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Math Help - about nature of roots

  1. #1
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    about nature of roots

    Friends kindly enlighten me on the following aspects:

    Is 3 + Sq. root of 2 - irrational?

    Whether irrational roots occur in pair?

    As an example (3+ Sq. root of 2) and (3 - Sq. root of 2)

    Hope I have made myself clear.

    Thanks.
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  2. #2
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    Re: about nature of roots

    Quote Originally Posted by arangu1508 View Post
    Friends kindly enlighten me on the following aspects:
    Is 3 + Sq. root of 2 - irrational?
    Whether irrational roots occur in pair?
    As an example (3+ Sq. root of 2) and (3 - Sq. root of 2)
    3+\sqrt{2} is an irrational number.
    If \mathbf{r} is rational and \gamma is irrational then \mathbf{r}+\gamma is irrational.

    If the coefficients of a quadratic are rational then any irrational root has a pairing.
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  3. #3
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    Re: about nature of roots

    Thank you Mr. Plato. Very useful. So the coefficients of a quadratic should be rational for the roots to be (irrational) pair.
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  4. #4
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    Re: about nature of roots

    If all coefficents of a quadratic equation are rational, then the all irrational roots appear in "conjugate pairs". Your statement, "So the coefficients of a quadratic should be rational for the roots to be (irrational) pair", however, is not necessarily true. An obvious example is \sqrt{3}x^2- 5\sqrt{3}x+ 2\sqrt{3}= 0 which has irrational coefficients but "irrational pair" roots, \frac{1\pm\sqrt{17}}{2}.
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  5. #5
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    Re: about nature of roots

    Quote Originally Posted by arangu1508 View Post
    Friends kindly enlighten me on the following aspects:

    Is 3 + Sq. root of 2 - irrational?

    Whether irrational roots occur in pair?

    As an example (3+ Sq. root of 2) and (3 - Sq. root of 2)

    Hope I have made myself clear.

    Thanks.
    Suppose j is any rational number, in this case 3. I am a math beginner, but since j + any rational number is equal to a rational number [j + rational number = rational number], 3 + any rational number = a rational number. I know obvious. Now comes the challenge, is 3 + an irrational number equal to an irrational number? When we found out that 3 + any rational number = a rational number, so 3 + any irrational number can't be a rational number or j + irrational number does not equal a rational number. Therefore, j + an irrational number must be irrational also.
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