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### 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.

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
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

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/
```
Associated Topics:
Middle School Negative Numbers

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