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Consider the following expression: $1/(\sqrt a_1 ± \sqrt a_2 ± ... \sqrt a_n)$ where $a_1 ... a_n$ are positive integers

I would like a general algorithm by which I can convert expressions of this form into expression of the following form: exp/k

where k is some positive integer and exp is an expression consisting of additions, subtractions, multiplications and square roots but no divisions.

Here is an example of a simple case:

$1/(\sqrt 2 + \sqrt 3) = (\sqrt 3 - \sqrt 2)/(3 - 2) = \sqrt 3 - \sqrt 2$

Here I multiplied the top and bottom by the conjugate. This technique can be used in the general problem for n≤4 by breaking the sum up into pairs of roots. I will illustrate this below in the case where all square root terms are added rather than having subtraction mixed in.

consider this denominator: $(\sqrt a + \sqrt b) + (\sqrt c + \sqrt d)$
multiplying by its conjugate: $(\sqrt a + \sqrt b) - (\sqrt c + \sqrt d)$ yields:
$(\sqrt a + \sqrt b)^2 - (\sqrt c + \sqrt d)^2$
$a+b+2\sqrt (ab) - c - d - 2\sqrt (cd) = $
$ A + (\sqrt B - \sqrt C)$ where $A= a+b-c-d$ and $B=4ab$ and $C=4cd$
multiplying by its conjugate $ A - (\sqrt B - \sqrt C)$ yields:
$ A^2 - (\sqrt B - \sqrt C)^2 =$
$ A^2 - B - C + 2\sqrt (BC) =$
$ D + \sqrt E =$ where $D=A^2-B-C$ and $E=4BC$
now multiply by its conjugate $ D - \sqrt E$ yields:
$ D^2 - E$ which is an integer

I would like to know how to generalize this to n=5 and beyond, or if this is not possible, why that is the case. Thanks

@ancientmathemetician:

I am not sure how to do this for the case n=8. Consider the following expression: $(\sqrt a_1 + \sqrt a_2 + \sqrt a_3 + \sqrt a_4) + (\sqrt a_5 + \sqrt a_6 + \sqrt a_7 + \sqrt a_8)$
I could multiply by its conjugate which would yield the following

$(\sqrt a_1 + \sqrt a_2 + \sqrt a_3 + \sqrt a_4)^2 - (\sqrt a_5 + \sqrt a_6 + \sqrt a_7 + \sqrt a_8)^2$

However once each squared term is distributed out there will be 6 new square root terms introduced rather than just the original 4. Below I will do the first term:

$(\sqrt a_1 + \sqrt a_2 + \sqrt a_3 + \sqrt a_4)^2 =$
$a_1 + a_2 + a_3 + a_4 + 2\sqrt (a_1a_2) + 2\sqrt (a_1a_3) + 2\sqrt (a_1a_4) + 2\sqrt (a_2a_3) + 2\sqrt (a_2a_4) + 2\sqrt (a_3a_4)$

as a result, I go from 8 roots in the denominator to 12 and a single non root term

If instead I broke up the expression into a group of 5 square roots and a group of 3 square roots I would end up with 10 roots from the group of 5 and 3 from the group of 3 totalling 13.

in either case the number of roots does not go down, which is the trick I have been exploiting. It seems to me a new trick of sorts is required to find the general solution.

Thanks

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  • $\begingroup$ One small comment: you mustn't overdo it. For example, if $D=\sqrt{E}$ you stop there and don't multiply by $D-\sqrt{E}=0$. $\endgroup$ – ancientmathematician Nov 23 '18 at 7:52
  • $\begingroup$ Hints: If I wanted to go along this path I'd do two things. (i) I'd persuade myself that the process applies equally well to $(b_1 \sqrt a_1 ± b_2 \sqrt a_2 ± ... b_n \sqrt a_n)$ with integral $b_i$, some possibly $0$; and (ii) I'd show I could do it for $n=2,4,8,16, \dots $. Then with the right number of zero $b_i$ I'd be done for all $n$. $\endgroup$ – ancientmathematician Nov 23 '18 at 7:57
  • $\begingroup$ Thanks for pointing out the case where D=√E, didn't think of that. but ultimately if such a case arizes then one can just end the algorithm early. I agree it would be worth doing the more general case with b√a instead of just √a because it captures the changes in sign. As for showing it can be done for the cases n=2,4,8,16... I am not sure how to do it for the n=8 case so if you want to demonstrate that I'll accept that as an answer $\endgroup$ – mathew Nov 23 '18 at 8:29
  • $\begingroup$ You are right to point out the way in which it won't work. So to save it I think you have to deal with the fully general case. That is, the general case of $b_0$, $b_0+b_1 \sqrt{a_1}$, $b_0+b_1 \sqrt{a_1}+b_2 \sqrt{a_2} + b_{1,2}\sqrt{a_1 a_2}$, $\dots$ - in other words, allow for all the cross terms from the outset. $\endgroup$ – ancientmathematician Nov 23 '18 at 10:15
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$$(\sqrt a+\sqrt b+\sqrt c)(s+t\sqrt a+u\sqrt b+v\sqrt c+w\sqrt{ab}+x\sqrt{ac}+y\sqrt{bc}+z\sqrt{abc})=1$$ The rational coefficients of the various square-roots give you eight equations in eight unknowns $s$ to $z$.
It looks like it works for any number of square-roots. A $2^N\times2^N$ matrix has to be inverted. I don't know how to tell if it's going to be singular.
The matrix is very sparse - mostly zeros - so this isn't the most efficient method. I think it shows there is an answer.

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