Re: JSH: Operator ambiguity, Escultura

From: KeithK (me_at_nomail.com)
Date: 11/29/04 

Date: Mon, 29 Nov 2004 01:31:31 -0700

"James Harris" <jstevh@msn.com> wrote in message
news:3c65f87.0411280652.17b14c6@posting.google.com...
> 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.
>
No, you're confusing the square root of a number (which has both a positive
answer and it's negation) with the sqrt() operator, which is _defined_ to
yield the positive root.

This definition prohibits the implied step above: i = sqrt(1)/sqrt(-1),
because this equation is false since 1/sqrt(-1) = 1/i = -i.

> 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.
>
No, i does not = +/-sqrt(-1). You're tilting at definitions again. The
_definition_ of 'i' is i = sqrt(-1).

KeithK

> 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

  • Re: Forth PARANOIA
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    (comp.lang.forth)
  • Re: Riemann surface
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    (sci.math)
  • Re: read string to float
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  • Re: the number of divisors, number theory
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  • Re: solution to nonlinear eq
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    (comp.soft-sys.matlab)