Adding Positive to Negative Numbers

Date: 06/01/99 at 08:51:25
From: jeff
Subject: It's too hard

How do positive numbers get added to negative numbers?

Please help!

Date: 06/01/99 at 12:49:03
From: Doctor Peterson
Subject: Re: It's too hard

Hi, Jeff.

You're not alone in needing help here!

The first word that came to my mind when I read your question was 
"subtract." That's what negative numbers do. The hard part is to know 
what to subtract from what, and what to do next.

Let's picture a number line - maybe imagine we've drawn one on the 
floor. The positive numbers are in front of you, so that if you're 
standing on the zero and I say "go 3 steps forward," you'll end up on 
the 3. The negative numbers are behind you. If I now say "go back 5 
steps," what will happen? It will take you 3 steps (if you don't trip) 
to get back to the zero, and there will be two steps left, so you'll 
end up on the -2.

Suppose I give the two commands in the opposite order. First you go 
back 5 steps and find yourself on the -5. Now you go forward 3 steps; 
that undoes 3 of your 5 backward steps, and again you end up on the 
-2.

What this shows is that

    +3 + -5 = -2

and

    -5 + +3 = -2

It doesn't matter which order you add them in. Either way, you have 
two opposite numbers fighting, and the larger one (the 5) wins. Since 
the 5 is negative, you'll end up with a negative number that is the 
difference between the sizes of the two numbers. So we subtract 3 from 
5 giving 2, and give it a negative sign because the negative number 
won.

Here's a more careful statement of what we just did:

    +3 + -5 = -5 + 3 = -(5 - 3) = -(2) = -2

That means that I saw that 5 > 3, (5 is greater than 3), so I put them 
in that order, then pulled out a negative sign from both numbers 
(using the distributive property) and did the subtraction.

If my explanation doesn't quite fit your needs, look through our 
previous answers to negative number questions. I'm sure you'll find 
one that puts it the way you can best understand:

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

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