Negative Numbers in Real Life

Date: 06/25/2008 at 00:40:03
From: Cody
Subject: concrete examples of negative numbers 

I'm finding it very difficult to imagine negative numbers outside of
math.  I've read some examples of their use here and elsewhere, but
I'm still flabbergasted. 

I understand negative temperatures, because they're based on the 
temperature of water.  The temperature of frozen water is 0ºC.
Anything below is negative.  But measurements are a concept.  My
trouble is with more concrete things. 

Mathematically, I can subtract 6 apples when I only have 5.  I can't
do this in reality.  I can use negative numbers in math, but outside
of measurements, I can't really grasp them in the real world.

For example, I can't have negative money.  After I'm bankrupt, I can't
spend any more money (without going to a bank or getting a credit
card).  But in the "world of math", I can take $600 from $500 and get
-$100. 

I'm looking for an example where negative numbers work in the "real
world" without measurements.  How can the math be accurate but impossible?

Date: 06/26/2008 at 09:56:50
From: Doctor Peterson
Subject: Re: concrete examples of negative numbers

Hi, Cody.

One thing that can help is to realize that mathematics is a world of 
its own that can be used to MODEL things in the real world, but, like 
any model, is not IDENTICAL to that real world.  Negative numbers are 
part of the "model world", not the real world.  So there are some 
situations where negative numbers make sense (and a negative answer 
to a problem is valid), and others where they do not (so that a 
negative answer just means there is no solution to the original 
problem).  In the case of temperatures, the 0 point (except in 
absolute temperature) is arbitrary, so that SOME negative values are 
possible, but others are not.

In the case of money, a negative answer may or may not be valid.  If 
you are just spending money from a basic checking account, a negative 
balance means that you are overdrawn--but it DOES still have meaning, 
because you now owe that much money to the bank.  If you have an 
account with automatic overdraft protection, the negative balance 
means that you have borrowed that much and have to pay it back.  So 
how to interpret the negative result depends on the situation; often 
positive and negative are just two sides of the same coin, each with 
its own interpretation.

The idea of negative numbers was often considered suspect even into 
the 1800's.  I've read a book by a mathematician of that time trying 
to present algebra in a way that didn't treat negative numbers as 
real; he called them fictitious, and presented them as just a 
shorthand for operations that should properly be done in reverse.  But
he admitted that using negative numbers made the work easier, 
always gave the right answer (when interpreted correctly), and 
unified what would otherwise have required several cases (depending 
on which number is greater, for example).  That is, negative numbers 
serve as a good, though imperfect, MODEL--you don't have to recognize 
the existence of the negative numbers themselves as concrete entities 
in order to make good use of them.  (And, by the way, even counting 
numbers are really an abstract concept too--you never saw a "three" 
by itself, did you?  ALL numbers live in the math world, not the real 
one.)

What we are doing when we use negative numbers is translating a real 
problem into a problem about, say, locations on a number line (which 
correspond to amounts of money you have or owe, say), solving that 
new problem, and then deciding how to translate the answer back into 
the real world problem.  The negative numbers live in this separate 
world of math, and may have various meanings or lack of meaning in 
the real world.


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

Date: 06/26/2008 at 14:53:24
From: Cody
Subject: Thank you (concrete examples of negative numbers )

Now THAT was a well put together answer.  It's more clear now, 
thanks.