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Math Help - Closure of Rationals

  1. #1
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    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
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    Quote Originally Posted by Tim28 View Post
    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 a+b we only use two numbers at once. When we write a_1+...+a_n we perform addition repeately. But it is still binary i.e. we add them one at a time. But when we have a_1+a_2+... then it is a limit of a finite sum i.e. the limit of a_1,a_1+a_2,a_1+a_2+a_3,.... 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.
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  3. #3
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    Thank you

    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.
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    Quote Originally Posted by Tim28 View Post
    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.)
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