Most unfortunately, it was proven that the digits are periodic. Its a complex formula, but they are periodic, where as the digits of sqrt (2) are algebraic, and

Most unfortunately, it was proven that the digits are periodic. Its a complex formula, but they are periodic, where as the digits of sqrt (2) are algebraic, and the digits of e are logarithmic. So without using the identity elements of the arithmetic group, you simply arrive at three very different classes of irrationality.