TITLE: Is the sum of all rational numbers between two integers infinity
QUESTION [6 upvotes]: If there are infinite numbers between two rational numbers then would that entail that the sum of all numbers, say between 1 and 2, be infinity?
I believe that this cannot be true and has to do something with area under a curve?

REPLY [4 votes]: Before you can answer this question you need a definition for what it means to "add up infinitely many numbers".
For the moment, assume those numbers are listed in some order:
$$
a_1, a_2, \ldots .
$$
Then mathematicians define the infinite sum to be the limit (if it exists) of the numbers
$$
a_1,  \quad  a_1 +  a_2, \quad  a_1 +  a_2 + a_3,  \ldots .
$$
Then, for example, you could show that the "infinite sum"
$$
1 +
\frac{1}{2} +
\frac{1}{4} +
\frac{1}{8} + \cdots
$$
is $2$. So sometimes the sum of infinitely many numbers is finite.
With that definition, you clearly can't sum all the rational numbers between $1$ and $2$ since all of them are greater than $1$, so those partial sums will grow without bound. You can't sum all the rational numbers between $0$ and $0.0001$ since infinitely many of them are greater than $0.00001$. So you are more or less correct - mathematicians prefer to say you can't sum them, not that the sum is infinity.
When you learn calculus (that "something with area under a curve") you will see how to add up more and more and more pieces without growing to infinity because at each stage the pieces are smaller and smaller and smaller.