Practical Applications of Negative Numbers

Date: 10/13/2002 at 01:07:55
From: Michael Weinberg
Subject: Practical application of negative numbers

I am preparing a unit on operations with negative numbers for a class 
of very bright and accelerated fifth graders. I would like them to 
have some of the theory and background on negative numbers other 
than the obvious temperature and checkbook examples. What is the 
history of negatives? Why were they invented, considering that you 
can't have -2 of a given object? Can they be used to count anything? 
What have they been used for? 

I would like to go deep, so any guidance you have would be most 
appreciated.  

Thanks.

Date: 10/13/2002 at 23:14:59
From: Doctor Peterson
Subject: Re: Practical application of negative numbers

Hi, Michael.

This site, listed in our FAQ under Math History, should be your first 
source for questions on history:

   MacTutor Math History Archive
   http://www-history.mcs.st-and.ac.uk/ 

In particular, you will find some relevant material under the history 
of zero:

   http://www-history.mcs.st-and.ac.uk/HistTopics/Zero.html 

Check the link on Brahmagupta for some further details on the earliest 
use. The Chinese also used negative numbers very early, writing them 
in black and positive numbers in red. Negative numbers were not taken 
seriously, though, until the time of Cardan and Stifel, whom you can 
look up there; and it was not until some time later that negative 
numbers were given "equal rights" with positives, so that one did not 
need to know ahead of time whether a value was positive or negative.

We have some discussion of this topic here (check out the links):

   Negative Number History
   http://mathforum.org/library/drmath/view/52593.html 

To answer your specific questions, I would say that negative numbers 
were invented not to count anything (unless you think of "counting" a 
debt, which is how Brahmagupta described the concept), but in order 
to make it much easier to work with equations. The best example I can 
think of, however, is beyond your students, unless you just show them 
the problems without going into the solutions. This is the solution 
of quadratic equations. Until the use of negative numbers, each kind 
of equation had to be treated separately: 

    x^2 + ax = b
    x^2 - ax = b

and so on. Once it was found that negative and positive numbers could 
be handled in the same way, the same methods could be used for all 
quadratic equations, since all could be expressed in the same general 
form by allowing variables to be have either sign:

    x^2 + ax + b = 0

(Note that if a and b are positive, this has no positive solutions, 
so they would not even have considered it.)

In my mind the most significant use of negative numbers is in 
coordinate systems. We can locate points going both left and right, up 
and down from an origin, which would be impossible using only positive 
numbers. We would have to find an origin for which all points of 
interest were on the same side. So allowing negative numbers frees us 
up to describe things that happen anywhere in space, such as orbits of 
planets or graphs of equations. (Similarly, by allowing negative 
temperatures, we don't have to use a scale that starts at the lowest 
temperature we can observe; that's how the Fahrenheit scale 
originated, as a way to avoid negative numbers.) So really the number 
line is the central concept in working with negative numbers. Without 
them, we can't name all the points on the line, but only "half" of 
them.

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

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