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As you know we teach our students, sum of rational numbers is also a rational number as a property of rational numbers when we introduce real numbers in the beginning of advanced level mathematics. But though the constant e is sum of rational numbers, it can be proven that e is irrational. Have you come across this issue from students and what would you suggest as a reason for this?
What you should be teaching your students is not that “the sum of rational numbers is rational”, but, “the sum of two rational numbers is rational” which you can extend, using the principle of mathematical induction, to “the sum of a *finite* number of rational numbers is rational”.
It should be clear that this claim cannot be extended to *infinite* sums blindly. In fact, by definition, an infinite sum is actually the limit of a finite sum (partial sums).
“e” is a counterexample for this. The infinite sum of rational numbers need not necessarily be a rational number.
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