Adding Negatives

Date: 12/04/97 at 15:12:20
From: Michael Shaw
Subject: Integers

My class disagrees on the answer to this problem:

-2 - 7/10

Some think the answer is -1 3/10 while others think that the answer is 
-2 7/10.

Can you please clarify?

Date: 12/12/97 at 11:06:26
From: Doctor Mark
Subject: Re: Integers

Hi Michael,

Well, the answer is - 2 7/10, but that's more of a definition than 
anything else.

As you know, when you write a mixed number, like, say, 1 2/3, it means 
to *add* the  integer (1) and the fraction (2/3), so it is really the 
same thing as 1 + (2/3), which is the same thing as 5/3, or, in 
decimals, 1.66666...  .

It turns out that this is a really ancient notation for addition.  
About 500 years ago, when people wrote two numbers next to each other, 
it meant "add," and the "one and two thirds" is a remnant of that old 
notation which persists to the present day.  

However,  A b/c (i.e., "A and b cths") does *not* always mean "add 
the integer (A) and the fraction (b/c)", and that's where the 
confusion lies. It only means "add the integer and the fraction" when 
there is *no* minus sign to the left of the mixed number.  

If you put a negative sign in front of (to the left of) a mixed 
number, that must mean to take the negative of the entire mixed 
number, because if it meant to add the fraction to the negative of the 
integer part, you would get crazy stuff  (I'll show you what kind of 
crazy stuff in  a minute).  

When you write - 2 7/10, you mean "the negative of the number 2 7/10"; 
that is, you mean:

- 2 7/10 = - (2 7/10) = - (2 + (7/10) ) = - 2 - (7/10)

So in this case, writing the fraction (7/10) next to the number (- 2) 
actually means "subtract," not "add."

Here's where the crazy stuff would pop up if you didn't use the 
definition above. What would happen if you wanted to *always* say that 
putting a negative sign in front of a mixed number meant to add that 
negative integer to the (positive) fractional part?  Let's try it:

Suppose that 

- 2 7/10 = ( - 2) + (7/10) 

Then we would get that this equals

= (7/10) - 2 = (7/10) - (20/10) = - (13/10) = - (1 and (3/10)) = 
  - 1 3/10.

But why stop there?  By the same reasoning, we would have to interpret
 - 1 3/10 as the sum of (-1) and the fraction (3/10):

- 1 3/10 = (- 1) + (3/10) = (3/10) - 1 = (3/10) - (10/10) = - 7/10.

So what is - 2 7/10?  Is it - 1 3/10, or - 7/10?  You see that this 
leads to problems!

Because of difficulties like this, we have to agree that putting a 
minus sign in front of a mixed number means "take the negative of the 
*entire* mixed number," and *not*  "add the (positive) fraction part 
and the negative integer part".  

However, I would say that the mistake is an extremely subtle one, and 
I think it would have caught a lot of people by surprise. If *you* 
were in my class and *you*  made a mistake like that, I would have 
told you that this was an extremely smart mistake to make, and that 
you were probably a pretty smart kid to make it!

Write back any other time you have questions.

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