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Math Help - Rational and Irrational numbers

  1. #1
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    Rational and Irrational numbers

    Now, don't think this is just an ordinary question! Here goes:

    a and b are irrational numbers
    a + b is rational
    a x b is rational

    Work out possible values for a and b where they are positive irrational numbers.



    I am sure that I have gotten down all the details, any help will be much appreciated, thank you!
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    Quote Originally Posted by Geometor View Post
    Now, don't think this is just an ordinary question! Here goes:

    a and b are irrational numbers
    a + b is rational
    a x b is rational

    Work out possible values for a and b where they are positive irrational numbers.



    I am sure that I have gotten down all the details, any help will be much appreciated, thank you!

    a = (1 + sqrt(2))
    b = -sqrt(2)
    then a and b are irrational numbers and...

    a + b = (1 + sqrt(2)) + (-sqrt(2)) = 1 which is rational

    a = sqrt(2)
    b = 1/sqrt(2)
    then a and b are both irrational and...

    a*b = sqrt(2)*1/sqrt(2) = 1 which is rational

    i should be using this oppurtunity to try out Latex, i don't know how to use it

    EDIT: o sorry, i forgot a and b must be positive, so the first example is incorrect, i'll come up with another one

    how about
    a = 10 - sqrt(5) + sqrt(2)
    b = sqrt(5) - sqrt(2)

    then a + b = 10 which is rational and both a and b are positive irrational numbers
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  3. #3
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    Quote Originally Posted by Jhevon View Post
    a = (1 + sqrt(2))
    b = -sqrt(2)
    then a and b are irrational numbers and...

    a + b = (1 + sqrt(2)) + (-sqrt(2)) = 1 which is rational

    a = sqrt(2)
    b = 1/sqrt(2)
    then a and b are both irrational and...

    a*b = sqrt(2)*1/sqrt(2) = 1 which is rational

    i should be using this oppurtunity to try out Latex, i don't know how to use it

    EDIT: o sorry, i forgot a and b must be positive, so the first example is incorrect, i'll come up with another one

    how about
    a = 10 - sqrt(5) + sqrt(2)
    b = sqrt(5) - sqrt(2)

    then a + b = 10 which is rational and both a and b are positive irrational numbers
    Many thanks but I don't think that is correct since
    10 - sqrt(5) + sqrt(2) times sqrt(5) - sqrt(2) is irrational :S
    thanks for trying, lol this is a really complex problem ay?

    is this a trick question and it might be impossible?
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  4. #4
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    Quote Originally Posted by Geometor View Post
    Now, don't think this is just an ordinary question! Here goes:

    a and b are irrational numbers
    a + b is rational
    a x b is rational

    Work out possible values for a and b where they are positive irrational numbers.
    a=1-\sqrt{2} \mbox{ and }b=1+\sqrt{2}
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  5. #5
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    Quote Originally Posted by ThePerfectHacker View Post
    a=1-\sqrt{2} \mbox{ and }b=1+\sqrt{2}
    erm... the question requires "positive" numbers and I believe that 1- sq. root 2 is negative?

    thanks for helping anyway
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  6. #6
    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by Geometor View Post
    Many thanks but I don't think that is correct since
    10 - sqrt(5) + sqrt(2) times sqrt(5) - sqrt(2) is irrational :S
    thanks for trying, lol this is a really complex problem ay?

    is this a trick question and it might be impossible?
    o, we have to come up with a pair of numbers that satisfy both condtions at the same time? i thought we could use different examples for each.

    EDIT: TPH came up with an example
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  7. #7
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    Quote Originally Posted by Geometor View Post
    erm... the question requires "positive" numbers and I believe that 1- sq. root 2 is negative?

    thanks for helping anyway
    So then change it a little bit,

     a = 2 - \sqrt{2} \mbox{ and } b = 2 + \sqrt{2}

    What is so hard?
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  8. #8
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    Quote Originally Posted by Geometor View Post
    Now, don't think this is just an ordinary question! Here goes:

    a and b are irrational numbers
    a + b is rational
    a x b is rational

    Work out possible values for a and b where they are positive irrational numbers.



    I am sure that I have gotten down all the details, any help will be much appreciated, thank you!
    Because their sum is rational a and b are of the form:

    a = A + x,

    b = B - x

    where A and B are rational and x is irrational.

    Then:

    <br />
a\ b = AB + (-A+B)x - x^2 = C<br />

    where C is rational.

    Now we have demanded that A, B and C be rational, but we may as well
    demand that they be integers (the derivation of rational solutions from
    integer solutions is fairly elementary and left to the reader).

    Anyway we have:

    <br />
x^2 + (A-B)x + (C- AB) = 0<br />

    and x is irrational. However:

    <br />
x=\frac{-(A+B) \pm \sqrt{(A-B)^2 - 4(C-AB)}}{2}=\frac{-(A+B) \pm \sqrt{(A+B)^2 - 4C}}{2}<br />

    which is irrational only if q= (A+B)^2 - 4C is not a perfect
    square.

    So here is our algorithm for finding solutions:

    Choose an integer K, and an integer C such that K^2 - 4C is not a perfect square,
    then choose A and B so that A+B=K, then:

    <br />
x=\frac{-(A+B) \pm \sqrt{(A+B)^2 - 4C}}{2}<br />

    is an irrational number such that if:

    a=A+x
    b=B-x

    then a and b are irrational and a+b is an integer as is a \times b

    Of course this does not guarantee that both a and b are positive.

    RonL
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    thanks for the help! sorry, can only thank one person a day I believe?
    I find this very confusing though :S
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  10. #10
    Grand Panjandrum
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    Quote Originally Posted by Geometor View Post
    thanks for the help! sorry, can only thank one person a day I believe?
    I find this very confusing though :S
    No you can thank as many as you want.

    RonL
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  11. #11
    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by Geometor View Post
    thanks for the help! sorry, can only thank one person a day I believe?
    how come?
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  12. #12
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    Quote Originally Posted by Jhevon View Post
    how come?
    oh woops, nevermind it's just that I can't thank on the same post so I got mixed up
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