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Math Help - Prove that both the sum and product of two rational numbers are rational

  1. #1
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    Prove that both the sum and product of two rational numbers are rational

    I'm new to proofs and I want to make sure that I'm doing them right, so I might be doing a lot of this "here's my proof, is it suitable?" posts.
    -Thanks for your help in advance

    My proof:
    1st statement: sum of two rational numbers is rational
    If a number is rational it can then be written in the form a/b.
    We are saying that the sum of these two rational numbers (a/b) and (c/d) (where a,b,c,d are integers) is also a rational number.

    (a/b)+(c/d) = ((ad+bc)/bd)
    -let k = (ad+bc) and j = bd
    Therefore ((ad+bc)/bd) = k/j where k and j are integers because the product of two integers is an integer and the sum of two integers is an integer.
    We've just proven that the sum of two rational numbers is a rational number.

    2nd statement: the product of two rational numbers is rational
    given (a/b) and (c/d) where a,b,c,d are integers
    (a/b)*(c/d) = (ac/bd)
    ac is an int because the product of two integers is an integer
    bd is an int because the product of two integers is an integer

    Therefore (ac/bd) is rational
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  2. #2
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    Re: Prove that both the sum and product of two rational numbers are rational

    Quote Originally Posted by evthim View Post
    I'm new to proofs and I want to make sure that I'm doing them right, so I might be doing a lot of this "here's my proof, is it suitable?" posts.
    -Thanks for your help in advance

    My proof:
    1st statement: sum of two rational numbers is rational
    If a number is rational it can then be written in the form a/b.
    We are saying that the sum of these two rational numbers (a/b) and (c/d) (where a,b,c,d are integers) is also a rational number.

    (a/b)+(c/d) = ((ad+bc)/bd)
    -let k = (ad+bc) and j = bd
    Therefore ((ad+bc)/bd) = k/j where k and j are integers because the product of two integers is an integer and the sum of two integers is an integer.
    We've just proven that the sum of two rational numbers is a rational number.

    2nd statement: the product of two rational numbers is rational
    given (a/b) and (c/d) where a,b,c,d are integers
    (a/b)*(c/d) = (ac/bd)
    ac is an int because the product of two integers is an integer
    bd is an int because the product of two integers is an integer

    Therefore (ac/bd) is rational
    1. Is acceptable if you have already shown that the sum and product of integers is always an integer.

    2. Is acceptable if you have already shown that the product of two integers is always an integer.
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  3. #3
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    Re: Prove that both the sum and product of two rational numbers are rational

    Another way to phrase that is that the set of integers is closed under addition and multiplication, that is, the product or the sum of two integers is always an integer.
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