Above, we defined as the particular real number satisfying

which gave us when . From this expression, we have, as ,

or

This is one way to define . Another way to arrive at the same definition is to ask what logarithmic base gives that the derivative of is . We denote by .

Numerically,
is a transcendental number (a type of irrational
number^{3.5}), so its decimal expansion never repeats.
The initial decimal expansion of
is given by^{3.6}

Any number of digits can be computed from the formula by making sufficiently small.

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Center for Computer Research in Music and Acoustics (CCRMA), Stanford University