Add Two Irrational Surds to Get Another Surd

[dodo]

Hi,
does somebody know an example of two surds that, added together, give another surd?

By 'surd' I mean here 'irrational surd', as opposed to $\sqrt 4 + \sqrt 9 = \sqrt 25$.

 

[DeadWolfe]

$\sqrt a + \sqrt a = \sqrt {4a}$

 

[dodo]

Cool. I need to be more specific: by 'example', I meant a numerical example. Particular surds, like $\sqrt 3$ or $7 \sqrt 66$. No unknowns.

 

[Moo Of Doom]

Just replace a by any positive real number, and you'll have one...

 

[Xalos]

It is impossible to get such a solution by adding two different surds or you could add the same surd to itself and get a surd as deadwolfe says.

 

[Moo Of Doom]

$\sqrt a + \sqrt {4a} = \sqrt {9a}$ also works, so the surds can be different.

 

[robert Ihnot]

We assume all positive integers. This problem is quite solvable using the quadratic equation on: $$\sqrt{a}+\sqrt{b}=\sqrt{c}$$

Which yields: $$c=(a+b) \pm 2\sqrt{ab}$$

Thus it follows that a and b must have a common factor, and otherwise are squares. The negative sign can not be used.

$$a=sm^2, b=sn^2, c=s(m+n)^2.$$

The solution then yields only: $$m\sqrt{s}+n\sqrt{s} =(m+n)\sqrt{s}$$

 

[dodo]

Thanks for all your answers; now I think I can pin down the motivation behind the question. Each irrational surd seems (if I'm not mistaken) to generate an ideal on R. When I google about this (not that I know shrlit), there is something called 'Dedekind domains', on which ideals can be uniquely expressed as a product of 'prime' factors.

So this collection of ideals (plus some 'nice' additions, like 0 and 1) begins to behave, it seems to me, like the ring of integers (note to myself: prove it is a ring). Now, one of the holy grails is to understand the relation between prime factors and addition (given the prime factorization of two integers, what is the prime factorization of their sum? - heavy open problem). And while there are plenty of examples of sums of integers to toy with, I can't find a single example of a sum of 'surd ideals'. Annoying, to say the least.

P.S.: Oh well, neither multiplication is an internal law, nor there are additive inverses. Bummer. It's still annoying.
P.P.S.: What am I saying, even addition is not internal; $\sqrt 2 + \sqrt 3$, if irrational at all, is not a surd, for the reasons in post#7.

Just let it go. I was just wandering about.