Operating on Negative Numbers

Date: 10/21/96 at 19:35:47
From: Jake
Subject: Integers

Hi Dr. Math,

I have lots of problems with multiplying and dividing integers. 
Do you have any tricks for me? I can add and subtract integers, but 
multiplication and division are really hard for me. The negative and 
positive numbers give me a really hard time.

Thank you.

Date: 10/29/96 at 11:26:10
From: Doctor Lim
Subject: Re: Integers

Dear  Jake,

One important thing to remember is the rules of operation.

Multiplication

1) plus x plus       = plus
    (2) x (2)        = 4

2) minus x minus     = plus  
     (-2) x (-2)     = 4

3) minus x plus      = minus
     (-2) x (2)      = -4

4) plus x minus      = minus
     (2) x (-2)      = -4


Division

1) plus /  plus      = plus
     (2) /  (2)      = 1 

2) minus / minus     = plus  
     (-2) / (-2)     = 1

3) minus / plus      = minus
     (-2) / (2)      = - 1

4) plus / minus      = minus
     (2) / (-2)      = -1
__________
 
Let me explain further.

3) minus x plus      = minus
     (-2) x (2)      = -4

4) plus x minus      = minus
     (2) x (-2)      = -4

These two are exactly the same, just as 2 x 3 = 3 x 2

This means that for example in -6 x 6, you have added 
negative six to itself six times.  You then get the 
answer of - 36.

It also means that you borrowed $6 from your friend 
six times - so how much do you have to pay your friend? 
6 x 6 which is  36.

This $36 will have to come out of your next allowance, 
so you will have a minus $36. 
__________

2) minus x minus     = plus  
     (-2) x (-2)     = 4

To find -6 x -6, we'd have to add negative six to itself negative six
times.  

If we add two negative sixes together (-6 x 2), we get -6 + -6 = -12.
If we add one negative six together (-6 x 1), we get -6.
If we add zero negative sixes together (-6 x 0), we get 0.

Notice that we are increasing by sixes each time.

If we add -1 negative sixes together (-6 x -1), we get -(-6) = 6.
If we add -2 negative sixes together (-6 x -2), we get -(-6) + -(-6) 
= 12.

You can see the pattern then, and how -6 x -6 = 36.
__________

So in the operation of 

  8 - (-5)
= 8 -1(-5)

-1 x -5 
To find -1 x -5, we'd have to add negative one to itself negative 
five times.  

If we add two negative fives together (-5 x 2), we get -5 + -5 = -10.
If we add one negative five together (-5 x 1), we get -5.
If we add zero negative five together (-5 x 0), we get 0.

Notice that we are increasing by five each time.

If we add -1 negative five together (-5 x -1), we get -(-5) = 5.

  8 - (-5)
= 8 + 5
= 13
__________

minus 1 x minus 5 = 5

  8 - (-5)
= 8 + 5
= 13
__________

In division of negative integers, it is much easier.

In a thermometer, for example, the temperature can be read from -200 
degrees to +200 degrees.

If the temperature drops -2 degrees each day 
a) what will be the temperature drop in 4 days ? 

 one day -------> - 2 degrees
four days ------>  -2 x 4 degrees = -8 degrees

The temperature will drop -8 degrees.
__________

If the temperature drops - 2 degrees each day 
b) how many days will it take to drop to -12 degrees ?

 -2 degrees -----> one day
-12 degrees -----> -12/-2 days = 6 days.

It will take 6 days for the temperature to drop -12 degrees.
__________

If the temperature drops - 12 degrees in four days
c) what is the temperature drop in one day?

four days ------> - 12 degrees
 one day  ------>  -12/4 = -3 degrees
               
In one day, the temperature drop will be -3 degrees.
__________

By putting the negative integers in things that you are familiar
with, you will be able to understand better.

When you get a problem like divide - 14 by 2, think of it as if you 
owe $14 each to 2 persons. How much do you have to pay each person?

-14 / 2 = -7

You owe each person $7, which is a negative 7 from your next 
allowance.


Got the idea? Don't worry about the method. Once you understand the 
basics and practise them often enough, it will be easy. The hardest 
step is always the first one.

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