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48th digit in the head (Posted on 2019-01-30) Difficulty: 3 of 5
What is the 48th digit after the decimal point of the number (21/2 + 31/2)1948.

You should solve this problem in your head.

No Solution Yet Submitted by Danish Ahmed Khan    
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Some Thoughts certainly not in my head (but spoiler present anyway) | Comment 1 of 4
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?

  Posted by Charlie on 2019-01-30 15:02:06
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