The square root of 2 is an irrational number, the first to have been proved irrational. Its decimal expansion begins 1.41421356237309504880168872420969807856... (sequence A002194 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 $$\sqrt{2} = \sum_{i=0}^\infty (-1)^{i+1} \frac{(2i-3)!!}{(2i)!!}.$$

Bibliography

1

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