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Interpreting the Difference Between Two Numbers

Date: 09/22/2006 at 17:07:54
From: Mary
Subject: What is the difference between 8 and 5? or between 5 and 8? 

Not using absolute values, if a question reads "What is the difference
between 15 and 12" or "What is the difference between 12 and 15," 
would your response be +3 and -3 respectively?  And would the 
expressions be X = 15 - 12 (for the first example) and X = 12 - 15 
(for the second example) with the idea that whatever number is stated 
first in the written expression is the first number in the equation?

Date: 09/22/2006 at 23:11:55
From: Doctor Peterson
Subject: Re: What is the difference between 8 and 5? or between 5 and 8?

Hi, Mary.

Elementary algebra books tend to twist the English language a bit 
here, to make things easier for the students.  In real life, the
difference between a and b is |a-b| (or |b-a|, which is the same);
differences are always positive.  But that would lead to ugly 
equations that students would struggle to solve; so they pretend that
when we say "the difference of a and b" we mean just a-b--in that 
order, even though if b were greater than a, the "difference" would be

(In fact, there are some cases in math where we _would_ allow a 
negative difference; the one that comes to mind is when we take
"successive differences" in a sequence to see how it is made, and
consider the difference from one term to the next to be the second
minus the first, regardless of which is greater.  But we wouldn't
quite use the same phrase as in the textbooks!)

Anyway, regardless of the usual rules in real life, for the sake of
problems in this course, your "thoughts" represent what they want you
to do.

Here's a similar question from our archives that you might find
interesting as it discusses the use of absolute value in these situations:

  The Difference of x and y... 

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

- Doctor Peterson, The Math Forum 

Date: 09/25/2006 at 12:27:24
From: Mary
Subject: Thank you (What is the difference between 8 and 5? or between
5 and 8? )

Thank you for your complete and usable answer to my question!  It was
most helpful for our discussion and understanding.
Associated Topics:
Elementary Definitions
Elementary Subtraction
Middle School Algebra
Middle School Definitions

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