Re: JSH: Operator ambiguity, Escultura

From: James Harris (jstevh_at_msn.com)
Date: 11/28/04 

Date: 28 Nov 2004 06:52:52 -0800

rupertmccallum@yahoo.com (Rupert) wrote in message news:<d6af759.0411271901.4a62a25@posting.google.com>...
> jstevh@msn.com (James Harris) wrote in message news:<3c65f87.0411271359.ca44e69@posting.google.com>...
> > I would like to pull out and highlight something interesting that E.
> > E. Escultura posted a few days ago, which I'd guess he's probably
> > talked about many times before, but I just noticed it and think it's
> > neat.
> >
> > First some more preamble as *by convention* as has been noted when I
> > brought up the subject of operator ambiguity before, sqrt(x) is taken
> > to be positive.
> >
> > So, by the convention, sqrt(4) = 2, and that's good as, -2(-2) = 4, so
> > if you say that sqrt(4) = 2 and sqrt(4) = -2, then 2 = -2, and 4 = 0,
> > which is not good.
> >
> > Naively then, you may believe that you can just say, take the positive
> > of the square root but as Escultura showed, that doesn't work:
> >
> > i = sqrt(-1) = sqrt(1/-1) = 1/i, giving -1 = 1. Contradiction.
> >
>
> If we make the convention that sqrt(-1)=i, then it's not true that
> sqrt(1/-1)=sqrt(1)/sqrt(-1).
>

So your assertion is that the substitution 1/-1 = -1 cannot be made?

The resolution is that the sqrt() operator gives TWO answers. It
always does, and convention can't force it to give only one answer.

So i = +/-sqrt(-1), where the +/- in front is a nod to the reality
that the result of using the square root operator is two solutions.

That's the operator ambiguity.

Naively you may believe that if you simply say, take only ONE answer
from the square root, and try to figure out some way to do it that you
can succeed, but you will always have a contradiction lurking.

The ambiguity extends to other operators like the cuberoot operator or
an infinity of other operators where you get more than one answer.

James Harris

Relevant Pages

  • JSH: Operator ambiguity, Escultura
    ... First some more preamble as *by convention* as has been noted when I ... brought up the subject of operator ambiguity before, ... You see, the ambiguity in the square root operator still remains, ...
    (sci.math)
  • Re: JSH: Operator ambiguity, Escultura
    ... > The resolution is that the sqrt() operator gives TWO answers. ... you're confusing the square root of a number (which has both a positive ... yield the positive root. ... > That's the operator ambiguity. ...
    (sci.math)
  • Re: Operator ambiguity, Escultura
    ... And then proclaiming that you have taken the positive square root. ... > brought up the subject of operator ambiguity before, ... > You see, the ambiguity in the square root operator still remains, ...
    (sci.math)
Loading