Tips for Negative and Positive Numbers

Date: 11/30/2001 at 17:47:46
From: Chris
Subject: Tips for negative and positive numbers

Integers are tricky I get multiplying and division but subtracting and 
adding them confuses me. I really get confused when adding or 
subtracting a negative and a positive. Can you give me tips on how to 
add and subtract integers?

Thank you.

Date: 11/30/2001 at 19:08:59
From: Doctor Achilles
Subject: Re: Tips for negative and positive numbers

Hi Chris,

Thanks for writing to Dr. Math.

When it comes to multiplication and division, it can get a little 
tricky. But you can use a pretty simple rule: count up the number of 
negatives. If you have an EVEN number of negatives (0, 2, 4, etc.) 
then the answer will be POSITIVE; if you have an ODD number of 
negatives (1, 3, 5, etc.) then the answer will be NEGATIVE.

What I do with addition and subtraction is a little different from 
what most people do. Let me take a roundabout way to get there and 
show you. I hope this will be helpful.

Addition is easier than subtraction for me. The reason is that 
addition is symmetric or commutative. (See the Dr. Math Glossary of 
Properties and select "commutative": 
   http://mathforum.org/dr.math/faq/faq.property.glossary.html   .)  

For example:

  1 + 2

is the same thing as:

  2 + 1

Subtraction is NOT symmetric:

  1 - 2

is NOT the same as:

  2 - 1

When you start dealing with negative and positive integers, and adding 
and subtracting, things start getting really hard. I don't like 
memorizing a lot of different rules; I only want to memorize one or 
two rules if I can. So these are the rules I know:

1) I know how to translate any addition or subtraction of integers 
   problem into an addition problem.

2) I know how to add positive and negative integers.

This way I don't have to learn how to subtract positive and negative 
integers and get confused with that.

The way to translate a subtraction problem to addition is this:

  Whenever you see a minus, change the sign integer that comes right 
  after it and make the minus into a plus.

So when you have a minus followed by a negative number, for example:

  3 - (-5) = ?

just CHANGE the -5 to 5 and change the minus to a plus:

  3 + 5 = ?

And when you have a minus followed by a positive number, for example:

  4 - 6 = ?

just CHANGE the 6 to -6 and change the minus to a plus:

  4 + (-6) = ?

The reason I like this form is that then I don't have to worry about 
which number comes before and after the + sign. All I have to know is 
that I have a 4 and a -6 and I'm adding them together. It doesn't 
matter to me whether the actual problem I have is:

  4 + (-6) = ?

or:

  (-6) + 4 = ?

I just need to know which two numbers I'm adding together and that's 
all.

Okay, so much for rule 1 (changing to addition). 

Now how do I add integers? (Rule 2).

All I do here is just count. I count (add up) how many positives I 
have and count (add up) how many negatives I have. Then I imagine that 
they are on two teams, fighting each other. Each time a negative finds 
a positive, they cancel each other out, and then I just see what's 
left over.

So with:

  2 + 3 = ?

I have 5 positives and no negatives, so when they get together there 
are no negatives so the 5 positives all make it and the answer is 5.

With:

  (-1) + (-3) = ?

I have no positives and 4 negatives, so when they get together there 
are no positives so the 4 negatives all make it and the answer is 
(-4).

With:

  4 + (-6) = ?

I have 4 positives and 6 negatives. When they get together, one 
negative cancels out one positive: that leaves me with 3 positives 
and 5 negatives. Then another negative cancels out another positive: 
that leaves me with 2 positives and 4 negatives. Then another 
negative cancels out another positive: that leaves me with 1 positive 
and 3 negatives. Then another negative cancels out another positive: 
that leaves me with 0 positives and 2 negatives. Now we're out of 
positives, so the negatives win and there are 2 of them left, so the 
answer is (-2).

Finally, with:

  (-3) + 8 = ?

I have 8 positives and 3 negatives. When they get together, one 
negative cancels out one positive: that leaves me with 7 positives 
and 2 negatives. Then another negative cancels out another positive: 
that leaves me with 6 positives and 1 negative. Then another 
negative cancels out another positive: that leaves me with 5 positives 
and 0 negatives. Now we're out of negatives, so the positives win and 
there are 5 of them left, so the answer is 5.

I hope these rules and examples were helpful.  If you're still stuck, 
please write back and we'll try to help you some more.

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