Negative Two Kids

Date: 09/27/2002 at 06:59:21
From: John Doe
Subject: Multiplying negatives

Let's say 3 x 2 = 6. Three is how many kids there are in a "group."  
Two is how many "groups" there are, and six is the answer. 

Now let's say 3 x (-2) = -6. How can you have negative two "groups" of 
three kids? 

How, Dr. Math?

Date: 09/27/2002 at 12:27:16
From: Doctor Ian
Subject: Re: Multiplying negatives

Hi John,

This is an _excellent_ question, and the answer is that in order to 
have a group of -2 kids, you have to decide what it means to have 2 
kids. Then -2 kids is the opposite of that.

It's easier to see this with things like directions. If I decide that 
'5 miles' means '5 miles east', then '-5 miles' means '5 miles west'.  

Similarly, if I decide that '5 feet' means '5 feet up', then '-5 feet' 
means '5 feet down'. 

So in one context, '2 kids' might mean that you _collect_ enough money 
to let 2 kids do something (attend a baseball game, for example).  
Then '-2 kids' would just mean that you have to _pay_ that same amount 
of money, instead of collecting it. 

In that context, '-2 * 3 kids' would mean that you have to pay for 3 
groups of -2 kids, or -2 groups of 3 kids, which are really the same 
thing.  

Probably the most important thing when dealing with negative numbers 
is to learn not to try to assign 'meanings' to the numbers themselves, 
but to keep in mind that putting the '-' sign in front of numbers just 
lets us skip writing things like 'east' and 'west', or 'paid' and 
'collected' after every number.

There is absolutely _nothing_ that can be done with negative numbers 
that can't be done without them, so long as we're willing to tag every 
positive number with the appropriate units.  The benefit of using 
'negative' numbers is that we save time by not having to write the 
units over and over.  

But every benefit comes at a price, and the price of this benefit is 
that we have to be careful to remember which units we've made positive 
and negative... and that sometimes we end up with expresions like '-2 
kids' that seem silly if looked at outside the context of a particular 
problem or situation. 

Does that make sense? 

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

Date: 09/28/2002 at 15:34:11
From: John Doe
Subject: Thank you (Multiplying negatives)

Thank you for clearing it up. Now I understand that there is no way to 
do that unless you think of a later point in time. As in a person is 
going to give money for 3 payments, but hasn't yet. (-5 x -3 = 15) So, 
I would later have +15 dollars. Thank you!