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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Sign up to join this communityIf 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?
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.