Sum of rational numbers The sum of a finite number of rational numbers is of course a rational number, but the sum of an infinite number of rational numbers might be an irrational number.  Can someone give me some intuition why this sum might be irrational?  I just "don't feel it."
 A: Dear I am thinking about it because I am also a student of Mathematics.
We know that $e$ is an irrational number.
The value of $e$ is
$$= 1 + \frac{1}{1} + \frac{1}{2!} + \frac{1}{3!} + \ldots$$
$$= 1 + \frac{1}{1} + \frac{1}{2} + \frac{1}{6} + \frac{1}{24} + \ldots$$
$$= 2.7182 \ldots$$ (Irrational Number)
So sum of infinite rational numbers may be irrational.
A: Any irrational number $x$ is the limit of a sequence of rational numbers $a_n$ (take for example the decimal expansion truncated after $n$ decimals, for $n=1,2,3,\dots$). Then
$$x = a_1 + (a_2-a_1) + (a_3-a_2) + \dots$$
is a sum of rational numbers, but it is irrational by construction.
A: Look at this.
So: $3.14159 \dots = 3 + \frac{1}{10} + \frac{4}{10^2} +\frac{1}{10^3} + \frac{5}{10^4}+ \frac{9}{10^5} + \dots$
The above expression is $\pi$.
A: An irrational number is a "gap" inside the rationals. The decimal expansion of any irrational number is an infinite sum converging to it. It gives better and better approximations to the irrational number, which is unfortunately just not "there".
There are many more examples of such limiting constructions. Sometimes you get new objects and sometimes you don't. An example is the delta function which can be approximated using bona fide functions. So there are functions arbitrarily "close" (in some sense) to the delta function, but the delta function itself is just "too good" to be an actual function.
As an aside, if these approximations are good enough, you can prove that the number is transcendental (Legendre's or Roth's theorem).
A: The problem is psychological: you think of the "infinite sum" of rational numbers as an obvious, intuitive concept, but it's not.  It has a precise mathematical meaning, and that precise mathematical meaning only works if you allow the sums to be real numbers (which themselves have a precise mathematical meaning).  The definitions which allow these "infinite sums" to make sense are much less trivial than someone who's never worked through them would think.
