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Added surds

  1. Jan 26, 2008 #1
    does somebody know an example of two surds that, added together, give another surd?

    By 'surd' I mean here 'irrational surd', as opposed to [itex]\sqrt 4 + \sqrt 9 = \sqrt 25[/itex].
  2. jcsd
  3. Jan 26, 2008 #2
    [itex]\sqrt a + \sqrt a = \sqrt {4a}[/itex]
  4. Jan 26, 2008 #3
    Cool. I need to be more specific: by 'example', I meant a numerical example. Particular surds, like [itex]\sqrt 3[/itex] or [itex]7 \sqrt 66[/itex]. No unknowns.
  5. Jan 26, 2008 #4
    Just replace a by any positive real number, and you'll have one...
  6. Jan 26, 2008 #5
    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.
  7. Jan 26, 2008 #6
    [itex]\sqrt a + \sqrt {4a} = \sqrt {9a}[/itex] also works, so the surds can be different.
  8. Jan 26, 2008 #7
    We assume all positive integers. This problem is quite solvable using the quadratic equation on: [tex]\sqrt{a}+\sqrt{b}=\sqrt{c} [/tex]

    Which yields: [tex]c=(a+b) \pm 2\sqrt{ab}[/tex]

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

    [tex]a=sm^2, b=sn^2, c=s(m+n)^2.[/tex]

    The solution then yields only: [tex]m\sqrt{s}+n\sqrt{s} =(m+n)\sqrt{s} [/tex]
    Last edited: Jan 26, 2008
  9. Jan 26, 2008 #8
    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; [itex]\sqrt 2 + \sqrt 3[/itex], if irrational at all, is not a surd, for the reasons in post#7.

    Just let it go. I was just wandering about.
    Last edited: Jan 26, 2008
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