Absolute Value as a Grouping Symbol?

Date: 07/25/2002 at 18:53:48
From: Andre Pearson
Subject: Absolute value as grouping symbol

I teach 8th grade Algebra in California.  I was teaching the Order of 
Operations. I explained that there are four grouping symbols: 
parentheses, brackets, braces, and the fraction bar.  

One student asked me if the absolute value bars are also grouping 
symbols. I told him no. But as I thought about it, in a practical 
sense, absolute value is a grouping symbol: with every equation/
expression in my algebra textbook that has an absolute value, you must 
solve what's in the absolute first!

I have two questions: 1) Are there any examples of a problem where you 
do not have to evaluate the absolute first? and 2) Would it would be 
wrong for me to teach the absolute value as a "pseudo" grouping 
symbol?

Thanks,
Andre Pearson
Downey, CA

Date: 07/25/2002 at 23:25:39
From: Doctor Peterson
Subject: Re: Absolute value as grouping symbol

Hi, Andre.

You actually missed another important grouping symbol (though it looks 
like the fraction bar, so you might have meant to include both): the 
vinculum. This is the bar over the top of a radical expression, and 
was actually used before parentheses in expressions like
      ___
    x y+z  meaning  x(y+z)

But yes, the absolute value bars do serve partly as a grouping symbol. 
That is, their primary meaning is to indicate an absolute value, but 
they incidentally require that whatever is inside must be evaluated 
first. Thus

    3|x+y|

is equivalent to

    3*abs(x+y)

where I have used functional notation, in which again the parentheses 
are primarily to identify the argument of the function "abs" but also 
serve to group. In a function with two arguments, I suppose you could 
call the comma separating the arguments a grouping symbol too: 
atan2(x,y).

In summary, I would go a bit beyond calling the absolute value a 
"pseudo" grouping symbol, and call it a symbol one of whose functions 
is to group an expression.

I don't think I understand your first question; the point is that you 
CAN'T evaluate an absolute value before evaluating its argument. I 
suppose you meant, first before doing something else outside of it. 
Of course you can distribute, as you can with parentheses:

    3|x+y| = |3x + 3y|

but in terms of actually evaluating an expression as it stands, you 
have to evaluate the argument, then the absolute value, then whatever 
it is used in.

Here is an answer that touched on these issues, which I found by 
searching for "absolute value grouping" . It's surprising what you can 
find in our archives if you look; but I remembered writing this one:

   Grouping Symbols
   http://mathforum.org/library/drmath/view/58406.html 

If you have any further questions, feel free to write back.

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/