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An injective function going from N to the set of algebraic numbers

  1. Apr 14, 2012 #1
    1. The problem statement, all variables and given/known data
    Prove that the set of algebraic numbers is countably infinite.


    2. Relevant equations
    If there exists a bijective map between N and a set A, N and A have the same cardinality


    3. The attempt at a solution
    Rather than coming up with a bijective map between S =the set of algebraic numers and N =natural numbers, I proved S is countable but i also have to prove that S is infinite.
    So, I wanted to design an injective function f:N to S.
    Can anyone come up with sucn an injective function?
     
  2. jcsd
  3. Apr 14, 2012 #2

    morphism

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    How about the map that sends n to n....???
     
  4. Apr 14, 2012 #3
    Ahh...
    $f(x,n)=x-n$\\
    $k(n) ={a:f(a,n)=0}$\\

    Then, $k(1)=1, k(2)=2,k(3)=3, ... k(n) = n$.\\
    this should be an injective map from N to the set of algebraic numbers...
     
  5. Apr 14, 2012 #4
    thank you so much!
     
  6. Apr 14, 2012 #5

    morphism

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    No problem.

    But really you shouldn't be thinking of maps and such: just try to think about why there are infinitely many algebraic numbers. Every rational number is algebraic. But there's more. Every number of the form ##\sqrt[n]{r}## with r rational is also algebraic. And there's more still..
     
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