Question and answer - Interesting
Question:
One night you see your daughter quietly gazing out the window at the stars. You ask, "Is everything OK?" She answers with a question: "I've been thinking. When I want to go from point A to point B, I can divide the journey up into segments. Let's say the first segment is half the distance to B. Then the second segment is half the distance again. If I keep this up, traveling half the remaining distance each time, I end up with an infinite number of segments. But since each segment has to take finite time, this means it takes infinite time to get to point B. It seems to me I've just proved you can't go anywhere! Obviously something's wrong here, right?" How do you respond?
Answer:
This very interesting young lady has hit on the idea behind Zeno's Paradox. This bugged the ancients to no end, and still boggles most peoples' minds to this day. Let's take the paradox at face value for a moment and assume we want to walk a mile, but by segments that are always half the remaining distance. Adding all these segments up, we see that the distance walked is D = 1/2 + 1/4 + 1/8 + 1/16 +… This is an example of an infinite series, and as you learn in early calculus, it is not the case that this sums to infinity.
Instead, it has a finite sum: 1. If you think about it, it's not so crazy: if I cut a finite length up into an infinite number of smaller and smaller pieces, I still have the sum I started with. So, rationalizing the paradox is actually pretty straightforward: let's say it takes 2 seconds to cross halfway from A to B. Then the next half segment will take 1 second. The next will take 0.5, the next will be 0.25, and so on. Add 'em all up and you get a finite 4 seconds! Whew — we actually can get from point A to point B!
PS: This article is taken from the GlobalSpec Newsletter dt. 16, May 2006
One night you see your daughter quietly gazing out the window at the stars. You ask, "Is everything OK?" She answers with a question: "I've been thinking. When I want to go from point A to point B, I can divide the journey up into segments. Let's say the first segment is half the distance to B. Then the second segment is half the distance again. If I keep this up, traveling half the remaining distance each time, I end up with an infinite number of segments. But since each segment has to take finite time, this means it takes infinite time to get to point B. It seems to me I've just proved you can't go anywhere! Obviously something's wrong here, right?" How do you respond?
Answer:
This very interesting young lady has hit on the idea behind Zeno's Paradox. This bugged the ancients to no end, and still boggles most peoples' minds to this day. Let's take the paradox at face value for a moment and assume we want to walk a mile, but by segments that are always half the remaining distance. Adding all these segments up, we see that the distance walked is D = 1/2 + 1/4 + 1/8 + 1/16 +… This is an example of an infinite series, and as you learn in early calculus, it is not the case that this sums to infinity.
Instead, it has a finite sum: 1. If you think about it, it's not so crazy: if I cut a finite length up into an infinite number of smaller and smaller pieces, I still have the sum I started with. So, rationalizing the paradox is actually pretty straightforward: let's say it takes 2 seconds to cross halfway from A to B. Then the next half segment will take 1 second. The next will take 0.5, the next will be 0.25, and so on. Add 'em all up and you get a finite 4 seconds! Whew — we actually can get from point A to point B!
PS: This article is taken from the GlobalSpec Newsletter dt. 16, May 2006
posted by Kulasekharan N @ 10:21 AM
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