(April 3, 2015 at 12:51 pm)robvalue Wrote: Try telling a non mathematician that 0.9recurring is 1. That's a long day
But even a very long day has a least upper bound.
The 0.0123456789 recurring still interests us.
0.0123456789 recurring = 0.0123456789 x (1 + 10^-10 + 10^-20 + ... )
= 0.0123456789 x 1/(1 - 10^-10)
= 123456789/9999999999
= 1/81 - 10/89999999991
For not only does 1/81 = 0.012345679 recurring come up with a missing 8 in its blocks, but so does
1/891 = 0.0011223344556677899 recurring (all doublets but the 8 which is single),
1/8991 = 0.00011122233344455566677788999 recurring (all triplets but the 8 which is double),
...and so on.
See Cheer & Goldston for details at http://www.math.sjsu.edu/~goldston/otherpub.pdf
