When you use
//N to get a numerical result,
Mathematica does what a standard calculator would do: it gives you a result to a fixed number of significant figures. You can also tell
Mathematica exactly how many significant figures to keep in a particular calculation. This allows you to get numerical results in
Mathematica to any degree of precision.
Doing any kind of numerical calculation can introduce small roundoff errors into your results. When you increase the numerical precision, these errors typically become correspondingly smaller. Making sure that you get the same answer when you increase numerical precision is often a good way to check your results.
to a fixed number of significant digits. Typing N[Pi] is exactly equivalent to Pi//N. 
turns out to be very close to an integer. To check that the result is not, in fact, an integer, you have to use sufficient numerical precision. 
