TITLE: $e$ to 50 billion decimal places
QUESTION [41 upvotes]: Sorry if this is a really naive question, but in my reading of a lot of textbooks and articles, there is a lot of mention of how many decimals we know of a certain number today, such as $\pi$ or $e$. An excerpt from my textbook:

In 1748, Leonard Euler used the sum of the infinite series of $e$ (mentioned in the book in a section about Taylor Series) to find the value of $e$ to 23 digits. In 2003, Shigeru Kondo, again using the series, computed $e$ to 50 billion decimals places

My question is why does it matter how many decimals we know? Isn't this just a huge waste of time? What could we ever do with so many decimal places? And, if $e$ can be represented as a sum of infinite series of $1/n!$, can't we just plug that into a computer that just loops the same equation but increasing $n$ every iteration, and find as many decimals of $e$ as we like?
(Once again, I realize this may be an ignorant/naive question, but I've always been curious about this)

REPLY [20 votes]: I'd just like to give two quotes from the book The SIAM 100-Digit Challenge: A Study in High-Accuracy Numerical Computing that might help explain motivation. Here is the one from their chapter on computing constants to 10,000 digits:

While such an exercise might seem frivolous, the fact is we learned a
  lot from the continual refinement of our algorithms to work efficiently at
  ultrahigh precision. The reward is a deeper understanding of the theory,
  and often a better algorithm for low-precision cases.

and here is something from the foreword written by David Bailey, one of the pioneers of experimental mathematics:

Some may question why anyone would  care about such prodigious precision,
  when in the “real” physical world, hardly any quantities are known to an
  accuracy beyond about 12 decimal digits. For instance, a value of π
  correct to 20 decimal digits would suffice to calculate the circumference
  of a circle around the sun at the orbit of the earth to within the width
  of an atom. So why should anyone care about finding any answers to 10,000
  digit accuracy?
In fact, recent work in experimental mathematics has provided an important
  venue where numerical results are needed to very high numerical
  precision, in some cases to thousands of decimal digit accuracy. In
  particular, precision of this scale is often required when applying integer
  relation algorithms to discover new mathematical identities. An integer
  relation algorithm is an algorithm that, given $n$ real numbers
  ($x_i,\quad 1\leq i\leq n$), in the form of high-precision floating-point
  numerical values, produces $n$ integers, not all zero, such that
  $a_1x_1+a_2x_2+\cdots+a_n x_n=0$.
The best known example of this sort is the discovery in 1995 of a new formula
  for π:
$$\pi=\sum_{k=0}^{\infty}\frac1{16^k}\left(\frac{4}{8k+1}-\frac{2}{8k+4}-\frac1{8k+5}-\frac1{8k+6}\right)$$
This formula was found by a computer program implementing the PSLQ integer
  relation algorithm, using (in this case) a numerical precision of
  approximately 200 digits. This computation also required, as an input
  real vector, more than 25 mathematical constants, each computed to 200-digit
  accuracy. The mathematical significance of this particular formula is that it 
  permits one to directly calculate binary or hexadecimal digits of 
  π beginning at any arbitrary position, using an algorithm that is very simple,
  requires almost no memory, and does not require multiple-precision arithmetic.