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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
number3.5), so its decimal expansion never repeats.
The initial decimal expansion of
is given by3.6
Any number of digits can be computed from the formula
by making
sufficiently small.
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