system
Given the following post, respond as if you're commenting on Reddit:
human
How can the sum of infinite rational numbers be an irrational number?

There are convergent series where every term is a rational number, and the sum (the limit of the sequence of partial sums) is an irrational number. An example is the serie formed by the sequence 1/n!, whose sum is e, an irrational number. 

On the other hand, the sum of two rational numers is always a rational number.

How is this possible?
gpt
This is actually how the reals are defined: the reals are defined by a collection of values such that for each value v some series of rationals that "Cauchy converges" (roughly, a series where all partial sums sufficiently far into the it are all arbitrarily closer to each other) cannot be separated from v.

For example, let's consider the series 3 + .1 + .04 + .001 + ... = 3.141... If we were to write out more digits, we'd get more digits of π. While for every rational r we can say that this number is some positive distance from r, we construct the reals so that there is now some value π we can associate with this series which we cannot say it is separated from. Of course, any real number (rational or irrational) can be written in as a series using its decimal expansion, so the same works for any other rational/irrational number.

Underlying all of this is the philosophical question of why we need the real numbers. Why aren't the rationals enough? As the Greeks found out long ago, even simple right triangles with length-1 legs cannot have the length of their hypotenuse expressed as a rational. Any rational number you pick is either too small or too big. We thus fill in the gap with real numbers: those values that can only be reached by taking smaller and smaller increments of the rationals infinite times. Now there are no more such holes where some value is somehow either too big or too small compared to all numbers. The reals are, in both a formal and intuitive sense, the completion of the rationals.


TL;DR: It's defined that way, because we want infinite sequences in the rationals that seem to approach a value (in the sense that the terms all bunch up in the end) to actually have a number to converge to. Those values they converge to are called the reals, which includes rationals and irrationals alike.