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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?

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    $\begingroup$ No, you cannot conclude from the fact that there are infinitely many rationals that the sum must be infinite. This is the error behind Zeno's paradox of the Achilles (e.g., $\sum \frac{1}{2^n}$ is finite, even though you are adding infinitely many positive rational numbers). $\endgroup$ – Arturo Magidin Nov 26 '18 at 1:21
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    $\begingroup$ However, the sum is potentially infinite anyway; certainly it is infinite for all rationals between $1$ and $2$, since this includes countably many rationals that are greater than or equal to $1$, so the sum is strictly larger than any positive integer. $\endgroup$ – Arturo Magidin Nov 26 '18 at 1:23
  • $\begingroup$ I was just about to type that last comment, @ArturoMagidin. Summing infinitely many numbers greater than $1$ must, almost by definition, be infinite. $\endgroup$ – The Count Nov 26 '18 at 1:24
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    $\begingroup$ And the rationals between $0$ and $1$ include all rationals of the form $\frac{1}{n}$, and since the harmonic series diverges... so, the answer is that the sum of all rationals between two integers is always infinite, but not for the reason you provide; and the sum of all rationals between any two rationals should also be infinite, verifiable by using a linear transformation to take, say, $[1,2]$ to $[q_1,q_2]$ in a way that maps rationals to rationals, and use it to put lower bounds on the sum. $\endgroup$ – Arturo Magidin Nov 26 '18 at 1:27
  • $\begingroup$ You want to get into thorny matters? Ask whether the sum of all rational numbers between $-1$ and $1$ is infinite. $\endgroup$ – Brian Tung Nov 26 '18 at 1:48
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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.

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  • $\begingroup$ "You can't sum all the numbers between [...]" See that OP asks for all the rational numbers. It doesn't seems obvious to me that your proof holds... $\endgroup$ – rafa11111 Nov 26 '18 at 1:35
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    $\begingroup$ @rafa11111 There are infinitely many rational numbers in the interval that are larger than the midpoint. I edited the answer. $\endgroup$ – Ethan Bolker Nov 26 '18 at 1:42
  • $\begingroup$ Typo: $1+\frac12+\frac14+\frac18+\cdots = 2$, not $1$. $\endgroup$ – Brian Tung Nov 26 '18 at 1:47
  • $\begingroup$ @BrianTung I fixed it thanks. You could have. $\endgroup$ – Ethan Bolker Nov 26 '18 at 1:48
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    $\begingroup$ @EthanBolker: I know. I like to allow the answerer to fix it first in the way they think best. If you hadn't come back to fix it, I'd have done it eventually. $\endgroup$ – Brian Tung Nov 26 '18 at 1:49

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