Answer to Question #153122 in Math for Elizabeth

Question

What are two different approximations for Euler’s constant ‘

Answer 1

The number e or Euler's number, is an important mathematical constant approximately equal to 2.71828.

The two possible approximations of e are:

1) The approximation found by R. Sabey in 2004 is correct to 18457734525360901453873570 decimal digits and is given by:$e = (1 + 9 ^{-4^{42}})^{3^{2^{85}}}$ ,

2) Another approximation found by Castellanos (1988ab) is good to 6 digits:

$e = 2 + \frac{54^2 + 41^2} {80^2}.$