Adding a Negative: Absolute Value and Parentheses

Date: 08/25/2001 at 13:53:11
From: Allison
Subject: I just don't get it

Dear Doctor Math,

Usually math is my best subject. This year I'm in the seventh grade 
and in a Pre-Algebra course. Our first lesson was on positive and 
negative numbers and I got that, but when we got into the absolute 
value symbols and the parentheses it went completely over my head!  

For example: 0-(-2). My  math teacher said that if you have two 
negative symbols that it equals one positive number; 0-(-(-2)). Well, 
that just confused me! So I asked him to try to explain it, but I just 
still didn't get it. Anyway, he said that the answer would be positive 
2.  Well that confused me even more because if we're supposed to be 
subtracting, than how did we end up adding? And if you're just going 
to turn (-2) into 2, than why don't they just come out and say it: 
0-2.  

I don't see the need for all of this other stuff and I don't 
understand what an absolute value is. If you have the absolute value 
of -20, isn't that the same as -20?  Help!

- Overwhelmed in Oklahoma

Date: 08/25/2001 at 23:10:32
From: Doctor Peterson
Subject: Re: I just don't get it

Hi, Allison. Thanks for a stimulating question! You've told me enough 
of your thinking to really help me think through how to make things 
clearer to you.

First, negative numbers can be confusing at first, so we get lots of 
questions about them. If my answer isn't enough, you can try looking 
through our archives to see if other things we've said about negative 
numbers help:

   http://mathforum.org/dr.math/tocs/negative.middle.html   

Also check out our FAQ on multiplying negative times negative:

   http://mathforum.org/dr.math/faq/faq.negxneg.html   

Now let's look at your questions.

First, what is an absolute value? Basically, it's what's left of a 
number if you take off the sign. So the absolute value of -20, |-20|, 
is 20, since we take off the "-". The absolute value of +20 is also 
20, since we take off the "+" as well (which doesn't change anything, 
of course).

We define it formally in more complicated sounding ways, such as "the 
number itself, or the negative of the number, whichever is not 
negative"; or, "the distance between 0 and the given number on the 
number line." But for many purposes, what I've said is what lies 
behind it all: we can work with negative numbers by thinking 
separately about the number's sign and its "size," which we call 
"absolute value."

Now, what is 0 - (-2)? First, we should ask what it even means to 
subtract a number from another. What is 3 - 2? Well, it's the number I 
have to add to 2 to get 3. The definition of subtraction is :

    a - b = c  when  a = b + c

In this case, we want

    0 - (-2) = ?  when  0 = (-2) + ?

So what do we have to add to -2 to get 0? Look on the number line:

      -3  -2  -1   0   1   2   3
    <--+---+---+---+---+---+---+-->
           o------>

To get from -2 to 0, we have to go right 2 units; that is, we have to 
add 2. To put it another way,

    -2 + 2 = 0, so 0 - (-2) = 2

This is really how we defined negative numbers in the first place. The 
negative of a number is the number you can add to it to get 0. (We say 
that -2 is the "additive inverse" of 2, meaning that their sum is 
zero.)

If you're not yet convinced that 2 is the right answer, play with 
these ideas and the number line until you're convinced.

Now what was all that that your teacher was saying about double 
negatives?

Once you get used to the idea of negative numbers this becomes second 
nature, so you can easily forget that it confuses people who are new 
to it. But here's the reasoning (using algebraic symbols, rather than 
all the talk I did to try to convince you). I'll write it out in two 
steps, and then explain them, so don't worry if you don't follow it 
the first time through:

    0 - (-2) = 0 + -(-2)
             = 0 + 2
             = 2

The first step is to replace subtraction with addition of the 
negative. The general rule is

    a - b = a + -b

Why should this be true? Take an example,

    3 - 2 = 3 + -2

This means that if I start at 3 on the number line and add -2, I end 
up at 1. I've taken 2 away from 3 by adding -2:

      -3  -2  -1   0   1   2   3
    <--+---+---+---+---+---+---+-->
                   +---------->
                        <------+

So adding the negative is the same as subtracting the original number. 
Essentially, a negative means taking something away; that's why we use 
the minus sign.

The second step in my explanation was to replace -(-2) with 2. Why 
should the negative of a negative be the original number? Well, the 
negative of a number means you flip the number over on the number 
line, from a positive

                      2
                +---------->

to a negative

     <----------+
          -2

If you flip it again,

                +---------->

you get back to the positive. So the negative of -2 is 2; 2 and -2 are 
negatives of one another.

Put this all together, and you have what your teacher was saying: 
subtracting -2 is the same as adding +2. It feels odd at first, 
because it says that you can get bigger by adding something. But 
aren't negative numbers kind of backward in the first place? You 
should expect things to work backward with them. That's really what 
they are all about.

Feel free to write back if you need more help with all this!

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