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Subtract two squares a-b in a sqrt numbersystem

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If a number system write out numbers as a sequense of SQRTs+int summed together make up the numbers how would one go on subtracting squares?

Using small exmaple we want to subtract 11^2-5^2
Well in this numbersystem it would of course just be written 11-5

Isn't there a geometrical or set theoretic way using the fact that we can calculate the area creating and adding two overlapping rectangles, and then we subtract area reused twice and end up with. 

(11*6+11*6)-(6*6) and computational wise it of course seem easier to first do the squaring into integers and then perform a straight subtraction.

But could there be gains the way i show for very big squares, or is it always a loss situation?

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Subtract two squares a-b in a sqrt numbersystem jonas.thornvall@gmail.com - 2013-10-20 09:24 -0700
  Re: Subtract two squares a-b in a sqrt numbersystem "Michael Haufe (TNO)" <tno@thenewobjective.com> - 2013-10-20 17:33 -0700
    Re: Subtract two squares a-b in a sqrt numbersystem jonas.thornvall@gmail.com - 2013-10-21 14:51 -0700
      Re: Subtract two squares a-b in a sqrt numbersystem "Chris M. Thomasson" <no@spam.invalid> - 2013-10-21 15:09 -0700

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