Picturing Negative Numbers

Date: 11/03/2002 at 19:19:03
From: James
Subject: Difficulty picturing negative numbers

If I have 4 apples and eat them, I get 0 apples. But obviously I can't 
eat another apple, so how do I get to minus 1? How do I get -4 apples? 
I can eat +4, but not -4 apples. 

Mum says that if I borrowed 4 apples from our neighbours I would then 
owe them 4 apples and that is represented by negative four. I 
understand that if I bought 10 more apples, I would have to give 4 to 
our neighbours (10-4 =6) and would be left with 6 apples for me. But 
I still can't picture these negative four apples. I can see positive 
four apples but I can't see negative four apples. Where are they? 

Also, I know there are minus degrees - on the Celsius and Farenheit 
scales, but there aren't on the Kelvin scale, which starts at absolute 
zero (-273 degrees Celsius,0 K) the lowest temperature you can have. 
You can't get a minus number from that! Or can you?

James

Date: 11/03/2002 at 23:13:43
From: Doctor Peterson
Subject: Re: Difficulty picturing negative numbers

Hi, James.

Not all real-life situations can be modeled with negative numbers, 
just as not all problems can be solved with fractions. You have to 
choose a problem where negative numbers fit.

Rather than eating apples, let's talk about buying apples. Suppose I 
run an apple store. If I have 4 apples in stock and sell 4 apples, I 
have 0 apples left that belong to me. (Even if they haven't been 
shipped yet, they're not mine anymore.) If instead someone had 
ordered 7 apples (to be delivered tomorrow), then I could ship 4 now, 
and owe 3 more to be shipped after the next picking. It's as if I 
had -3 apples, since the next 3 apples I get will be shipped and 
leave me with 0 apples. That is, I can add 3 to what I own now, and 
the sum will be zero.

You can't see these "negative" apples, except on your records. But you 
might be able to represent them visually. Suppose you have all your 
apples lined up on a shelf; and suppose you put a tag on each apple 
that has been sold. Then you start with four apples:

    o o o o
   ---------------

Now you sell seven, so you put seven tags on them; three tags have to 
lie on the shelf waiting for an apple:

    _ _ _ _
    o o o o _ _ _
   ---------------

You can either ship the 4 apples now, and be left with three tags
reminding you to ship more when you have them:

            _ _ _
    --------------

or you can wait and ship them all when they are available. In any 
case, the fact that you have three more tags than apples means that 
you have -3 apples that belong to you. So 4 - 7 = -3.

So in this picture, tags represent negative apples, apples that you 
owe. Tags on apples represent apples that belong to someone else; 
the tags cancel out the apples, as if they weren't there at all: 
1 + -1 = 0.

As for negative temperatures, you're right that you can't cool an 
object to below absolute zero (though I think I've heard that it is 
possible to get a negative Kelvin temperature somehow). That's just 
another example where negative numbers don't occur in real life. It 
has nothing to do with whether negative numbers make sense in 
themselves.

If you have any further questions, feel free to write back.

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