Using a calculator and then, for more precision, UBASIC, the part after the decimal point of even powers of (sqrt(2)+sqrt(3)) begin with a series of 9's that's ever increasing in length.
In a calculator
2 9.898979485566356
4 97.98979485566358
6 969.9989690710702
8 9601.99989585503
10 95049.99998947932
12 940897.999998939
14 9313929.99999992
16 92198402.00000002
18 912670090.0000011
20 9034502498.00002
the even powers show this up to a point. The question was, Is the conversion to zeros an artifact of rounding error, or do differences beyond the 9's add up to tip the fractional part into having leading zeros. Actually UBASIC confirms that the continuation of 9's goes much further (higher even powers); but I wouldn't know where it ends, or if it ends.
I don't know the reason, or whether this continues, but I took the problem to Wolfram Alpha. That shows 970 digits before the decimal, then 969 9's before "random" digits start. So the answer to the question is 9. I just don't know why. Is the 970 digits before the decimal coincidentally close to the 969 9's after the decimal, or does that match continue?
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Posted by Charlie
on 2019-01-30 15:02:06 |
certainly not in my head (but spoiler present anyway)
