About πe
The value of
πe is one of the questions that started this investigation. It would
be fitting, then, to focus on this one value now that an infinitely differentiable extension
has been found. Here is a summary of what is known to be exact, using the approximations
described in my
paper:
- πe ~ 37,149,###,###.###
|
digits known to be exact
|
- πe ~ 37,149,805,###.###
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could be this high
|
- πe ~ 37,149,795,###.###
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could be this low
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The reason for these values can be seen in the values of successive approximations of
the inverse super-logarithm, as defined in my
paper. An
exact value in this context, is defined to be the digits within the decimal representation of
the values of the
n-th approximation and 2
n-th approximation for the same input,
that agree when rounded to the nearest place value that would allow agreement. The rounding
system used here, however is subjective, a 5 can either be rounded up or down, other than that
normal half-rounding is used:
- slog−1e(π)10
|
= 37,105,406,757.569
|
- slog−1e(π)20
|
= 37,155,268,624.635
|
- slog−1e(π)30
|
= 37,152,290,690.852
|
- slog−1e(π)40
|
= 37,150,849,430.350
|
- slog−1e(π)50
|
= 37,150,331,380.039
|
- slog−1e(π)60
|
= 37,150,112,554.576
|
- slog−1e(π)70
|
= 37,149,986,051.500
|
- slog−1e(π)80
|
= 37,149,912,712.494
|
- slog−1e(π)90
|
= 37,149,874,928.208
|
- slog−1e(π)100
|
= 37,149,852,157.218
|
- slog−1e(π)110
|
= 37,149,835,758.346
|
- slog−1e(π)120
|
= 37,149,825,450.226
|
- slog−1e(π)130
|
= 37,149,819,264.787
|
- slog−1e(π)140
|
= 37,149,814,532.744
|
- slog−1e(π)150
|
= 37,149,810,983.642
|
- slog−1e(π)200
|
= 37,149,804,230.685
|
- slog−1e(π)300
|
= 37,149,802,030.621
|
- slog−1e(π)400
|
= 37,149,801,852.585
|
