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Proof by Induction-Sqrt(n)

  1. Sep 4, 2010 #1
    Proof by Induction--Sqrt(n)

    1. The problem statement, all variables and given/known data
    Prove that if a line of unit length is given, then a line of length sqrt(n) can be constructed for each n.


    2. Relevant equations
    N/A


    3. The attempt at a solution

    So I'm not really sure where to begin...I assumed that a unit length is the representation of the natural numbers (1, 2, 3...n). And then I drew a triangle with unit length 1 on the legs and then constructed the hypotenuse to be sqrt(2). And then I drew a triangle with unit length 1 on a leg and unit length 2 on a leg and then I constructed the hypotenuse to be sqrt(5). But I don't know how to:

    i) Write this as a formal proof by induction, or
    ii) How to find some sqrt(n) with unit lengths, like sqrt(3) or sqrt(4).
     
  2. jcsd
  3. Sep 4, 2010 #2

    Office_Shredder

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    Re: Proof by Induction--Sqrt(n)

    What if you use the sqrt(2) line that you just constructed to make the sqrt(3) line somehow?
     
  4. Sep 4, 2010 #3
    Re: Proof by Induction--Sqrt(n)

    So then (sqrt(2))^2+1^2 = c^2
    c=sqrt(3)

    Ok, that makes sense. Is there a way to generalize this as a rule or equation to prove with induction?

    So far I have:

    (1^2)+(1^2)=c^2
    (1+1) = c^2
    c=sqrt(2)

    (sqrt(2))^2+1^2 = c^2
    2+1 = c^2
    c=sqrt(3)

    (sqrt(2))^2+(sqrt(2))^2=c^2
    2+2 = c^2
    c = 2

    etc etc?

    How would I prove that I can find sqrt(n) for all n, such that n is a natural number?
     
  5. Sep 4, 2010 #4

    Office_Shredder

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    Re: Proof by Induction--Sqrt(n)

    Well, it says to use induction. We can use the line with length sqrt(1) to make a line of length sqrt(2). We can use the line with length sqrt(2) to make a line with length sqrt(3). Given a line of length sqrt(n), can you make a line with length sqrt(n+1)? And how does that help you with a proof by induction?
     
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