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Math Help - How can an irrational number be finite?

  1. #1
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    How can an irrational number be finite?

    Hi, I'm an IB student from Mexico and I'm dying to solve something.
    As part of a homework i must fin the exact value of an irrational number (our teacher told us it would be a trinomial) but as far as I'm concerned an irrational number can't be exact.
    The number is:
    [IMG]file:///C:/Users/ANAMAR%7E1/AppData/Local/Temp/moz-screenshot-2.png[/IMG]The root of: the root of 1 plus the the root of 1 and so on...
    I wasn't able to type it here bit it looks as if the first root is "housing" the next one and so on.

    Thanks for you help and I really hope someone knows how to solve this!
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  2. #2
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    Quote Originally Posted by stressedIBstudent1 View Post
    Hi, I'm an IB student from Mexico and I'm dying to solve something.
    As part of a homework i must fin the exact value of an irrational number (our teacher told us it would be a trinomial) but as far as I'm concerned an irrational number can't be exact.
    The number is:
    [IMG]file:///C:/Users/ANAMAR%7E1/AppData/Local/Temp/moz-screenshot-2.png[/IMG]The root of: the root of 1 plus the the root of 1 and so on...
    I wasn't able to type it here bit it looks as if the first root is "housing" the next one and so on.

    Thanks for you help and I really hope someone knows how to solve this!
    first, note that image files will not show as a link to your hard drive. you have to upload it as an attachment.

    let x = \sqrt{1 + \sqrt{1 + \sqrt{1 + ...}}}

    x^2 = 1 + \sqrt{1 + \sqrt{1 + \sqrt{1 + ...}}}

    x^2 = 1 + x

    x^2 - x - 1 = 0

    remember the quadratic formula?

    you'll get two irrational roots ... one is the value of your "nested" radical.

    also, see the following article ...

    Golden ratio - Wikipedia, the free encyclopedia
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  3. #3
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    Thank you veeery much!
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  4. #4
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    Hello, stressedIBstudent1!

    A slightly different approach . . .


    \text{Evaluate: }\sqrt{1 + \sqrt{1 + \sqrt{1 + \hdots}}}

    \text{We have: }\;x \;=\;\sqrt{1 + \underbrace{\sqrt{1 + \sqrt{1 + \hdots }}}_{\text{This is }x}}

    So we have: . x \;=\;\sqrt{1 + x}

    Square both sides: . x^2 \;=\;1 + x \quad\Rightarrow\quad x^2 - x - 1 \:=\:0

    And we get the same quadratic equation that skeeter found.

    Follow his advice and get the answer: . x \;=\;\frac{1 + \sqrt{5}}{2}
    . . which happens to be the Golden Mean, \phi.

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  5. #5
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    You really explain things in a simple way, thank you very much!
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    How can an irrational number be finite?
    The wording might better have been 'how can an irrational number be algebraic?'
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  7. #7
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    Quote Originally Posted by TheCoffeeMachine View Post
    The wording might better have been 'how can an irrational number be algebraic?'
    I think the OP was confusing "exact" as a number with only a finite decimal representation.
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