about it, I guess ‘nerdy applications of nerdy theories’ would be a good descriptor of Zeno and his pals originally discussing this stuff in ancient Greece.
1/2 + 1/4 + 1/8 + 1/16 + 1/32 + 1/64 + 1/128 + 1/256 + 1/512 + 1/1024 + etc… = 1
1/3 = 0.333…
2/3 = 0.666…
3/3 = 0.999… = 1.0
Math is kind enough to act as if something separating two units is infinitely small, just assume it doesn’t exist. The difference between 0.99999(to infinity) and 1 is effectively nothing.
So Zeno loses. Another cheap knockoff of Xenu.
Can you elaborate on that? My assumption on such issues was always simply that, since you can ALSO infinitely divide units of time, you’re traversing an infinite number of points of space in an infinite number of points of time while traversing a finite distance in a finite period. No fancy maths involved.
If there’s a version WITH fancy maths, I’m all ears!
Not to open up bring back ghosts of old threads, but the difference isn’t “effectively nothing.” It is nothing. 0.999… is equal to 1.
In real analysis, there is no such thing as an infinitesimal element, that is, a positive number that’s smaller than any other positive number. So as pulykamell points out, we don’t “assume” the difference between 0.9~ and 1 doesn’t exist because it simply doesn’t.
On the other hand, it is possible to develop nonstandard models of analysis in which infinitesimal elements exist. It’s then possible to use them to develop the calculus using infinitesimals instead of limits. I’ve never worked with these models, so I won’t try to say more.
Danger! Danger!
Reference to past threads here, please do not rehash the whole thing here…
You can see that if you have the right perspective, Zeno’s paradox is easily reconciled, as you have done above. What takes a mathematical education is getting that perspective.
It isn’t so much that you need fancy math to work this out; indeed, anybody can tell that you can make it across the room and out the door. We didn’t need fancy math tools to figure out that motion is possible; we had to find an underlying philosophy of infinity and the infinitesimal before our math could reflect that obvious reality. The process of discovering and setting up a rigorous foundation for the calculus (and real numbers) helped physicists and mathematicians hone their ideas about infinity, continuity and motion. So when people ask question about things like Zeno’s paradox, it isn’t that they need math to tackle the question, they need the philosophy of infinity that modern math has developed.
Whole books about our modern understanding of these topics have been written, but I can try to expand on it later if you want, though I’m hardly the most qualified on this board. For some reason I’ve read a lot of pop science books on this topic in my life, in particular “Everything and More” by David Foster Wallace.
Isn’t this something to do with the Planck length and the Planck time? In other words when you get down to these infinitessimally small measurements of distance and time you just can’t divide things further because that’s the point the physical laws of the universe up and quit on you. (And who can blame them?)
What if it’s 0.999… the length of a treadmill?
and one of the treadmills has a prize behind it.
I don’t want to rehash this either but I did see an incredibly simple explanation of it once that anyone can understand.
1/3 = .9999…
1/3 + 1/3 + 1/3 = 1
Since 1/3 = .9999… then it is the same result if you substitute .9999… for 1/3 in any equation which makes sense.
Crap. I screwed that up. .9999… should be .3333… like below.
1/3 = .3333…
1/3 + 1/3 + 1/3 = 1
Since 1/3 = .3333… then it is the same result if you substitute .3333… for 1/3 in any equation which makes sense.
( Scary voice on the phone from 1408) Even if you leave the room, you can never leave the room! (/scary voice on the phone from 1408)
You just have to go the whole .99999999999999999999999999 yards