Comments on: Math Puzzle

By: Swing Trading

Swing Trading — Wed, 28 Oct 2009 18:34:08 +0000

Nice one. I have stumbled and twittered this for my friends. My friends will enjoy reading it also.

By: Aditya

Aditya — Sat, 23 Aug 2008 19:45:26 +0000

Thanks! Well, this is what blogs are all about eventually- Conversations.

By: Mona

Mona — Thu, 14 Aug 2008 15:52:29 +0000

Interesting post and the responses are even more interesting.

By: Aditya

Aditya — Sat, 10 Nov 2007 06:36:20 +0000

Anonick- Thanks for the response… I appreciate it when someone takes the effort of explaining subtleties like these.

By: Anonick

Anonick — Sat, 10 Nov 2007 04:48:28 +0000

I always get afterthoughts after hitting the submit button, so forgive me while I elaborate on the rigourousness of the infinite series (which you may know, but Quixote doesn’t, by the looks of it).

You take the sum of 9/10^n till a finite n. Now, if I give you a positive real number e, and you can find an integer N such that the sum of 9/10, 9/10^2, 9/10^3, … 9/10^N is within e of 1, and if you can always find such an integer N given a number e, then your series converges to 1, that is, the sum of infinite terms is 1. Infinity has no meaning in maths, aside from being defined as a limit.

I’ll give an example. Suppose I want to find an N such that N 9’s written after the decimal point would get me to within 0.00034 of 1. Simple: 4 9’s, since 1 – 0.9999 = 0.0001 < 0.00034 . You can see that if I want the sum of N terms to be within any range of 1, I can always write enough 9’s to do the trick. “Convince yourself” (in the colourful language of textbook authors) that infinite 9’s get to 1.

By: Anonick

Anonick — Sat, 10 Nov 2007 04:35:52 +0000

There is a way to verify the “subtle error” pointed above, and since you’ve had 3+ years of calculus, I need only state that 0.9999999… is not a number per se, it is a representation of an infinite series, which converges to, guess what? 1! If you have a problem with 0.9999…, you should have a problem with every infinite series. I have cleared this problem out of my mind by reasoning that the pure mathematicians know what they’re doing .

By: Aditya

Aditya — Mon, 13 Aug 2007 14:48:43 +0000

OMG… Have I heard of Zeno’s paradox? Not in a very, very long while! Zeno’s paradoxes used to flummox me back in school. There was a time between the 6th and 8th grades (I can’t remember when exactly) where I found the one about the tortoise and the hare absolutely fascinating. Not so much now after 3+ years of calculus. I did indeed think of including a simplified version above but changed my mind in the end

By: Anne

Anne — Mon, 13 Aug 2007 13:04:38 +0000

Interesting. Heard of Zeno’s paradox?

By: Quixote

Quixote — Sun, 05 Aug 2007 18:16:48 +0000

The subtle error in both puzzle examples is that the values given are only approximately equal to each other. An infinitely recurring decimal, by definition, never reaches an absolute value. So, though x may equal .9999…., x then is only an approximation of one (1), not the absolute value. In the second example, .333…. is not genuinely equal to 1/3, only infinitely close to it. Equal and infinitely close to equal are not the same things even though the difference may not be possible to measure.

This means that in the strict, categorical sense the “solutions” to your mathematical puzzles are not true. They’ve already been fudged by the first propositions.

(By the way, thanks for the visit to my site.)