Absolute Value: Magnitude of a Number

Date: 11/24/97 at 23:36:12
From: Ann L. Triplett
Subject: Absolute value

I have just visited this Web site for the first time, and view it as 
an important potential resource. I have just started teaching 
developmental (remedial) math part time at a community college (it's 
my second semester), so I am actually teaching  K-12 material. I find 
that I don't have enough experience to field some of my students' 
questions about how they could use this in real life. (Ironic,
since I have worked since 1981, first at Federal Express and then as a
consultant, now part time - so I SHOULD have some experience using 
MATH in real life/business). 

They stumped me today - I could not think of a single time I have used
the concept of ABSOLUTE VALUE in my work - any ideas where they may 
run into it ?

Ann Triplett - Arlington, TX

Date: 12/08/97 at 14:04:52
From: Doctor Mark
Subject: Re: Absolute value

Hi Ann,

Since I teach remedial math at my state college (twice per term), I 
feel your pain...

Absolute value is one of those things that really doesn't have any
applications, in a strict sense. Said another way, it has lots of
applications.  Let's see why.

The best way I have found of describing absolute value is that the 
absolute value of a number is just the number, ignoring the sign.  
That is, it's sort of like being colorblind.  People who are 
colorblind don't see colors, and people who are "absolute valuers" are 
"minus sign blind": they don't see minus signs. Of course, this only 
holds for single numbers, not combinations like 5 - 3, or variable 
quantities like - x.

The main mistake that students make regarding absolute value (which I
actually prefer to call the "magnitude," since it's shorter) of a 
number is that they treat it by analogy: since  |-5| = 5, it must 
follow that |5| = - 5.  That is, they think that absolute value means 
"change the sign," not "ignore the sign."

There are just not that many situations in which you are supposed to
"ignore" the sign of a number, because the sign of a number is just as
important as the number next to the minus sign (which, of course, is 
the absolute value of the number!).

On the other hand, there are, in some sense, lots of situations in 
which the absolute value rears its head.  The reason that its use is 
not apparent is that we denote the sign of a number by some word, 
like "increase by," "decrease by," or "fell by," followed by the 
absolute value of the number. For instance, suppose the temperature 
went from 55 degrees to 32 degrees. We say that "the temperature 
dropped by 23 degrees".  What we mean is that the change in 
temperature [which is *always* (final temperature) - (initial 
temperature)] was 32 - 55 = - 23 degrees. But we express that as "it 
dropped (the "-") by 23 (the absolute value) degrees." That is, in 
English, we use words to denote the sign of the number, and absolute 
value to designate the magnitude of the number.

Similar considerations apply to profit/loss, income/expense, and so 
on. So in some sense, the absolute value is used all the time.

You might consider subscribing to the <math-teach> mailing list, also 
run by the Math Forum (for how to subscribe, see "About this 
discussion"):

   http://mathforum.org/kb/forum.jspa?forumID=206   

There you can interact with other college faculty, and with K-12 
mathematics teachers, and post your questions. I am a pretty active 
member of that group, and if you post any pedagogical/why do we need 
this/what's this good for types of questions there, you will probably 
get responses from me.

-Doctor Mark,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/