Compute tan (1/55555) (angle is in degree) on a calculator.
in the denominator you can use any number of 5’s, the more the better. You should
get an interesting number. Is it a coincidence, can you explain it?
Microsoft'a calculator gives 1.8000180001800018000180001800018e-5, and 1.8000000001800000000180000000018e-10 for tan(1/5555555555).
However, WolframAlpha gives the more interesting tan(9/(5 (10^5- 1))) = 0.000018000180003744076321598376237840569439716110575219807...
Again, the continued fraction form is instructive: [0; 55554, 1, 166663, 1, 277773, 1, 388883, 1, 499993, 1, 611103, 1, 722213, 1, 833323, 1, 944433, 1, 1055543, 1, 1166653, 1, 1277763, 1, 1388873, 1, 1499983, 1, 1611093, (many terms) ...18610923, 1, 18722033, 1, 18833143, 1, 18944253, 1, 19055363, 1}] with a pleasing regularity.
We might also consider tan(9/(5 (10^100-1))) [0. (100 zeroes)18(100 zeroes)18(100zeroes) 3744(100 zeroes)7632(100 zeroes)...]*
Continued fraction expansion: [0; (100 5s)4, 1, 1(100 6s)3, 1, 2(100 7s)3, 1, 3(100 8s)3, 1, 4(100 9s)3, 1, 6(100 1s)03, 1, 7(100 2s)13, 1, 8(100 3s)23, 1]*
The expansion's brevity suggests the limits of WolframAlpha are also being tested here; and indeed the continued fraction form for tan(9/(5 (10^1000- 1))) is given with just 2 terms [0; (1000 5s)4, 1].*
There are further traps for the unwary calculator, for example tan(9/(5 (10^50- 1))) includes this misleading string:
1428571428571428571428571428571428571
44396464380342857142857142857142857142857
14285714289008715169353142857142857142857
14285714285714285721343847464521142857142857
142857142857142857142857
As for the question, I guess that for such small angles the calculator assumes that tan(x) = x, since 1/55555= 0.0000180001800018...
*where '100' and '1000' mean 'a lot' and 'even more' respectively, since I didn't count them individually.
Edited on June 5, 2019, 6:21 am
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Posted by broll
on 2019-06-05 05:57:51 |
Again, quite interesting
