# period

## 1 Examples

###### Example 1.

The transcendental number  $\pi$ is a period since we can write

 $\displaystyle\pi=\int\limits_{x^{2}+y^{2}\leq 1}~{}dx~{}dy.$
###### Example 2.

Any algebraic number  $\alpha$ is a period since we use the somewhat definition that integration over a 0-dimensional space is taken to mean evaluation:

 $\displaystyle\alpha=\int_{\{\alpha\}}x$
###### Example 3.

The logarithms of algebraic numbers are periods:

 $\displaystyle\log\alpha=\int_{1}^{\alpha}\frac{1}{x}~{}dx$

## 3 Inclusion

With the existence of a non-period, we have the following chain of set inclusions:

 $\displaystyle\mathbb{Z}\subsetneq\mathbb{Q}\subsetneq\overline{\mathbb{Q}}% \subsetneq\mathcal{P}\subsetneq\mathbb{C},$

where $\overline{\mathbb{Q}}$ denotes the set of algebraic numbers. The periods promise to prove an interesting and important set of numbers in that nebulous between $\overline{\mathbb{Q}}$ and $\mathbb{C}$.

## 4 References

Kontsevich and Zagier. Periods. 2001. Available on line at \urlhttp://www.ihes.fr/PREPRINTS/M01/M01-22.ps.gz.

Title period Period 2013-03-22 13:55:43 2013-03-22 13:55:43 mathcam (2727) mathcam (2727) 11 mathcam (2727) Definition msc 11F67 period representation algebraic domain