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
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