The first row will have infinite numbers.some_random_panda wrote:
I thought that this was interesting and so i removed it from the post and put it in this one...
Say, you have infinite number of columns leading down of infinite numbers in this pattern and infinite rows of infinite numbers leading across in this pattern...
0 1 1/2 1/4 1/8 1/16... =2
-1 0 1 1/2 1/4 1/8... =1
-1/2 -1 0 1 1/2 1/4... =1/2
-1/4 -1/2 -1 0 1 1/2... =1/4
-1/8 -1/4 -1/2 -1 0 1... =1/8
. . . . . . .
. . . . . . .
. . . . . . =4 <--------rows equal 4
=-2 =-1 =-1/2 =-1/4 =-1/8 =-1/16... -----> =-4 <------------columns equal
-4
yet the table clearly equals 0!
Anyone know why if you add it horizontally, vertically or as a concept that it is different?
The second row will have infinity + 1 numbers.
The longest row will have infinity + infinity numbers.
In your second proof, ordering numbers by the size of their sets, infinity + infinity is not equal to infinity. In fact, infinity + 1 doesn't equal infinity.
Assuming your second proof is true:
What this means is that each row and column will have one more element than the previous row/column.
Let's look at your addition of rows. The first row total is 2, the second is 1...the last I assume is 0, and the row goes something like this
[0 - 1/Very large number - ........ - 1/4 - 1/2 - 0 + 1/2 + 1/4 + ....... + 1/Very large number + 0]. Note this has twice the number of elements in it than the first rwo.
Remembering that column 2 has one more number than column 1, and column 3 has two more than column 1 etc, there are incomplete rows beneath this last one you've added. So, while you've added an infinity number of rows to get to E = 4, you've missed the other infinity number of rows which are incomplete (incomplete meaning that it doesn't represent all the columns). These incomplete rows contain all negative numbers, and their sums will cancel out your positive sum of 4.
You were right. The boxed bit does equal 0 because it is symmetrical. But notice, Region A = - Region B, so they cancel each other out. Hence the table does = 0. The only reason you got 4 and -4 was because, using finite maths to solve this question, you didn't take into account the incomplete rows of Region B.
Another way of looking at this question is that because in normal everyday maths infinity + infinity = infinity is impossible, hence this question, where infinity + infinity must equal to infinity, cannot be solved by everyday maths.
Last edited by Vub (2007-06-28 22:51:49)