So they're walking normally, not taking smaller and smaller steps? In that case, there is an event that ends the exercise - the person reaches the wall.
I havent done anything like this in a long time, but isn't it something like this: Please Register or Log in to view the hidden image! Ok, i left out a few steps.. Please Register or Log in to view the hidden image! I may have gotten the primitive wrong though. Like I said, I haven't done this in ages Please Register or Log in to view the hidden image!
No, the person is taking smaller and smaller steps, just taking each step more quickly. His velocity relative to the wall remains the same. This is not my own idea, I read a similar version somewhere on a math site, but I don't have a link. I had read the gedanken of crossing the room here on sciforums a few years ago, then a similar timed version much later at a science website.
How many metaphorical flies must die to entertain you? Is there no limit?! Turn from your murderous ways! Join me at http://www.mathematicswithoutcruelty.org/ You can make a difference!
Yes, that's correct. I tried searching for the site, but no luck. If I remember correctly, what it boils down to is that as the distance covered in each step becomes smaller, so does the time required to cover that distance. An almost infinitely small distance could be covered in an almost infinitely small amount of time. The total sum of all the time increments would necessarily equal twenty seconds because the walker's speed doesn't change as he moves across the room.
All good. This applies to the fly as well, right? As the length of each trip taken by the fly becomes smaller, so does the time required to cover that distance. The fly keeps flying until the trains meet in a finite time, taking infinity trips to do so.
This thread is stepping close to the bad old joke about the mathematician, the physicist, and the beautiful woman. About time to close this thread?
It's closed enough for all practical purposes. Please Register or Log in to view the hidden image! Not yet.
Yes, I suppose it is very similar to the fly gedanken. To go off-topic just a little, another thought struck me. How about the .999999.... = 1 deal? When the 9's stretch out to infinity, the result is a whole 1. When the steps stretch out to infinity, why wouldn't the walker reach the wall, and in a finite amount of time? Wouldn't it be inconsistent to state otherwise?
That's right. The walker does indeed reach the wall in twenty seconds, taking an infinite number of steps. Zeno's arrow does reach the target. Achilles does pass the tortoise. The fly does get crushed by the trains. That's the resolution of Zeno's paradox - an infinite series can add up to a well-defined finite number.
The "0.999... deal" is on-topic here. After all, the infinite series in question is just a constant times 0.111... (base 2). Mathematics would be in deep trouble if the infinite series did not yield the same number as the "fast way".
Pete, Remember, I said the fly gedanken was similar to the person crossing the room. The difference is that the fly can not make an infinite number of trips. After 26 trips, the distance between the trains is smaller than the fly's body. Please Register or Log in to view the hidden image! Please Register or Log in to view the hidden image!
Because there are more variables in real life than in the mathematical formulas, for instance the size of ones shoes. At a certain point reality prevents you from halving the distance Please Register or Log in to view the hidden image!
thats true. so in real life the fly wont travel 150 miles. If you make it a closed system based on the trains' paths, and once they crash it's done, there's no more fly. Graph it all relative to time, and you'll see you'll get a different answer. About 137.5 miles. Here's an interesting picture of Zeno of Elea, with a graph depicting what we're talking about Please Register or Log in to view the hidden image! Please Register or Log in to view the hidden image!
Thats an awful big difference though.. 12.5 miles. I was thinking more along the lines of that the fly need time to land, turn and take off. Also wind would come into play and to what extent the fly is actually traveling in a straight line. These things are uncertainties and cant be calculated.
If only time to land, turn and take off are taken into account maybe the difference would be 20 or 30 yards or so. I think any difference greater than a 100 yards would be a stretch.
