Re: A little lesson for sqrt(144) year olds.

From: Dirk Van de moortel (dirkvandemoortel_at_ThankS-NO-SperM.hotmail.com)
Date: 07/08/04 

Date: Thu, 08 Jul 2004 16:43:36 GMT

"David W. Cantrell" <DWCantrell@sigmaxi.org> wrote in message news:20040708122529.273$TD@newsreader.com...
> "Dirk Van de moortel" <dirkvandemoortel@ThankS-NO-SperM.hotmail.com> wrote:
> > "David W. Cantrell" <DWCantrell@sigmaxi.org> wrote in message
> > news:20040708114506.706$cW@newsreader.com...
> > > mold-guardian@comcast.net (Myxococcus xanthus) wrote:
> > > > "Androcles" <androc1es@nospamblueyonder.co.uk> wrote in message
> > > > news:<w76Hc.921$3J7.8430614@news-text.cableinet.net>...
> > > > > <stephen@nomail.com> wrote in message
> > > > > news:ccikb4$9u5$1@msunews.cl.msu.edu...
> > > >
> > > > > | It is amazing anyone could have the slightest problem
> > > > > | understanding Dirk's point. Apparently in the world you and
> > > > > | Androcles inhabit there is no ambiguous way to write the positive
> > > > > | number x such that x*x=2.
> > > > > |
> > > > > | The equation x^2=N has two roots. Those two roots are sqrt(N)
> > > > > | and -sqrt(N).
> > > > > | Apparently you and Androcles think the two roots are sqrt(N) and
> > > > > | sqrt(N) and have no way of distinguishing the two roots from each
> > > > > | other,
> > > > >
> > > > > That is quite correct. There is no way to distinguish the two roots
> > > > > from each other, just as there is no way to determine the sign of x
> > > > > from the symbol alone. If you can tell me the sign of x, do so, but
> > > > > please do not insist sqrt(N) is positive.
> > > > >
> > > > > | or of specifying which root you mean.
> > > > >
> > > > > That is NOT correct, I CAN specify which root I mean.
> > > >
> > > > Androcles:
> > > > The radical symbol designated by the ascii expression "sqrt" is, by
> > > > universal definition and convention, taken to mean only the principal
> > > > square root.
> > >
> > > That statement is, unfortunately, not true. Many common definitions and
> > > conventions in mathematics fail to be universal. That is the case here.
> > > [Below, I shall use "V" as an ersatz radical sign (with index 2
> > > understood) and similarly "3V" when the index is to be 3.]
> > >
> > > Evidence:
> > >
> > > Webster's NewWorld Dictionary of Mathematics (1989) says
> > > "Root of Number
> > > A _square root_ of a real number b is a solution (real number) of the
> > > equation x^2=b; a _cube root_ is... These roots are denoted by Vb, 3Vb,
> > > ... For example, V9 is 3 or -3, and V-4 is not a real number. When
> > > there are two roots, the positive root is called the principal root.
> > > The equations V(a^2) = a, 3V(a^3) = a,... hold when a is positive and
> > > the radical denotes the principal root."
> > >
> > > The HarperCollins Dictionary of Mathematics (1991) states, under the
> > > entry for square root, "a number or quantity that when multiplied by
> > > itself is equal to a given number or quantity, usually written Vx in
> > > arithmetic expressions, and x^(1/2) in algebraic expressions", and
> > > under the entry for principal value, "the principal value of V4 is 2
> > > although -2 is also a root."
> > >
> > > I don't know if more recent editions of these dictionaries have been
> > > changed in this regard. I would hope so.
> > >
> > > > Evidence:
> > > > ----------------------------------
> > > > Any nonnegative real number x has a unique nonnegative square root r;
> > > > this is called the principal square root and is written r = x^(1/2)
> > > > or r = sqrt(x). For example, the principal square root of 9 is
> > > > sqrt(9) = +3, while the other square root of 9 is -sqrt(9) = -3. In
> > > > common usage, unless otherwise specified, "the" square root is
> > > > generally taken to mean the principal square root.
> > > > http://mathworld.wolfram.com/SquareRoot.html
> > >
> > > It's interesting that you quote that as your first bit of evidence
> > > because I happen to have written it. (Granted, Eric does not credit me
> > > for having done so in that entry. Nonetheless, the wording is
> > > essentially mine, and I doubt that Eric would contest that.) I wrote
> > > that passage with the most common usage in mind. I wish that that uasge
> > > were universal.
> > >
> > > I wish I had noticed this thread earlier. Perhaps my comments could
> > > have reduced the senseless vitriol somewhat.
> >
> > David, when you read a technical or scientific article and see
> > an equation that contains a square root, what do you think it
> > means? A positive number, a negative number, of a number
> > that depends on the context?
>
> To some extent, the answer depends on the date of the article. Certainly,
> in a recent article, I would expect that the radical sign denotes the
> principal value. But I don't think we can relegate the alternative usage
> to the "archaic" category quite yet. The two dictionaries I'd mentioned
> above are still in print, I think.

Can you perhaps cite some less recent technical article where
the appearance of the radical sign denotes 'the negative root'?
I'd be interested.

In the Webster quote:
     "... For example, V9 is 3 or -3, and V-4 is not a real number. When
     there are two roots, the positive root is called the principal root.
     The equations V(a^2) = a, 3V(a^3) = a,... hold when a is positive
     and the radical denotes the principal root."

First they say that
        V9 is 3 or -3
and then they say that
        The equations V(a^2) = a holds when a is positive
        and the radical denotes the principal root.

The second obviously implies V(a^2) = -a when a is negative,
so the thing V of the first statement must be something other
than the principal root.
Do you think that, notation-wise and confusion-wise, Webster
has done a decent job there? Would you do it the same way?

Dirk Vdm

Relevant Pages

  • Re: A little lesson for sqrt(144) year olds.
    ... >> denotes the principal value. ... I've never seen it denote just the negative root. ... the positive root is called the principal root. ... > so the thing V of the first statement must be something other ...
    (sci.math)
  • Re: A little lesson for sqrt(144) year olds.
    ... >> denotes the principal value. ... I've never seen it denote just the negative root. ... the positive root is called the principal root. ... > so the thing V of the first statement must be something other ...
    (sci.physics)
  • Re: A little lesson for sqrt(144) year olds.
    ... >> denotes the principal value. ... I've never seen it denote just the negative root. ... the positive root is called the principal root. ... > so the thing V of the first statement must be something other ...
    (sci.physics.relativity)
  • Re: A little lesson for sqrt(144) year olds.
    ... >> an equation that contains a square root, ... In the Webster quote: ... the positive root is called the principal root. ... so the thing V of the first statement must be something other ...
    (sci.math)
  • Re: A little lesson for sqrt(144) year olds.
    ... >> an equation that contains a square root, ... In the Webster quote: ... the positive root is called the principal root. ... so the thing V of the first statement must be something other ...
    (sci.physics)