The Need for Computers
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With the vast range of mathematical tools available to the scientist, why are computers necessary? It is true that a scientist will likely try to represent the problem he is trying to solve using traditional (pencil and paper) mathematics. This approach often leads to solutions that are easy to understand - each step in the process unfolds as the calculations proceed. Then, the results of the calculations are often put into graphical form using computer techniques. The problem with this approach is that many physical problems simply do not lend themselves to hand (or, as we like to say, analytical) calculations. The difficulty may be that available mathematical techniques simply cannot do the job, or that the sheer quantity of calculations forbid meaningful results in reasonable times. Here's a simple example:

Solve the equation 3x + 7 = 10. Easy enough, x = 1. You can do it in your head - it took about 10 seconds.

Now solve the simultaneous equations 2x + 4y = 2; 3x - 2y = 1. Not bad, x = 1/2 and y = 1/4. It took about a minute.

Next, try the simultaneous equations 2x - 2y + 5z = 8   x - 3y + 2z = 4 3x + 7y + 5z = 8. The solutions are x = 9/5, y = - 1/5, z = 4/5 and, after I corrected some simple errors in addition, it took about 9 minutes to get the solutions. To the right is a plot of the number of variables vs. the time it took to solve for them. You can extrapolate to the case of more variables. Clearly, here's a case where a computer would come in handy. If the number of variables becomes large, computers become necessary. Do such large arrays of equations show up? Yes, indeed. In the calculation of nuclear properties, for example, the number of variables can easily be in the thousands. Notice that's not really hard to solve the equations, it's just time consuming (and boring). Computers never get bored. Also note another valuable use of computers - they allow you to draw nice pictures!