Stuttering calculator

Ok, take your average desk calculator.

type :  10 / 3 = = = =

Any reasonable explanation for the sequence you get?

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grg99Asked:
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GwynforWebCommented:
Clearly 10 is being repeatedly by 3. For binary operations eg a*b,  a/b, this is the way I think it works.  I believe they have 2 registers holding the left hand value (displayed) and 1 holding the right hand. When = is hit  (left) operator (right) is evaluated and placed in the display (ie the  left register) and the right  is unchanged. Repeatedly hitting = continuously divides the display value by the right value in this case 3
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grg99Author Commented:
yesbut how to explain the interesting numbes that pop up:


3.3333333333333333333333333333333

1.1111111111111111111111111111111

0.37037037037037037037037037037037

0.12345679012345679012345679012346

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aburrCommented:
What's so special about that. Your last lint is missing an 8.

Seriously, you have found an interesting set of digits. I do not beleive there is any special reasoning behind them.
It is interesting to note that whenever you divide two intergers you will get an answer in which the digits will eventually repeat themselves.
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Andrea ErcolinoCommented:
its simpler, just divide 10 / 81 ;-)))
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aburrCommented:
but you miss the other interesting strings
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Andrea ErcolinoCommented:
0,098765432098765432098765432098765... = 8 / 81 ;-)))
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Andrea ErcolinoCommented:
can be shown that the length of the repeating substring in <<1 / n>> is a divisor of the totient of <<n>>

in this case, <<9>> is a divisor of <<Phi( 81 ) = 81 - 27 = 54>>
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Andrea ErcolinoCommented:
some primes <<p>> have the longest repeating substring <<p - 1>>, being <<7>> the smallest one
<<1 / 7 = 0,14285714285714285714285714285714...>>
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JR2003Commented:
It's easy to see why these patterns occur if you try them using log division.
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JR2003Commented:
It's easy to see why these patterns occur if you try them using LONG division.
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grg99Author Commented:
some primes <<p>> have the longest repeating substring <<p - 1>>, being <<7>> the smallest one
<<1 / 7 = 0,14285714285714285714285714285714...>>

IIRC:
What's even more interesting, TWO sevenths is: .285714285714....., FOUR sevents is .57142857....

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GwynforWebCommented:
Multiply 37,037 by any single number (1-9), then multiply that number by 3. Every digit in the answer will be the same as that first single number. (this is failry easy to work out but still cute)
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GwynforWebCommented:
Thanks for the points :-)
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