Negative Numbers

The concept of a "negative" number has often been treated with
suspicion.  The ancient Chinese calculated with colored rods, red 
for positive quantities and black for negative (just the opposite 
of our accounting practices today) but, like their European counter-
parts, they would not accept a negative number as a solution of a 
problem or equation.  Instead, they would always re-state a problem 
so the result was a positive quantity.  This is why they often had 
to treat many different "cases" of what was essentially a single 
problem.

This practice persists to the present day.  For example, here's a
quote from an official US government form:

        If line 61 is more than line 54, subtract line
        54 from line 61.  This is the [positive] amount
        you OVERPAID.

        If line 54 is more than line 61, subtract line
        61 from line 54.  This is the [positive] amount
        you OWE.

This is eriely similar to the way the ancient Egyptians expressed
their arithmetical problems.  (In acknowledgement of this cultural
debt, there appears a Masonic/Egyptian pyramid on every US dollar
bill.)  It's unclear whether negative numbers will ever be fully
accepted on a completely equal footing with positive quantities, 
i.e., magnitudes. (Of course, the notion of a negative _magnitude_
leads immediately to imaginary numbers.)

Interestingly, the above form does not provide any guidance on how to
proceed if line 61 EQUALS line 54.  This may suggest that the concept
of zero has not yet been fully assimilated.  In fact, many ancient
cultures did not even regard "1" as a number (let alone 0), because
the concept of "number" implied plurality.

As recently as the 1500s there were European mathematicians who
argued against the "existence" of negative numbers by saying 

         Zero signifies "nothing", and it's impossible
         for anything to be less than nothing.

On the other hand, the Indian Brahmagupta (7th century AD) explicitly
and freely used negative numbers, as well as zero, in his algebraic
work.  He even gave the rules for arithmetic, e.g., "a negative number
divided by a negative number is a positive number", and so on.  I
believe this is considered to be the earliest [known] systemization 
of negative numbers as entities in themselves.