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Agincourt

Prove me wrong please

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This post is a little off topic of game math, but it''s been plagueing me for awhile now. And if my math proffesor spoke any english I''d ask him. Here it is - if 4 + 4 = 8; divide each 4 in half like this - (2 + 2) + (2 + 2) = 8; and repeat - 1+1+1+1+1+1+1+1=8; and repeat some more until you reach 1/infinite. so the problem looks like 1/inf + 1/inf + ... inifinite time = 8; so, (and this is where i''m probably wrong) can''t you equate this to - (1/inf)^inf = 8 ?? and 1/inf = 0, right? so if you multiply 0 by itself infinite times, can it not equal any number? obviously not, or I''d just right 0^inf for all my homework problems :-) so which assumption am I wrong on? I''m a little unclear on all the nuances of the infinite concept. Use the WriteCoolGame() function Works every time

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no, 1/inf IS NOT = to 0

1/inf APPROACHES 0. ie; it is so bloody close but it isn''t quite there yet.


a rule:
Any finite number divided by infinity is a number infinitesimally larger than, but never equal to, zero.


Oh yeah, when dealing with infinite numbers, use a pen and paper and remember that you have limits in it such as x -> infinity. Luckily I don''t have to deal with them or imaginary numbers anymore. Atleast I hope not. Unfortunately I still have to do statistics.



Beer - the love catalyst
good ol'' homepage

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You can't repeat the step so much that it reaches 1/inf that is impossible it will never be 1/inf. By the way 1/inf does equall zero. If you have 0.0000000 and the zeros go on forever where is the space for that last "1"?

Edited by - medovids on January 10, 2002 12:21:55 AM

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As Dredge-Master points out, the limit of 1/n as n approaches positive infinity is zero. Not everyone has a background in Calculus, but this result itself is not very hard to derive.

The original question, however, is malformed. When you have the equation 4 + 4 = 8, the transformation (2 + 2) + (2 + 2) = 8 is equivalent to 2 * (4 / 2) + 2 * (4 / 2) = 8. Taking this one step further, we get (2/2)*4 + (2/2)*4 = 4 + 4 = 8.

What exactly are you doing when you "divide each 4 in half"? Not much, I''m afraid. In essence, you''re replacing one expression (which evaluates to 4) with another expression (which also evaluates to 4). This reasoning alone should be sufficient to convince you that there is nothing wrong.

You needn''t resort to a (Calculus-based) proof of convergence.

medovids: The value (1 / infinity), taken just like that, is undefined. I''m afraid the problem is more difficult than you think.

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Infinity is not a number. As an example the interval (0,1) has an infinite number of numbers as does the interval (0,2), but (0,2) has more numbers than (0,1), i.e. (0,2) is a larger interval than (0,1). Infinity is not equal to infinity. As another example the limit as n goes to infinity of 2^n and 3^n are both infinity but they approach infinity at differant rates, i.e. as n increases by one one of them doubles and the other triples.

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The actual fault in the text by Agincourt has already been mentioned. He/she assumes 1/inf = zero, when it''s infact undefined. It aproaches zero.

Anyways, here''s a general description of the problem, and a "proof" that the result is always 8.

First the basic case. Rewrite it as a sum.
  
1
2
---
\ 4
4 + 4 = | ----- = 2 * 4 = 8
/ 0
--- 2
n=1


Then make a general case, where you divide the 4 by two, and sum it two times as many times.

  
k
2
---
\ 4 k 4 1 4
| ----- = 2 * ----- = 2 * ----- = 2 * 4 = 8
/ k-1 k-1 0
--- 2 2 2
n=1


As you can see, the result is always 2*4, and is independent of k (which is the number of times you divide the original 4).

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Hey...CanLt ‡ be negative??? in that case, u r 0wnzed...

Hence, if we divide a positive number by a negative, we get a negative in return...If you add upp all the values, you get -8...

So the answer is that the sum = +-8

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Normally I disapprove of homework/school related posts that are not related to game development.

This post is acceptable for two reasons. First, Agincourt showed that he had thought about the problem himself and was not merely looking for an answer. Second, it provoked some interesting theoretical discussion. I believe the responses to this post could help Agincourt build his problem solving skills and I do approve of that sort of discussion here.

Graham Rhodes
Senior Scientist
Applied Research Associates, Inc.

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LilBudyWizer said it right. The = operation isn''t even appropriate in such a case. Neither is dividing a number by a non-number. You might as well talk about 1/banana.

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quote:
Original post by BS-er
LilBudyWizer said it right. The = operation isn''t even appropriate in such a case. Neither is dividing a number by a non-number. You might as well talk about 1/banana.


Yes but everyone knows that bananas aren''t numbers (I hope ), but not everyone knows that infinity isn''t.

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