square root of 2

The square root of 2 is an irrational number, the first to have been proved irrational. Its decimal expansion begins 1.41421356237309504880168872420969807856… (sequence http://www.research.att.com/ njas/sequences/A002193A002194 in Sloane’s OEIS) Its simple continued fraction is

1+\frac{1}{2+\frac{1}{2+\frac{1}{2+\frac{1}{2+\ldots}}}},

periodically repeating the 2. Some call this number Pythagoras’ constant.

There are several different ways to express $\sqrt{2}$ as an infinite product. One way is

\sqrt{2}=\prod_{i=0}^{\infty}\frac{(4i+2)^{2}}{(4i+1)(4i+3)},

another is

\sqrt{2}=\sum_{i=0}^{\infty}(-1)^{i+1}\frac{(2i-3)!!}{(2i)!!}.

References

1 Flannery, David. The square root of 2 : a dialogue concerning a number and a sequence. New York: Copernicus, 2006.