Rational numbers that form a group under addition

[Pippi]

Rational numbers form a group under addition. However, a sequence of rational numbers converges to irrational number. Presumably, group theory does not allow adding an infinite number of rational numbers. This is not indicated in the textbook definition of a group. I might be looking in vain, but can someone suggests a possible explanation why group operation is defined as such?

 

[mathwonk]

you seem to be asking why adding a finite number of terms is considered more basic than adding an infinite number of terms. the answer seems too obvious to respond to. hence no answers.

if you are asking for the history of the definition of a group, it started apparently with galois and legendre? trying to understand solution systems of algebraic equations. the key was to study the permutations of solutions. composing two permutations yields another permutation, the first example of a group operation (on two elements).the idea behind your question is very intelligent since it observes that infinite sums allow one to pass out of the realm of rationals. indeed the limitations of finite addition, in not allowing the study of irrationals, is one motivation for introducing infinite sums. ok we know how to add finitely many rationals, and we always get rationals. mow what happens if we try to add an infinite number of rationals?

 

[Pippi]

I am not asking what is more basic. For one, group theory does not explicitly say adding an infinite number of terms is NOT allowed. Just look at those textbook definitions. Second, if the definition does not explicitly say so, why can't I?

 

[mathwonk]

i think you should look again. there is nothing in the definition of a group that says how to add an infinite number of terms. notice that you need a notion of convergence to do so. it is really an approximation process, not an addition process.

 

[Pippi]

Alright. Thank you for answering my question!