Practical Applications of Absolute Value

Date: 09/26/2000 at 13:25:49
From: Anne Marie Garrett
Subject: Practical applications of absolute value

My 8th grade students would like to know the real world applications 
of the concept of absolute value. I have explained it as a distance to 
zero on the number line.  They are not convinced of its importance and 
why we need it.  Now that they've asked and I've thought about it, I'd 
like to know too!

Date: 09/26/2000 at 15:10:16
From: Doctor Douglas
Subject: Re: Practical applications of absolute value

Hi Anne Marie,

Thanks for sending your question to Ask Dr. Math.

Your choice of explanation as a "distance" is a very good one, since
that kind of quantity is one that doesn't depend on its sign.  I think 
that your students may be objecting to the "distance from zero" 
because that's a mathematics example.

Here are some "real-world" applications that I've come up with by
brainstorming with Dr. Tony here at the Math Forum.

1. Distances in real life:  suppose you go three blocks east, then six
   blocks west, then eleven blocks east again.  Now we can ask two 
   questions: Where are you relative to where you started?  This 
   requires us to retain the sign information, and is not answered by 
   the absolute value. The other obvious question is "How far did you 
   go?" Now every student in your class should add 3 + 6 + 11, each of 
   them doing at least one absolute value operation in their minds 
   (for the -6). 

   Of course distances are useful in many real world applications, 
   such as navigation and transport ("Do we have enough fuel to get 
   there AND back?"), architecture, engineering and science, and 
   sports ("How many consecutive 15-yard penalties can the referees 
   call before it becomes 'half-the-distance-to-the goal-line", if we 
   start at midfield?").

2. Suppose you are driving a car. Going too fast is obviously a hazard
   and might earn a speeding ticket. Going too slow is also a hazard, 
   and can earn a ticket also. What matters is how different one's 
   speed is from what everyone else is doing. This type of 
   "difference" is fundamental to all sorts of concepts in statistics, 
   where the absolute value is used in various ways of quantifying how 
   well or how poorly one thing predicts another. Statistics is used 
   in many important real world applications also, including medicine 
   and finance.

3. Suppose you are exchanging currency, say dollars and pesos. The 
   bank or exchange will charge a commission based on how much is
   exchanged (sometimes there will be a flat fee as well). This 
   commission is applied no matter whether you buy dollars or buy 
   pesos. In your pocket, it could be D dollars and P pesos, and D 
   could go up (or down) while P goes down (or up). But the commission 
   on a given transaction is the absolute value of either D or P, 
   times the exchange rate.

I hope that these examples helps to get you started.

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