Thread: Closure of Rationals

Euler's Theorem that 1+1/4+1/9+1/16+...1/k^2 +...=(pi^2)/6 says to me that the rationals are not closed under addition in the infinite case. The rationals are closed for a finite set. Many years have passed since I studied group theory. Please elaborate on this distinction

Originally Posted by Tim28

Euler's Theorem that 1+1/4+1/9+1/16+...1/k^2 +...=(pi^2)/6 says to me that the rationals are not closed under addition in the infinite case. The rationals are closed for a finite set. Many years have passed since I studied group theory. Please elaborate on this distinction

It is closed! Addition is a binary operation. When we write we only use two numbers at once. When we write we perform addition repeately. But it is still binary i.e. we add them one at a time. But when we have then it is a limit of a finite sum i.e. the limit of . And limits have nothing to do with group theory. Thus, just because rationals are not closed under infinite sums (limit of sums) does not mean they are not closed under addition.

An excellent answer. I feel much better. The book, "Journey through Genius" by William Dunham (Normally a fine expositor; I have three of his books.) should have indicated a limit. Then again, Euler himself may not have had the modern understanding of limit.

Originally Posted by Tim28

Then again, Euler himself may not have had the modern understanding of limit.

Whether he had or had not is irrelavant because Euler died in 1783 and group theory did not exist yet.
Group theory begins during the time of Galois (1811-1832).
But you also need to remember the group theory of the 19th century was different than today.
Back in those days group theory was done in terms of permutation groups.
It was Frobenius who gave the a more modern definition in 1887.
(Which happens to be identical to the old definition as Cayley's theorem demonstrates. Though nicer.)