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### Spaces and the Number Line

```Date: 02/26/2003 at 11:37:08
From: Holly
Subject: Minus Numbers

If you say, "How many degrees are there between -2 and 4?" would you
count the 0 degrees and answer 7 degrees or would you not count the
```

```
Date: 02/26/2003 at 12:08:24
From: Doctor Peterson
Subject: Re: Minus Numbers

Hi, Holly.

When we subtract two numbers, we aren't counting numbers between
them, but measuring the distance between. In this case, the number
line (or thermometer) looks like this:

|<--------------------->|
-+---+---+---+---+---+---+---+---+---+---+-
-5  -4  -3  -2  -1   0   1   2   3   4   5

The distance is the number of SPACES (units), like this:

|<--------------------->|
-+---+---+---+---+---+---+---+---+---+---+-
-5  -4  -3  -2  -1   0   1   2   3   4   5
\__/\__/\__/\__/\__/\__/
1   2   3   4   5   6

So the distance, and the difference, is 6 degrees.

If you want to do it by counting, then you want to count every number
after the first and including the last:

|<--------------------->|
-+---+---+---+---+---+---+---+---+---+---+-
-5  -4  -3  -2  -1   0   1   2   3   4   5
^   ^   ^   ^   ^   ^
1   2   3   4   5   6

What you are really doing here is counting each space as you go past
it, so it's the same as what I did above. This is like counting how
far you go in your car by counting each time the odometer clicks
another mile; you don't count both ends, because when you start you
haven't gone anywhere yet. You only count the miles you have actually
passed.

- Doctor Peterson, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
Elementary Temperature
Middle School Negative Numbers