and every sum of fractions is a fraction itself.

That's the flaw. Not true. It's true of every finite sum of fractions but as the number of terms necessary becomes infinite, to add up the terms and express an irrational as a ratio would require an infinitely large integer in the numerator or denominator.

It seems strange but there are plenty of instances like this in analysis where a property which holds for all finite sums does not hold for an infinite sum.

This is how it can be seen that the rational numbers are not complete: That is, we can construct a Cauchy series of rational numbers which does not converge to a rational number.

**Edited:**

err, excuse me,

since you guys seem to know a LOT more than I do, if I were to have issues with math could I pop a few questions here?

Sure.