Nasser M. Abbasi <firstname.lastname@example.org> wrote: > > I remember reading that some math people, 100 or 200 years ago, > decided that sqrt (of non-negative, non-complex values) was single valued > function and its value is the non-negative root (ie. principal > square root).
Beware that mathematicians have Humpty Dumpty attitude: : When I use a word. it means just what I choose it to : mean -- neither more nor less. Another mathematician may use the same word with different meaning (of course, in good math text meaning is explained).
Before complex numbers were invented only nonnegative square root was in use.
Currently there are three sligtly different conventions:
- real one: sqrt can be only applied to non-negative numbers and produces non-negative values. This one is used in schools before pupils learn about complex numbers, sometimes in real analysis and in theory of real fields
- algebraic: y = sqrt(x) is basically abbreviation for equation y^2 = x. Using functional notation is justified by fact that in algebraic context it normally does not matter which root you use. In fact, given a single root there is no algebraic way to distinguish it from the other one. Only dependent roots, like sqrt(2)*sqrt(3) versus sqrt(6) allows you to see difference. In some context dependent roots are just illegal input, in other context case reasoning is needed.
- complex analytic: sqrt is a multivalied function. When we need a single branch this branch is choosen so that resulting function is analytic
You may ask: Is not the principial branch of squre root the "true" one? My opinion is that using principial branch for numerical computation is unavoidable imperfection morally not much different from using floating point instead of true reals. Simply, for numeric computations we choose alternative which can be implemented with reasonable effort and frequently gives correct results. But ultimately it is up to programmers to make sure that correct branch is choosen. For example FriCAS numeric code for elliptic functions makes heavy use of square roots. To ensure nice branch cuts for elliptic integrals I had to compute appropriate branch of square root and this branch is different than principal branch.