Theoretically, there is no mathematical point that is beyond further fractioning, right? Simply continue to make the number at the bottom of the fraction go on to infinity.
Let’s say I hold a rubber ball out in front of me and drop it. Then I measure the amount of time between when I dropped it and when it hits the ground, each point in its journey in fractions of the total amount of time it takes to hit and bounce. We know we have a finite point in time - the ball bounces. What is the last possible fraction before the ball actually bounces, and why can’t it, too, be divided? (We KNOW that it can’t, because the ball does bounce.)
This may have a very fundamental answer that escapes me. I became a newspaper reporter precisely so I could avoid numbers and math like the plague. (Then, of course, the first thing they make you do at your new job is a local government budget story.)
You can’t measure the last possible fraction before it hits the ground because it always can be divided again. Let’s say you divide up the time into 1/12,000. Before the ball hits the ground, the fraction will be 11,999/12,000. If you want to get even closer, go to 23,999/24,000. You can keep getting closer to infinity. When you want the ball to touch the ground, just make the fraction 24,000/24,000 and your there. The fraction can be infinitely divided.
I realize I’m generalizing here, but as in most cases, I don’t care.
-Dave Barry
Ever thought about being a philosopher, Keith? You’ve innocently stumbled onto Xeno’s paradox. Xeno the Eleatic was a fifth century pre-Socratic Greek philosopher who argued that no runner could ever finish a race. In order to make it to the finish line, the runner would first have to make it to the point half-way between the start of the race and the finish. Once the runner reached that point, there would be another point halfway to the finish line (three-quarters from the start). However far the runner goes, there is always another halfway point a finite distance away. Since there are an infinite number of halfway points, and since crossing the distance between each point takes a finite amount of time, Xeno reasoned that the runner would never reach the finish line (or in your example, the ball will never reach the floor).
Of course we know both of these statements are false. I can get anywhere I want to go (assuming my seat doesn’t get bumped) even though I have to travel an infinite number of fractions of the total distance to get there. This is why it’s called a paradox. Mathematicians now know (and have for a very long time, I would guess) that this is because the series of fractions 1/2+1/4+1/8+… adds up to one. This is made even more confusing in the “Does it have an Asymptote?” thread. But don’t worry about it, Keith. The point is that even though there really is no smallest fraction, the ball will eventually bounce, because each smaller fraction of distance is travelled in an equally smaller amount of time. And since the sum of all those fractions, even if you make them infinately small, is still a finite distance to the floor, it also takes a finite time to fall.
Also, you have to remember that what you are measuring is not distance, it is distance over time. The ball is in motion, and if you halve the distance, you must halve the time as well. Thus, yes, you can slice the distance in half an infinite number of times, but the time change also will be sliced. So you get far enough and the ball doesn’t reach the next point because in time doesn’t reach the next part.
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