On Tuesday, 30 July 2013 15:29:43 UTC+1, rickhg12hs wrote:
>
> sage: var('a b')
> (a, b)
> sage: assume(a, 'real')
> sage: assume(b, 'real')
> sage: bool( sqrt((a+b)^2) == sqrt(a^2) + sqrt(b^2) )
> True
> sage:bool( (sqrt((a+b)^2) == sqrt(a^2) + sqrt(b^2)).subs(a=1,b=-1) )
> False
> sage:
>
> Why the strange equality?
>
> Because square root is multivalued. Consider
sqrt(1-z)*sqrt(1+z)=sqrt(1-z^2) and sqrt(z-1)*sqrt(z+1)=sqrt(z^2-1).
More academically, consider
Bradford,R.J. & Davenport,J.H.,
Towards Better Simplification of Elementary Functions.
Proc. ISSAC 2002 (ed. T. Mora), ACM Press, New York, 2002, pp. 15-22.
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