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/
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