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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- Jan 2nd 2012, 08:55 AMarangu1508about 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. - Jan 2nd 2012, 09:06 AMPlatoRe: about nature of roots
$\displaystyle 3+\sqrt{2}$ is an irrational number.

If $\displaystyle \mathbf{r}$ is rational and $\displaystyle \gamma$ is irrational then $\displaystyle \mathbf{r}+\gamma$ is irrational.

If the coefficients of a quadratic are rational then any irrational root has a pairing. - Jan 2nd 2012, 09:15 AMarangu1508Re: 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.

- Jan 2nd 2012, 12:21 PMHallsofIvyRe: 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 $\displaystyle \sqrt{3}x^2- 5\sqrt{3}x+ 2\sqrt{3}= 0$ which has irrational coefficients but "irrational pair" roots, $\displaystyle \frac{1\pm\sqrt{17}}{2}$. - Jan 2nd 2012, 05:04 PMokokjaeRe: about nature of roots
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.