Talking About Zero, Absolute Zero and Negative Numbers

Date: 11/12/2003 at 22:30:58
From: T. Ball
Subject: Absolute Zero, Zero, & Negative numbers

Zero is a tricky number to explain, especially when children are 
interested in why we have negative numbers.  It is difficult to 
conceptualize taking something away from nothing.  It becomes even 
trickier when thinking about absolute zero.  Are there any useful
strategies other than temperature to introduce negative numbers, and 
also to teach about zero, its properties, and how they differ from 
absolute zero?

Date: 11/13/2003 at 08:49:13
From: Doctor Peterson
Subject: Re: Absolute Zero, Zero, & Negative numbers

Thanks for writing to Dr. Math!

I'm not sure what you are asking about absolute zero; I'm only
familiar with that term as a temperature, and as far as numbers 
themselves are concerned, absolute zero is just zero.  The only 
special thing I see about it is how it fits into the scale: in the 
case of Kelvin temperature, we have a scale with a definite starting 
point, and negative temperatures do not exist (though I understand 
that this is not really quite true if you dig deep enough into the 
physics of temperature!).  On the other hand, in scales where no 
(known) lowest value exists, negative numbers have to be allowed, and 
zero becomes not an absolute end point, but a mere reference point 
along the scale from which quantities are measured in both directions.
 If the concept of absolute zero temperature had not been discovered,
then we could not have a Kelvin scale, and all temperature scales
would have to, at least theoretically, allow for negative temperatures.

There are some situations where only positive numbers make sense, and 
in those cases we have the equivalent of a Kelvin scale, with an 
absolute zero.  For example, a person's height can't be negative; no 
one can be less than zero meters tall!  But altitude, which on the 
surface sounds the same, does allow negatives; I can be 100 meters 
above sea level, or 100 meters below sea level (a negative number).  
Again, we can find an alternative scale for altitude that has an 
"absolute zero", namely the distance from the center of the earth; so 
altitude referenced to sea level is just a convenience for people who 
live near sea level (as Celsius is convenient for people who "live" 
in the zone where water is liquid).  But not all scales can be made 
absolute; coordinate systems for space offer no (known) "farthest 
left" location, so negative numbers are required no matter what you 
do.  Which leads into my next comment ...

Apart from temperature, I think the only good way to introduce 
negative numbers is with a number line (of which temperature is just 
a familiar example we can point to in a child's environment, 
especially when they live in a cold climate).  If you look at what we 
say about negative numbers, such as

  Positive and Negative Integer Rules
    http://mathforum.org/library/drmath/sets/select/dm_pos_neg.html 

you will find a lot of references to the number line.  The basic idea 
is that, if we want to locate every point on a line by associating it 
with a number, positive numbers just aren't enough!  In order to put 
any numbers on the line in numerical order we have to have a zero
point from which we start counting; and in order to label points in
both directions, we need negative numbers.  Once you have that idea of 
negative numbers as labels for points to the left of zero on a number 
line, everything else falls into place.

After building that view of negative numbers, you can return to 
counting situations and ask whether negative numbers ever make sense 
there.  That's when ideas of "owing" or "debt" arise as uses for 
negative numbers.  But probably those ideas won't be clear without the 
number line model to make the concept concrete.  In a sense, we are 
then modeling counting in terms of the number line, extending the 
idea of number to allow for negatives that result from subtracting 
more than you have.  Such modeling is really the essence of  mathematics.

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

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