We know that is irrational, and have known this since Lambert's proof of 1771 (see ). We have known that is transcendental since Lindemann's proof of 1882 . We also know that is not a Liouville number. Mahler proved this in 1953. An irrational number is Liouville if, for any n, there exist integers p and q so thatLiouville showed these numbers are all transcendental. In fact we know that for p,q integral with q sufficiently large. This irrationality estimate, due to Chudnovsky and Chudnovsky  is certainly not the best possible . It is likely that 14.65 should be replaced by for any . Almost all transcendental numbers satisfy such an inequality. We know a few related results for the rate of algebraic approximation. The results may be pursued in  and .
We know that is transcendental. This follows by noting that and applying the Gelfond-Schneider theorem . We know that is transcendental. This result is a consequence of the work that won Baker a Fields Medal in 1970. And we know a few more than the first two hundred million digits of the decimal expansion for (Kanada, see Section 3).
The state of our ignorance is more profound. We do not know whether such basic constants as , or are irrational, let alone transcendental. The best we can say about these three particular constants is that they cannot satisfy any polynomial of degree eight or less with integer coefficients of average size less than . This is a consequence of some recent computations employing the Ferguson-Forcade algorithm . We don't know anything of consequence about the simple continued fraction of , except (numerically) the first 17 million terms, which Gosper computed in 1985 using Sum 1. Likewise, apart from listing the first many millions of digits of , we know virtually nothing about the decimal expansion of . It is possible, albeit not a good bet, that all but finitely many of the decimal digits of are in fact 0's and 1's. Carl Sagan's recent novel Contact rests on a similar possibility. Questions concerning the normality of or the distribution of digits of particular transcendentals such as appear completely beyond the scope of current mathematical techniques. The evidence from analysis of the first thirty million digits is that they are very uniformly distributed . The next one hundred and seventy million digits apparently contain no surprises.
In part we perhaps settle for computing digits of because there
is little else
we can currently do. We would be amiss, however, if we did not emphasize that the
extended precision calculation of has substantial application as a test of the
``global integrity'' of a supercomputer. The extended precision calculations
described in Section 3 uncovered hardware errors which had to be corrected before
those calculations could successfully run. Such calculations, implemented as in
Section 4, are apparently now used routinely to check supercomputers before they leave
the factory. A large-scale calculation of is entirely
; it soaks into
all parts of the machine and a single bit awry leaves detectable consequences.