Was this calculus question answered correctly?
| Friday, Sep 03 2010 02:04 PM
Last Updated Friday, Sep 03 2010 02:04 PM
Q: In regard to your article "Californian conversations with ... German Robledo," I believe you will be getting more than just my input on the answer to the calculus question that was printed in the paper. I'm no mathematician, but I like math and believe that the erroneous answer can lead many readers to wrong thinking.
The question that Mr. Robledo states reads: "From calculus, what would you get if you added "half" plus "half of half" plus "half of half of half" plus "half of half of half of half" and so on (1/2 + 1/4 + 1/8 + 1/16 + ... ) and kept going for infinitely many iterations?"
And the answer reads: "They add up to one. Start with a square piece of paper and tear it in half. Put half aside, it represents the first term. Tear the remaining half in half and put it aside (second term). Continue this pattern. Notice that you can always take half of a quantity, so theoretically you will never run out of paper. The solution becomes obvious when you realize that all of the "half pieces" came from the original whole square. So all the pieces add up to one."
I believe the answer should read more like: "They add up to almost one -- being closer to one with the more iterations -- but will never reach exactly one."
That is most easily seen if one were to visualize that piece of paper that was used as an illustration in the answer. If one just kept tearing in half the smallest piece of paper left -- infinitely -- there would always be something left -- infinitely.
Anyway, I could be wrong, but if I am, I would like to I learn why I am. I'm likely not the only one that caught this -- it's pretty easy to see.
-- Alvin Gregorio
A: Robledo responded:
I am glad your reader likes math and is interested enough to send a response.
It is difficult to find a calculus problem that is non-threatening, has the feel of a puzzle and is appropriate for a wide audience. The actual problem involves working with limits to infinity. That approach would be appropriate for a calculus classroom discussion or a math journal article, but not for an article intended for a general audience.
Most students like this problem because it is counterintuitive. When asked for an initial guess they often give a large number. They have a "wow" moment when they go through the activity and see that the answer approaches one.
Another fun problem goes like this: You are 8 feet from a wall. You are instructed to move forward half the remaining distance every second. How many seconds will pass before you reach the wall? (Answer: You will never reach the wall because technically you can always find half of any number: 8, 4, 2, 1 1/2, 1/4, 1/8 ... Problems like this are associated with a Greek philosopher named Zeno and are known as Zeno's Paradoxes. It is worth searching for more about Zeno's paradoxes if you like these types of problems. Enjoy.
