## IMT Interval Length

In IMT, on page 5, it is stated that the length of any interval ${I}$, is given by

$\displaystyle |I|=\underset{N\rightarrow\infty}{\lim}\frac{1}{N}\#(I\cap \frac{1}{N}\mathbb{Z})$

where ${\frac{1}{N}\mathbb{Z}=\{\frac{n}{N}:n\in\mathbb{Z}\}}$ and ${\#A}$ denotes the cardinality of a finite set ${A}$. For smaller values of ${N}$ this can be directly tested using a spreadsheet. For larger values of ${N}$ we can just calculate the range of values of ${n}$ which lie within the interval and obtain the result from there. For example, for the open interval ${(0.6243783194,1.0568000000)}$ we obtain the following

As ${N}$ increases the value obtained approaches the actual value ${(0.4324216806)}$.