What is Mathematical Modeling?

Date: 11/08/2002 at 10:10:49
From: Joy McCarnan
Subject: Math modeling vs. calculations

Hello, Dr. Math. 

I work for a software corporation that sells engineering software that 
helps designers in various industries, particularly due to its 
capability for backsolving. I came across your Web site while 
searching the Internet for information on mathematical modeling - what 
it is, its uses, examples, etc. I am looking specifically for 
documentation that would describe the difference between calculations 
and mathematical modeling. 

I have found quite a bit of information about mathematical modeling, 
but not really a solid yet simple source of consistent facts and 
examples. Often math modeling is described in terms of a means to an 
end, such as a way to make computer animations or simulate atomic 
fission. Yours was one of the more helpful sites I came across, but I 
am mathematically challenged to some extent and perhaps did not catch 
all that I should have. In addition to that problem, I am also seeking 
to pack this information into an vocabulary understanding level of 
middle-school age. 

If you could send any more concise and reliable resource links, or if 
you have easily accessible information to share, it would be very much 
appreciated. 

Best regards,
 - Joy

Date: 11/08/2002 at 12:15:00
From: Doctor Ian
Subject: Re: Math modeling vs. calculations

Hi Joy,

The short answer to your question is this: Calculations become 
'models' when you regard their results as having some meaning in the
world. This applies even to something as simple as balancing your
checkbook, or trying to figure out if you should stop for gas.

Objects and situations in the real world are complicated, with lots of
details. Numbers, on the other hand, are very simple. With a few
exceptions (zero, one, pi, e), when you've seen one, you've seen them
all. The same operations work on large and small numbers, on positive
and negative numbers, on rational and irrational numbers. 

So if you can reduce a situation to a description that uses only
numbers (and sometimes variables to represent numbers whose values you
haven't figured out yet), you can forget about the details, and
concentrate on the part of the problem that interests you. And you get 
to use the same set of tools on a wide variety of problems. 

There's another very important aspect of numbers that makes them
tremendously valuable: Unlike objects in the real world, they're free.  
You can make up all the numbers you want, and do whatever you want to 
them, and the only cost to you is for pen and paper, or a calculator 
or computer, or whatever you use to keep track of them. 

There is a very funny _Calvin and Hobbes_ in which Calvin asks his
father how you figure out the load limit on a bridge. His father 
replies that you build the bridge, and then drive heavier and heavier
trucks over it until it breaks. Then you weigh the last truck that
made it safely, and rebuild the bridge.

He's kidding, of course but (as is so often the case in _Calvin and
Hobbes_) he's also making an important point. To go out and try things 
in the real world can be difficult and expensive. If we can get the 
same answers by simulating a world and then using that simulated world 
as a place to try out our ideas, we can save time, money, effort... 
and even lives.  Before a commercial pilot gets to fly a real 
airliner, he spends thousands of hours 'flying' it in a simulator. And 
if he accidentally 'lands' the plane in San Francisco Bay, no one gets 
hurt.  

How do we build these simulated worlds? We build them using numbers.  
Not every mathematical calculation is as complicated as trying to 
predict what happens when you move some gizmo in the cockpit of a 
Boeing 747, but the ones that qualify as 'models' share this same 
general idea, i.e., that you can use numbers to represent objects in 
the world, and operations on numbers to represent possible actions 
taken on or by those objects. Maybe the problem you're solving 
involves choosing the best shape for a garden, or figuring out how 
many pens and pencils can be purchased for a certain amount of money. 
You could solve the problems by building (and re-building, and re-
building) the garden, or by getting some pens and pencils and trying 
out all the different combinations... but it's much easier to use 
numbers to simulate the objects, and play with the numbers instead.

In the end, that's all that 'mathematical modeling' is: simulating 
objects and actions by using numbers, so that you can solve problems
in minutes, for free, without even getting out of your chair, instead
of solving them in days or months or years, at great expense, and with
great effort.  

We tend to think of it as something very complicated, but there's
really no lower bound on complexity that has to be met in order to
qualify as a 'model'.  If I have 6 boxes, each of which has a volume
of 12 cubic feet, can I fit them in my car, which has a cargo space of
65 cubic feet? I let the numbers represent actual physical space, and
I compare numbers instead of comparing spaces directly. Guess what? 
That's a model. 

Does this help? 

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

Date: 11/08/2002 at 13:51:58
From: Joy McCarnan
Subject: Thank you (Math modeling vs. calculations)

Thank you, Dr. Ian. I shared your explanation with the company 
president. He and I are in the process of compiling some material on 
math modeling in order to convey it to some contacts at our subsidiary 
office in India. The clarity and universal nature of your answer has 
helped me understand the concepts better, and it's given the president 
some presentation ideas. We both really enjoyed and benefited from the 
fast and helpful reply.

Joy McCarnan
Universal Technical Systems, Inc.
Rockford, IL