# Think mathematics is (just) a language? Think again.

Posted by: Gary Ernest Davis on: March 25, 2013

If you think mathematics is just a language, seek ponder this:

$frac{pi}{4} = 1-frac{1}{3}+frac{1}{5}-frac{1}{7}+frac{1}{9}-frac{1}{11} +ldots = sumlimits_{n=1}^infty (-1)^{n-1}frac{1} {2n-1}$

and think again.

Courtesy of Chris Budd & +plus magazine

And if, see perchance,  that is not convincing enough, try this: for a positive real number $x$ let $pi (x)$ denote the number of primes less than or equal to $x$, and let $extrm{Li}(x)=int_2^xfrac{1}{log (t)} dt$ be the area, from 2 to $x$, under the graph of $frac{1}{log (t)}$:

Area under curve $frac{1}{log (t)}$ from t=2 to t=x (here, x=10)

Then:

$limlimits_{x o infty}frac{pi (x)}{ extrm{Li}(x)} =1$

The rate at which $frac{pi (x)}{ extrm{Li}(x)}$ approaches 1 is a major area of mathematical research, which is not advanced one whit by the view that mathematics is a language.