"Subtraction" and "Negative"--Same Sign, Different Concepts?

Date: 12/14/2005 at 06:51:33
From: Chris
Subject: Difference between 'subtraction' and 'negative'?

Is there a difference between the operation "-" when used in the
expression "y = a - b" as compared with its meaning in the 
expression y = -b?  

I don't doubt the truth of the statement

  0 = -1  -(-1)

but the reasoning I have seen has never quite convinced me, because it 
seems that the - sign is being used to mean two different things.  On 
a number line these two things are something like

  1)    -b => Rotate the following number by 180 degrees.

  2) a - b => Continue (or count) in the direction of the following
              number, rotated by 180 degrees.

Now while these are fairly similar they are not the same thing.

Taking another approach, we are allowed say "y = a x b" but we are not 
allowed to say "y = x b".  So "-" is being used in a "syntactically" 
different way to "x".

This indicates to me that we are allowing "-" to be used to mean two
different things.  I think that, as it happens, the rules for 
combining the two different meanings of the sign allow us to get away
with saying "-(-) = +" without distinguishing between the two, because
it just happens to work out like that.

The reason I'm so hung up on this is that I think if we temporarily
used new signs - say, p and n, for positive (no rotation) and negative
(180 degree rotation)--and continued to use + and - for the operations 
of counting on or counting with rotaion, it all becomes a lot easier 
to explain what is actually happening.

That is, "pa - pb" can be shown to be the same as "pa + nb" and also
it is much easier to grasp that "pa - (nb) = pa + pb".

Is there anything to all this?

Date: 12/14/2005 at 11:37:24
From: Doctor Peterson
Subject: Re: Difference between 'subtraction' and 'negative'?

Hi, Chris.

You're exactly right that "minus" and "negative" are two different 
operations; that's why they have different names (and we _try_ to get 
kids to use the right names ;-). 

"Minus" is a binary operation (performed on two numbers), while
"negative" is a unary operation (performed on a single number).  They
have the same symbol because they are _very_ closely related, and it
doesn't cause any trouble to use the same symbol. 

However, some texts have done just what you suggest, and used 
different symbols for them; one common choice is a raised "-" to mean
negative, so an equation might look like
              _
  a - b = a +  b

This does help some students to get a better sense of the distinction 
before they move on to the normal notation.

I might also add that scientific calculators have separate keys for 
the two operations; on those that most closely mimic written math 
notation, the "negative" key is commonly labeled "(-)" to distinguish 
it from the "minus" key, "-". 

Computer programming languages don't need to make that distinction,
since they can determine the meaning from context; I'm not sure why
calculators don't do the same, but it's probably due to little
differences in the way users think of keys on a calculator vs. 
characters on a page. 

Regardless of whether we use distinct notations for the two 
operations, it is important to distinguish the operations themselves. 
The negative is called the "additive inverse", and is defined by

  a + -a = 0

That is, the additive inverse of a is the number you can add to a to 
get 0.  Having defined that, we actually _define_ subtraction in terms 
of this:

  a - b = a + -b

That is, we define subtraction as _meaning_ addition of the additive 
inverse; so, as you said, subtraction means "turn around on the number 
line and walk the given distance in the opposite direction".  This is 
the connection between the two operations, and the reason it is 
helpful to use the same symbol.  When I read "a - b", I see it as "a + 
-b", because addition has important properties, such as commutativity, 
that subtraction lacks, making it very useful to forget about 
subtraction and use only addition.

(Note that if we used a very different symbol, like your "n", it would 
be harder to see the connection and to learn to see it this way: it is 
not obvious that a - b = a + nb.)

There is a similar issue with regard to multiplication and division. 
If we wanted, we could use the division symbol "/" to indicate the 
multiplicative inverse (also called the reciprocal):

  a * /a = 1

We define division as multiplication by this inverse:

  a / b = a * /b

in much the same way as for subtraction.  For some reason this has 
never caught on, as far as symbols are concerned.  Similarly, although 
we can use "+" as a unary operator (which has no effect on a number, 
as +a = a, meaning 0+a), we don't happen to use multiplication, "*", 
in the same way, so that *a = a, meaning 1*a.  It wouldn't hurt to do 
so, but has not been found useful!

Getting back to negatives, the fact that

  -(-a) = a

(effectively, turning around twice and ending up in the same 
direction) allows us to easily rewrite

  -1 - (-1)

as

  -1 + -(-1) = -1 + 1 = 0

This is much harder to explain in terms of subtraction only; by seeing 
subtraction as adding the negative, it becomes relatively simple.

In summary, your thoughts are valid, and not really new; seeing things 
this way is actually essential to learning algebra well.  So you're in 
good company!

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

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

Date: 12/14/2005 at 12:14:54
From: Chris
Subject: Difference between 'subtraction' and 'negative'?

Fantastic!  To some small degree I have been hung up on this for years
and despite moderate searching I have never come across any of these
texts, so it is quite a relief for me to find that it is a valid way
of thinking.

Thanks again for your time.

Date: 12/14/2005 at 22:58:25
From: Doctor Peterson
Subject: Re: Difference between 'subtraction' and 'negative'?

Hi, Chris.

I wanted to confirm my experience that some (many?) texts use the
raised negative sign, so I tried finding references on the web.  Here
is the only explicit reference I found:

    http://homepages.ius.edu/MEHRINGE/T101/Notes/Section4-1.htm 

I didn't find enough evidence to determine how widespread this is, or
whether it is a current phenomenon.

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

Date: 01/03/2006 at 09:11:06
From: Chris
Subject: Thank you (Difference between 'subtraction' and 'negative'?)

Thanks for this.  Visually, the look of the expressions which mix the 
raised "-" and normal "-" are exactly right for me and express the 
meanings quite intuitively.

The distinction between  5 - 6 and 5 + ^-6 is clear, as  is the fact 
that they give the same result.

Regards,
Chris.