Division by a Negative and a Definition of Division

Date: 09/18/2006 at 03:04:18
From: saravanakumar
Subject: what does a number divided by a negative number signify

If 6 is divided by 3 it means that 6 is being divided into 3 equal 
parts.  But what does 6 divided by -3 mean?

Date: 09/19/2006 at 12:03:25
From: Doctor Ian
Subject: Re: what does a number divided by a negative number signify

Hi,

You can do yourself a big favor by letting go of the definition of
division you're using now.  It's one that is often used to help young
students think about division, but it's not ideal, as you've brought
up in your question.  So, what 

  6 / 3 = 2

really MEANS is simply that

      6 = 2 * 3

That is, division is just the inversion of a multiplication.  Using
that definition, we can think about all sorts of cases that don't make
sense using your definition.  For example, if

    pi
  ------- = x
  sqrt(2)

then

       pi = x * sqrt(2)

and we don't have to think about what it might mean to break something
like pi into "the square root of two equal parts", which kind of gives
me a headache just thinking about it! 

And it extends to your example, too:

  6 / -3 = x

means the same as 

       6 = -3 * x

which is true only if x = -2, so

  6 / -3 = -2

Try playing with variations on this, e.g., 

   -6 / 3 = x

       -6 = x * 3

       -6 = -2 * 3

   -6 / 3 = -2

which means that

  6 / -3 = -6 / 3

i.e., it doesn't matter where the sign is. 

Does that make sense?  Let me know if you need more help.

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