Adding Positives and Negatives

Date: 12/01/98 at 17:48:12
From: Jaime
Subject: Solving Equations

Hi,

I am having a lot of trouble with my homework, and I was wondering if 
you could help me understand it. Here are the problems:

   -31 + (-46) = b 
   a = 16 + (-36)
   74 + (-63) = d

Thanks so much, 
Jaime

Date: 12/01/98 at 18:16:56
From: Doctor Santu
Subject: Re: Solving Equations

Hello, Jaime!  We'll try to help.

It's not perfectly clear what is wanted here in the problems, but I can 
guess. If I'm wrong, you can write back.

It looks as if you're at the point where they're getting you to add a 
combination of positive and negative numbers. So the first problem is
just: "add a negative 31 and a negative 46, and call the answer 'b'."

Does this sound like what's going on? If it is, then here's how you do 
it. (If it isn't, write back and try and explain carefully what is 
going on in your class, so that we have an idea about their particular 
angle on it.)

When you're adding a combination of positive and negative numbers, 
here's what helps me.

I think of a positive number as going a distance to the right. For 
instance, if you start like:

   ............x............

then 4 would be:

   ............!...x........

See how you've moved to the right 4 places. The ! is where you started 
from.

I think of a negative number as going to the left. For instance, if we 
go back to the starting point:

   ............x............

what would a -7 look like?

   .....x......!............

Okay?  It's seven places to the left.

Now, when you want to add two positive numbers, you just add them.  
4 + 6 is where you'd end up if you first went 4, and then went 6 more, 
which is at 10.

When you want to add two negatives, like (-4) + (-6), it's like going 4 
to the left, and then another 6 to the left, you end up at -10.

So, adding two negatives is just as easy as adding two positives. Just 
add them, but remember that the answer is negative. So your first 
problem is not hard at all since you're adding a negative 31 and a 
negative 46. Go for it.

What about combinations of positives and negatives?

This is the only interesting situation. Both the remaining problems 
are of this type. What happens?

What happens if you add -8 and a 12? One number takes you one 
direction, and the other number takes you in the other. You can easily 
see that you end up a distance of 4 from where you started. On which 
side? It depends on which number was bigger. If the negative number was 
bigger, it wins, and you wind up on the left. If the positive number 
is bigger, *it* wins, and you wind up on the right!

In the case of -8 + 12, who wins? The 12, so you only go a distance of 
4 (the difference between the numebrs), and since 12 is bigger, it's 
on the right. The answer is 4.

How about 56 + (-61)? Here the -61 wins against the 56. The difference 
is 5. Answer: -5.

What about (-98) + 108? Answer: 10.

Hope this helps!

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