Editing E (mathematical constant) (section)

== Computing the digits ==
One way to compute the digits of {{mvar|e}} is with the series<ref name="Finch-2005">{{Cite book |last=Finch |first= Steven R. |url=http://worldcat.org/oclc/180072364 |title=Mathematical constants |date=2005 |publisher=Cambridge Univ. Press |isbn=978-0-521-81805-6 |oclc=180072364}}</ref>
<math display=block>e=\sum_{k=0}^\infty \frac{1}{k!}.</math>

A faster method involves two recursive functions <math>p(a,b)</math> and <math>q(a,b)</math>. The functions are defined as <math display=block>\binom{p(a,b)}{q(a,b)}= \begin{cases} \binom{1}{b}, & \text{if }b=a+1\text{,} \\ \binom{p(a,m)q(m,b)+p(m,b)}{q(a,m)q(m,b)}, & \text{otherwise, where }m=\lfloor(a+b)/2\rfloor .\end{cases}</math>

The expression <math display=block>1+\frac{p(0,n)}{q(0,n)}</math> produces the {{mvar|n}}th partial sum of the series above. This method uses [[binary splitting]] to compute {{mvar|e}} with fewer single-digit arithmetic operations and thus reduced [[bit complexity]]. Combining this with [[fast Fourier transform]]-based methods of multiplying integers makes computing the digits very fast.<ref name="Finch-2005" />