I was looking around on the internet until I stumbled upon this equation.
How does this actually work? It is quite amazing how the number ascend and then descend by ones.
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It is just a coincidence. You can yourself multiply out the numbers on pen and paper to see the reason – Manthanein Feb 16 at 2:07
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This looks like a trivial application of the Central Limit Theorem. – MikeMathMan Feb 16 at 3:43
Just multiply it out with the grade school pen-and-paper algorithm:
111111111 x 111111111
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111111111
111111111
111111111
111111111
111111111
111111111
111111111
111111111
111111111
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12345678987654321
Each digit of the result comes from summing the digits in one column -- that is, counting how many ones there are. Since this is at most 9, there are no carries between columns.
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1+1 - nice picture! To the OP, note that we can use this same picture to cook up versions of this in arbitrary bases. For example, in base we'll want the end result to be ""; thinking about the picture above, this will happen when we square "." – Noah Schweber Feb 16 at 2:13
There are occurrences of , namely , if , and , namely , if . If this means the decimal representation of is .