Re: [Phys-l] the square root(s)

[John Denker <jsd@av8n.com>]

On 07/16/2009 12:13 PM, Steve Highland wrote:

> The class I met today had a dispute going (good!) over what they saw as a
> contradiction in the book.  It defined the square roots of a number a to be
> the solutions of
>
> X^2 = a
>
> -- so that both the positive and negative values qualify.
>
> But then it said the value of {radical sign}a was just the positive value.
> So the text defined "square root" written out in words one way and square
> root written as a radical a different way.   It seemingly made a distinction
> between "a square root" and "the square root."
>
> Man, that's confusing.  Does anybody have a better way out of this language
> dilemma?

The kids are right.

My approach to explaining this is to say that despite what you
may have heard, mathematicians are human.  
  a) Sometimes they make localized trivial mistakes (such as typos).
  b) Sometimes they make pernicious mistakes that are not so easily
   repaired.

The _terminology_ is particularly likely to show human foibles.

Specifically, the terminology for square roots is a mess.  If for
example we have x = y^2 = 4, then:
 The square root of x is +2 (not -2).
 √4 is +2.
 4^(1/2) is +2.
 The square roots of x are +2 and -2.
 +2 is the square root of x.
 -2 is not the square root of x.
 -2 is a square root of x.
 +2 is also a square root of x.
 y might or might not be the square root of x.
 Either y or -y (but not both) is the square root of x.
 The solution to the equation y^2 = 4 is the solution set {+2, -2}
 The solution to the equation y^2 = 4 is y = ±√4 
 The solution to the equation y^2 = 4 is y = ±2.
 -2 is a solution to the equation y^2 = 4.
 +2 is also a solution to the equation y^2 = 4.
 ... et cetera ...............


I _think_ if you work hard enough you can make all of the above be 
consistent, but at best it's like walking through a minefield.


It gets even more complicated when 
 1) You extend it to complex numbers:  what is "the" square root
  of -4i?
 2) You extend it to cube roots and higher roots:  what is "the"
  cube root of -8?  Are you sure?  What rule did you use?  Is
  your rule consistent with your answer to question (1)?
 3) You extend it to transcendental functions.  Example:  what 
  is "the" solution to sin(x) = 0.1?
 4) Combinations of the above.


If you think the terminology for square roots is bad, wait until you
see the terminology for partial derivatives, or the terminology for
conditional probabilities.

===============================


To calm down the kids, I tell them that ideas are primary and fundamental,
while terminology is secondary.  Terminology is important only insofar as 
it helps us contemplate and communicate the ideas.

The kids don't necessarily believe me the first time(s) I tell them that.
Remember that the kids have been exposed to a gazillion exercises and
tests that emphasize rote regurgitation of terminology to the exclusion
of actual thinking.  How can I tell them not to worry about it, when the
tests require them to worry about it?

If I see one more T/F question that asks whether "the" solution to y^2 = x
is "the" square root of x I'm gonna start screaming and throwing things.


Constructive suggestions:  
 -- When reading, try to read past the terminology to get the _meaning_.
  In the real world, terminology problems are the least of your problems.
  Often you get an equation such as y^2 = x unencumbered by any language
  about "the" square root(s).  You need to understand the physics to know
  whether you need to keep the positive root *or* the negative root *or*
  both.
 -- When writing, there are often many choices as to the terminology.
  Choose something that is (relatively) unambiguous.  For example, if
  you want both square roots, write ±√x including the explicit ± sign,
  or talk explicitly about the solution _set_ (not just "the" solution).
  If you want only the positive square root, talk explicitly about the 
  positive square root (aka principal square root).

================

As for the more general question about how to teach kids who are 
algebra-torture survivors, I'll discuss that in a separate note,
later.