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sum k^3*1000^(-k) for k=1 to infinity ( = 334667000/332001998667 = 0.001 008 027 064 125 216 343 512 730 ...)
Also see if you can guess which one this is: 40920041/997002999 = 0.041 043 047 053 061 071 083 097 113 131 151 173 197 223 251 281 313 347 383 421...
Mmmh. Primes.
Earlier quoted context omitted.
If you'd like to continue the pattern beyond 52 digits, just keep adding 9s to the original fraction... 1/9999999999998 = 1.0000000000002 0000000000004 0000000000008 0000000000016 0000000000032 0000000000064 0000000000128 0000000000256 0000000000512 0000000001024 0000000002048 0000000004096 0000000008192 0000000016384 0000000032768 0000000065536 0000000131072 00000002621440... × 10^-13
interestingly, this works the other way around too: 1/98 = 0. 01 02 04 08 16 32 65... and even further? 1/8 = 0.125
If I'm not mistaken. We should utilize the self-similarity much more often.
Earlier quoted context omitted.
On most pocket calculators, 11111111×= will yield 12345678. People are often surprised that that sequence is accepted. To me, it exposes something about the calculator's internal architecture. It's also a useful self-test if you think the battery might be going.
The sequence is a shortcut accepted by the vast majority of regular calculators for most of operations. It simply takes the second operand to be the same as the first and repeated pressing of the = key repeats the operation ad infinitum. Ie. 1+== is 3. I have written an iOS calculator app and had very interesting times trying to find and mimic these shortcuts. I have thought for a long time they had to follow from so…
Earlier quoted context omitted.
On most pocket calculators, 11111111×= will yield 12345678. People are often surprised that that sequence is accepted. To me, it exposes something about the calculator's internal architecture. It's also a useful self-test if you think the battery might be going.
i'm interested now - what does that expose about the internal architecture for you?
Different operations take noticeably different amounts of time; a "timing attack" like those used for cryptanalysis might yield clues to what's in the black box.
The way new digits appear on the display when typed in suggests it might be implemented as a shift register. It would be interesting to look at high speed video of the display when the answer to a long computation appears; do the answer digits appear (rapidly) one at a time? Do they shift in from the left? Three caveats: (1) I've never noticed it happening; (2) LED displays are almost always multiplexed, but you could probably see through that; and (3) probably wouldn't work on an LCD because too slow. I used to have a vacuum fluorescent display calculator, though; IIRC it was not multiplexed.
There are a few articles on the web about the architecture of calculators, including the Busicom [1] and Sinclair [2]. Personally, I want to hear more about zoul's research---how did you do it?
[2] http://files.righto.com/calculator/sinclair_scientific_simul...
Earlier quoted context omitted.
The sequence is a shortcut accepted by the vast majority of regular calculators for most of operations. It simply takes the second operand to be the same as the first and repeated pressing of the = key repeats the operation ad infinitum. Ie. 1+== is 3. I have written an iOS calculator app and had very interesting times trying to find and mimic these shortcuts. I have thought for a long time they had to follow from so…
This site [1] reveals the secret to that pocket calculator shortcut and a few others, and also provides useful algorithms for calculating things such as square roots and logarithms. [1] http://tedmuller.us/Math.htm
The old Sinclair pocket calculators had some known arithmetic inaccuracies.
Feynman, R. P. 'Surely You're Joking, Mr. Feynman!': Adventures of a Curious Character. New York: W. W. Norton, 1997.
http://www.youtube.com/watch?v=daro6K6mym8
If you wanted to extend this, make it something like 1/999999998 instead.
The pattern will break down once you get past 8192, which is 2^13. That means that the pattern continues for an impressive 52 significant figures (well, it actually breaks down on the 52nd digit, which will be a 3 instead of a 2). The reason it works is that 9998 = 10^4 - 2. You can expand as 1 / (10^n - 2) = 1/10^n * 1/(1 - 2/10^n) = 1/10^n * (1 + 2/10^n + 2^2 /10^2n + 2^3 /10^3n + ...) which gives the observed patt…
The pattern will break down once you get past 8192, which is 2^13. That means that the pattern continues for an impressive 52 significant figures (well, it actually breaks down on the 52nd digit, which will be a 3 instead of a 2). The reason it works is that 9998 = 10^4 - 2. You can expand as 1 / (10^n - 2) = 1/10^n * 1/(1 - 2/10^n) = 1/10^n * (1 + 2/10^n + 2^2 /10^2n + 2^3 /10^3n + ...) which gives the observed patt…