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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."

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As Hans observes below, any real number is an infinite sum of rationals, so if you have any intuition for why there are irrational real numbers, there you go. I feel like adding: it is OK not to have intuition about infinite sums. (Maybe even that it is counterproductive to have expectations about what infinite sums ought to do, or ought not to do, until you have a great deal of experience with them.) A lot of stuff about infinite sums cannot be "felt" at first; only learned. If you "don't feel it", that might even be a sign that you understand it better than someone who does :) –  anon Dec 24 '10 at 23:05
    
Now I most certainly fell it. To be honest I read that sum of rational numbers might be irrational yesterday (I havent thought about it earlier). The example that I found was that $\sum\limits_{i=1}^{\infty}\frac{1}{F_{i}}$, where F means fibonacci number, is irrational. From this example it wasn't clear for me why is it true, but the 'decimal' example that Trevor wrote is convincing. –  Tomek Tarczynski Dec 24 '10 at 23:26
    
But still some things about rational and irrational numbers are not so obvious. For example: There is a rational number between every two irrational numbers, so how is it possible that there is 'so much more' irrational numbers than rational. –  Tomek Tarczynski Dec 24 '10 at 23:30
    
Infact, an irrational number or a real number is defined as a limit of a sequence of rational numbers. –  user17762 Jan 8 '11 at 22:23
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5 Answers

up vote 16 down vote accepted

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 $\pi$.

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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.

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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.

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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.

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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).

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