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 
nought and answer 6 degrees?

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.

Does that help answer your question?

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