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Several years ago, we had a fun discussion in this forum about the Monty Hall problem. Numberphile has two Youtube videos about the subject:
http://www.youtube.com/watch?v=4Lb-6rxZxx0
http://www.youtube.com/watch?v=7u6kFlWZOWg
The second video explains it in a way I can understand. Good stuff!
What really blows my mind is this Numberphile video:
http://www.youtube.com/watch?v=w-I6XTVZXww
They add up all positive intergers from one to infinity. What do you think the sum would be? Infinity? Nope. The sum is actually -1/12.
You can find a lot of articles about this particular video, including Scientific American, NYT and Slate.com (the Slate writer has multiple corrections where he kept changing his mind back and forth), by doing a Google search for "-1/12" including the quotation marks.
Intuit that.
http://www.youtube.com/watch?v=w-I6XTVZXww
They add up all positive intergers from one to infinity. What do you think the sum would be? Infinity? Nope. The sum is actually -1/12.
You can find a lot of articles about this particular video, including Scientific American, NYT and Slate.com (the Slate writer has multiple corrections where he kept changing his mind back and forth), by doing a Google search for "-1/12" including the quotation marks.
Intuit that.
>They add up all positive intergers from one to infinity. What do you think the sum would be? Infinity? Nope. The sum is actually -1/12.
Things aren't this simple when you're working with infinities. You can't just shift things around like this and still get meaningful results.
Consider this sequence:
And try:
It's as easy as that to show a contradiction when you assume that normal rules of addition can apply to infinite sets. In reality, you need to expand the definition of addition to apply it to infinite sets, which is usually done via the concept of limits.
- Greminty
Things aren't this simple when you're working with infinities. You can't just shift things around like this and still get meaningful results.
Consider this sequence:
| N = 1 + 1 + 1 + 1 ... |
And try:
| N - N | = | 1+ 1 + 1 + 1 + 1 ... | |
| - [ 1 + 1 + 1 + 1 ... | (shifted slightly, like they did in the video) | ||
| N - N | = | 1+ 0 + 0 + 0 + 0 ... | |
| 0 | = | 1 |
It's as easy as that to show a contradiction when you assume that normal rules of addition can apply to infinite sets. In reality, you need to expand the definition of addition to apply it to infinite sets, which is usually done via the concept of limits.
- Greminty
